Accessibility Checklist
Element	Requirements	Pass?
Headings	Content is organized under headings and subheadings that are used sequentially.	Yes
Images	Images that convey information include alternative text descriptions. These descriptions are provided in the alt text field, in the surrounding text, or linked to as a long description.	Yes
Images	Images and text do not rely on colour to convey information.	Yes
Images	Images that are purely decorative or are already described in the surrounding text contain empty alternative text descriptions. (Descriptive text is unnecessary if the image doesn’t convey contextual content information.)	Yes
Tables	Tables include row and/or column headers that have the correct scope assigned.	Yes
Tables	Tables include a title or caption.	No
Tables	Tables do not have merged or split cells.	Yes
Tables	Tables have adequate cell padding.	Yes
Links	The link text describes the destination of the link.	Yes
Links	Links do not open new windows or tabs. If they do, a textual reference is included in the link text.	Yes
Links	Links to files include the file type in the link text.	Yes
Audio	All audio content includes a transcript that includes all speech content and relevant descriptions of non-speech audio and speaker names/headings where necessary.	N/A
Video	All videos include high-quality (i.e., not machine generated) captions of all speech content and relevant non-speech content.	N/A
Video	All videos with contextual visuals (graphs, charts, etc.) are described audibly in the video.	N/A
H5P	All H5P activities have been tested for accessibility by the H5P team and have passed their testing.	N/A
H5P	All H5P activities that include images, videos, and/or audio content meet the accessibility requirements for those media types.	N/A
Formulas	Formulas have been created using LaTeX and are rendered with MathJax.	Yes
Formulas	If LaTeX is not an option, formulas are images with alternative text descriptions.	Yes
Font	Font size is 12 point or higher for body text.	Yes
Font	Font size is 9 point for footnotes or endnotes.	Yes
Font	Font size can be zoomed to 200% in the webbook or eBook formats.	Yes

How can I use the different formats?
Format	Internet required?	Device	Required apps	Accessibility Features	Screen reader compatible
Online webbook	Yes	Computer, tablet, phone	An Internet browser (Chrome, Firefox, Edge, or Safari)	WCAG 2.0 AA compliant, option to enlarge text, and compatible with browser text-to-speech tools	Yes
PDF	No	Computer, print copy	Adobe Reader (for reading on a computer) or a printer	Ability to highlight and annotate the text. If reading on the computer, you can zoom in.	Unsure
EPUB	No	Computer, tablet, phone	An eReader app	Option to enlarge text, change font style, size, and colour.	Unsure
HTML	No	Computer, tablet, phone	An Internet browser (Chrome, Firefox, Edge, or Safari)	WCAG 2.0 AA compliant and compatible with browser text-to-speech tools.	Yes

Positive or negative?	Math experience
Negative	My teacher in elementary school lined the whole class up in a row and made us play a multiplication game. I could see which question was mine, and I didn’t know the answer so I had to figure it out on my fingers before my turn came up. I got the answer right, but I was so nervous that I would be teased because I didn’t know the answer off the top of my head. I still don’t know my times tables.
Positive	In high school, I could use a calculator to figure out the simple multiplication problems, and then I could figure out the tougher problems without worrying about knowing my times tables.
Negative	Now that I am upgrading my math, I feel nervous every time I even think about opening the book. I want to get all the answers right, and I know that I won’t be able to. I really need everything to be right so that I know that I am getting it.

Unit 1: Multiplication

Topic A: Multiplying Larger Numbers

It is usually easier to multiply larger numbers if they are written underneath each other. The bottom number is called the multiplier.

To find the product of a one digit multiplier, use as many of these steps as you need to complete each multiplication question:

Step 1: Multiply the ones digit in the large number by the one-digit multiplier.
Step 2: Multiply the tens digit in the large number by the multiplier.
Step 3: Multiply the hundreds digit in the large number by the multiplier.
Step 4: Multiply the thousands digit in the large number by the multiplier.

Example A

62 × 4 =

Step 1: 4 × 2 ones = 8

$\begin{array}{rr}&62\\\times&4\\\hline&8\end{array}$

Step 2: 4 × 6 tens = 24 tens = 2 hundreds and 4 tens

$\begin{array}{rr}&62\\\times&4\\\hline&248\end{array}$

The product of 62 × 4 is 248.

Exercise 1

Find the product. Check your work using the answer key at the
end of the exercise.

$\begin{array}{rr}&72\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&81\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&21\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&80\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&73\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&71\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&40\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&90\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&82\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&40\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&92\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&83\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&90\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&71\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&53\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&30\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&81\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&61\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&70\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&41\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&90\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&60\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&92\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&81\\\times&4\\\hline\\\end{array}$

Answers to Exercise 1

Example B

523 × 3 =

$\begin{array}{rr}&523\\\times&3\\\hline&1569\end{array}$

Step 1: 3 × 3 ones = 9 ones

Step 2: 3 × 2 tens = 6 tens

Step 3: 3 × 5 hundreds = 15 hundreds = 1 thousand and 5 hundreds

The product of 523 × 3 is 1569.

Example C

901 × 8 =

$\begin{array}{rr}&901\\\times&8\\\hline&7208\end{array}$

Step 1: 8 × 1 one = 8 ones

Step 2: 8 × 0 tens = 0 tens

Step 3: 8 × 9 hundreds = 72 hundreds = 7 thousands and 2 hundreds

The product of 901 × 8 is 7208.

Exercise 2

Find the product. Check your work using the answer key at the
end of the exercise.

$\begin{array}{rr}&601\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&423\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&641\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&922\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&820\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&211\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&803\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&542\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&813\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&610\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&901\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&711\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&720\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&910\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&801\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&932\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&731\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&701\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&521\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&632\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&720\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&942\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&710\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&601\\\times&8\\\hline\\\end{array}$

Answers to Exercise 2

4207
1269
1282
3688
3280
1055
1606
1084
2439
2440
5406
5688
2160
4550
4806
2796
1462
3505
1042
1896
2880
1884
6390
4808

Renaming and Carrying

In the times tables, the products are often two digit numbers. You must rename and carry. The amount you carry is added onto the product of the next column.

Example D

$\begin{array}{rr}&68\\\times&7\\\hline\end{array}$

Step 1: Multiply the ones by the multiplier.

7 × 8 ones = 56 ones = 5 tens + 6 ones

Write the 6 ones in the product.

Carry the 5 tens to the tens column and make a note of it for yourself. You might write it above the tens column.

$\begin{array}{rrr}&_5\hspace{0.5em}\\&68\\\times&7\\\hline&6\end{array}$

Step 2: Multiply the tens by the multiplier.

7 × 6 tens = 42 tens

Now add on the 5 tens that you carried.

42 tens + 5 tens = 47 tens = 4 hundreds and 7 tens

$\begin{array}{rrr}&_5\hspace{0.5em}\\&68\\\times&7\\\hline&476\end{array}$

Exercise 3

Find the product. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&16\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&62\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&37\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&14\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&36\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&92\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&48\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&17\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&26\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&54\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&58\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&45\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&56\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&47\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&39\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&75\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&38\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&82\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&98\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&29\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&47\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&74\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&56\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&98\\\times&4\\\hline\\\end{array}$

Answers to Exercise 3

Example E

$4\times224=$ $\begin{array}{rr}&224\\\times&4\\\hline\end{array}$

Step 1: 4 × 4 ones = 16 ones = 1 ten and 6 ones

Write the 6 ones in the product and carry the one ten.

$\begin{array}{rr}&_1\hspace{0.5em}\\&224\\\times&4\\\hline&6\end{array}$

Step 2: 4 × 2 tens = 8 tens

8 tens + 1 ten we carried = 9 tens

$\begin{array}{rr}&_1\hspace{0.5em}\\&224\\\times&4\\\hline&96\end{array}$

Step 3: 4 × 2 hundreds = 8 hundreds

$\begin{array}{rr}&_1\hspace{0.5em}\\&224\\\times&4\\\hline&896\end{array}$

Example F

$4\times456=$ $\begin{array}{rr}&456\\\times&5\\\hline\end{array}$

Step 1: 5 × 6 ones = 30 ones = 3 ten and 0 ones

The 0 must be written to hold the ones place. Carry the 3 tens.

$\begin{array}{rr}&_3\hspace{0.5em}\\&456\\\times&5\\\hline&0\end{array}$

Step 2: 5 × 5 tens = 25 tens

25 tens + 3 tens = 28 tens = 2 hundreds and 8 tens

Write the 8 tens in the product. Carry the 2 hundreds.

$\begin{array}{rr}&_{2\text{ }3}\hspace{0.5em}\\&456\\\times&5\\\hline&80\end{array}$

Step 3: 5 × 4 hundreds = 20 hundreds + 2 hundreds = 22 hundreds = 2 thousands and 2 hundreds

$\begin{array}{rr}&_{2\text{ }3}\hspace{0.5em}\\&456\\\times&5\\\hline&2280\end{array}$

Exercise 4

Find the products. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&648\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&240\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&457\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&404\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&106\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&156\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&407\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&239\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&118\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&425\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&139\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&565\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&248\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&604\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&239\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&576\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&857\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&478\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&674\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&629\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&893\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&583\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&952\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&293\\\times&7\\\hline\\\end{array}$

Answers to Exercise 4

5832
1680
3656
808
954
624
3256
1434
1062
2975
1112
1130
744
3624
956
1728
5142
4302
4718
5032
4465
4664
8568
2051

Example G

$\begin{array}{rr}&2408\\\times&9\\\hline\end{array}$

$\begin{array}{rr}&_3\hspace{0.7em}_7\hspace{0.5em}\\&2408\\\times&9\\\hline&21672\end{array}$

Step 1: 9 × 8 ones = 72 ones = 7 tens and 2 ones

Step 2: 9 × 0 = 0 tens. 0 tens + 7 tens = 7 tens

Step 3: 9 × 4 hundreds = 36 hundreds = 3 thousands and 6 hundreds

Step 4: 9 × 2 thousands = 18 thousands. 18 thousands + 3 thousands = 21 thousands

Exercise 5

Find the products. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&4103\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&6087\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&3280\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&7034\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&8456\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&4758\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&4735\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&5402\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&5394\\\times&4\\\hline\\\end{array}$
$\begin{array}{rr}&2034\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&8652\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&6392\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&4187\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&1376\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&3297\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&9628\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&7689\\\times&8\\\hline\\\end{array}$
$\begin{array}{rr}&5160\\\times&3\\\hline\\\end{array}$
$\begin{array}{rr}&4256\\\times&7\\\hline\\\end{array}$
$\begin{array}{rr}&5491\\\times&5\\\hline\\\end{array}$
$\begin{array}{rr}&8032\\\times&9\\\hline\\\end{array}$
$\begin{array}{rr}&8645\\\times&6\\\hline\\\end{array}$
$\begin{array}{rr}&6453\\\times&2\\\hline\\\end{array}$
$\begin{array}{rr}&8129\\\times&4\\\hline\\\end{array}$

Answers to Exercise 5

32824
24348
19680
35170
16912
33306
14205
48618
21576
16272
43260
44744
25122
12384
6594
28884
61512
15480
29792
27455
72288
51870
12906
32516

Topic A: Self-Test

Mark /14 Aim 11/14

Find the products. (6 marks)
1. $\begin{array}{rr}&62\\\times&4\\\hline\\\end{array}$
2. $\begin{array}{rr}&31\\\times&4\\\hline\\\end{array}$
3. $\begin{array}{rr}&423\\\times&3\\\hline\\\end{array}$
4. $\begin{array}{rr}&734\\\times&2\\\hline\\\end{array}$
5. $\begin{array}{rr}&8342\\\times&2\\\hline\\\end{array}$
6. $\begin{array}{rr}&5231\\\times&3\\\hline\\\end{array}$
Multiply these numbers. (4 marks)
1. $\begin{array}{rr}&44\\\times&7\\\hline\\\end{array}$
2. $\begin{array}{rr}&69\\\times&8\\\hline\\\end{array}$
3. $\begin{array}{rr}&207\\\times&9\\\hline\\\end{array}$
4. $\begin{array}{rr}&184\\\times&6\\\hline\\\end{array}$
Find the products. (4 marks)
1. $\begin{array}{rr}&2834\\\times&5\\\hline\\\end{array}$
2. $\begin{array}{rr}&4037\\\times&6\\\hline\\\end{array}$
3. $\begin{array}{rr}&9241\\\times&8\\\hline\\\end{array}$
4. $\begin{array}{rr}&3652\\\times&4\\\hline\\\end{array}$

Answers to Topic A Self-Test

Find the products.
1. 249
2. 124
3. 1269
4. 1468
5. 16684
6. 15693
Multiply these numbers.
1. 308
2. 552
3. 1863
4. 1104
Find the products.
1. 14170
2. 24222
3. 73928
4. 14608

Topic B: Two- and Three-Digit Multipliers

When the multiplier is more than one digit, you use the same process and get partial products. You repeat the steps until you have multiplied by every digit, then add the partial products together.

Multiplying by Two-Digit Multipliers

Example A

$24\times23=$

Part 1: Multiply by the ones digit in the multiplier.

Multiply 3 ones by 24 using the method you already know. The first partial product is 72.

$\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\end{array}$

Part 2: Multiply by the tens digit in the multiplier. First, put a 0 to hold the ones place in your partial product. We are multiplying by a ten, so we hold the ones place.

Step 1: Multiply 2 tens by 4 ones = 8 tens

Write the 8 tens under the tens in your first partial product. It is very important to keep the columns straight – ones under one, tens under tens.

Step 2: Multiply 2 tens by 2 tens = 4 hundreds

Write the 4 hundreds in your partial product. The second partial product is 480.

$\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\\&480\end{array}$

Part 3: Add the partial products together, being sure to add ones to ones, tens to tens, hundreds to hundreds. The sum is the final product.

Draw a line under the partial products. Add. Check your addition.

$\begin{array}{rr}&_1\hspace{0.5em}\\&24\\\times&23\\\hline&72\\+&480\\\hline&552\end{array}$

Example B

$36\times425 =$

Part 1: Multiply by the ones digit in the multiplier. 6 × 425 = 2550

$\begin{array}{rr}&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\end{array}$

Part 2: Multiply by the tens digit in the multiplier. First put a 0 to hold the ones place in the second partial product.

Step 1: 3 tens × 5 tens = 15 tens = 1 hundred and 5 tens
Write the 5 tens in the second partial product and carry the 1 hundred. Now you can see why it is best to cross out the numbers you carry as soon as you have added them to the product.
Step 2: 3 tens × 2 tens = 6 hundreds
6 hundreds + 1 hundred (carried) = 7 hundreds. There is nothing to carry.
Step 3: 3 tens × 4 hundreds = 12 thousands

$\begin{array}{rr}&_1\hspace{0.5em}\\&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\\&12750\end{array}$

Part 3: Add the partial products together.

$\begin{array}{rr}&_1\hspace{0.5em}\\&_1\hspace{0.15em}_3\hspace{0.5em}\\&425\\\times&36\\\hline&2550\\+&12750\\\hline&15300\end{array}$

Tip: Keeping the columns straight with ones under ones, tens under tens, hundreds under hundreds is very important. Working on large-squared graphing paper using one digit per square is often helpful.

tens × tens = hundreds
tens × hundreds = thousands

Exercise 1

Multiply, being very careful to keep the columns straight when you write your partial products. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&84\\\times&12\\\hline&168\\+&840\\\hline&1008\end{array}$

$\begin{array}{rr}&73\\\times&12\\\hline\\\end{array}$
$\begin{array}{rr}&50\\\times&42\\\hline\\\end{array}$
$\begin{array}{rr}&62\\\times&31\\\hline\\\end{array}$
$\begin{array}{rr}&61\\\times&42\\\hline\\\end{array}$
$\begin{array}{rr}&91\\\times&53\\\hline\\\end{array}$
$\begin{array}{rr}&92\\\times&31\\\hline\\\end{array}$
$\begin{array}{rr}&91\\\times&49\\\hline\\\end{array}$
$\begin{array}{rr}&72\\\times&48\\\hline\\\end{array}$
$\begin{array}{rr}&53\\\times&30\\\hline\\\end{array}$
$\begin{array}{rr}&41\\\times&53\\\hline\\\end{array}$
$\begin{array}{rr}&42\\\times&94\\\hline\\\end{array}$
$\begin{array}{rr}&80\\\times&86\\\hline\\\end{array}$
$\begin{array}{rr}&31\\\times&79\\\hline\\\end{array}$
$\begin{array}{rr}&54\\\times&40\\\hline\\\end{array}$
$\begin{array}{rr}&61\\\times&48\\\hline\\\end{array}$
$\begin{array}{rr}&60\\\times&31\\\hline\\\end{array}$
$\begin{array}{rr}&55\\\times&73\\\hline\\\end{array}$
$\begin{array}{rr}&84\\\times&56\\\hline\\\end{array}$
$\begin{array}{rr}&53\\\times&38\\\hline\\\end{array}$

Answers to Exercise 1

876
2100
1922
2562
4823
2852
4459
3456
1590
2173
3948
6880
2449
2160
2928
1860
4015
4704
2014

When the multiplier has a zero in the ones place, use this shortcut.

Example C

$\begin{array}{rr}&48\\\times&80\\\hline&3840\end{array}$

Step 1: 0 ones × 48 = 0. Place one zero in the product and that will hold the ones place.
Step 2: Multiply by the tens digit and write the product beside the zero.

Example D

$\begin{array}{rr}&97\\\times&20\\\hline&1940\end{array}$

Exercise 2

Find the products. Use the shortcut for multipliers with a zero in them. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&76\\\times&70\\\hline&5320\end{array}$

$\begin{array}{rr}&52\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&91\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&83\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&49\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&61\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&16\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&36\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&398\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&432\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&863\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&907\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&503\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&452\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&943\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&248\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&6287\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&9025\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&8907\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&300\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&9075\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&3952\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&1528\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&7106\\\times&70\\\hline\\\end{array}$

Answers to Exercise 2

3640
6370
5810
3430
4270
1120
2520
27860
30240
60410
63490
35210
31640
66010
17360
440090
631750
623490
21000
635250
276640
106960
497420

How are you doing? Ask your instructor for help if you are not comfortable with multiplying.

Multiplying by Three-Digit Multipliers

To multiply by three digit multipliers, use the same method with one more part.

Example E

$417\times368 =$

$\begin{array}{rr}&417\\\times&368\\\hline&3336\\&25020\\+&125100\\\hline&153456\end{array}$

Part 1: Multiply by the 8 ones.
Part 2: Multiply the 6 tens; hold the ones place with 0.
Part 3: Multiply by the 3 hundreds. Put 00 to hold the ones and tens places in the third partial product.
- Step 1: 3 hundreds × 7 ones = 21 hundreds = 2 thousands and 1 hundred. Write the 1 hundred and carry the 2 thousands.
- Step 2: 3 hundreds × 1 ten = 3 thousands. 3 thousands + 2 thousands (carried) = 5 thousands.
- Step 3: 3 hundreds × 4 hundreds = 12 ten thousands.
Part 4: Add the partial products.

Exercise 3

Find the products. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&416\\\times&213\\\hline\\\end{array}$
$\begin{array}{rr}&375\\\times&291\\\hline\\\end{array}$
$\begin{array}{rr}&361\\\times&475\\\hline\\\end{array}$
$\begin{array}{rr}&275\\\times&863\\\hline\\\end{array}$
$\begin{array}{rr}&984\\\times&469\\\hline\\\end{array}$
$\begin{array}{rr}&489\\\times&578\\\hline\\\end{array}$
$\begin{array}{rr}&498\\\times&123\\\hline\\\end{array}$
$\begin{array}{rr}&267\\\times&854\\\hline\\\end{array}$
$\begin{array}{rr}&613\\\times&368\\\hline\\\end{array}$
$\begin{array}{rr}&725\\\times&547\\\hline\\\end{array}$
$\begin{array}{rr}&269\\\times&912\\\hline\\\end{array}$
$\begin{array}{rr}&752\\\times&697\\\hline\\\end{array}$
$\begin{array}{rr}&983\\\times&357\\\hline\\\end{array}$
$\begin{array}{rr}&835\\\times&148\\\hline\\\end{array}$
$\begin{array}{rr}&386\\\times&296\\\hline\\\end{array}$

Answers to Exercise 3

88608
109125
171475
237325
461496
282642
61254
228018
225584
396575
245328
524144
350931
123580
114256

You know to hold the ones place with a zero if the multiplier has a zero in the ones place. Use the same skill if the multiplier has a zero in the tens place.

Example F

$927\times405 =$

$\begin{array}{rr}&927\\\times&405\\\hline&4635\\+&370800\\\hline&375435\end{array}$

Part 1: Multiply by the 5 ones.
Part 2: Multiply by the 0 tens.
Hold the ones place with a 0; 0 × 927 = 0; Place one zero in the tens place in the second partial product.
Part 3: Multiply by the 4 hundreds. The ones and tens places are already held by zeros. Start this partial product in the hundreds place on the same line.
Part 4: Add the partial products.

Exercise 4

Find the products. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&_2\hspace{0.15em}_2\hspace{0.5em}\\&698\\\times&301\\\hline&698\\+&209400\\\hline&210098\end{array}$

$\begin{array}{rr}&923\\\times&403\\\hline\\\end{array}$
$\begin{array}{rr}&830\\\times&108\\\hline\\\end{array}$
$\begin{array}{rr}&482\\\times&206\\\hline\\\end{array}$
$\begin{array}{rr}&432\\\times&205\\\hline\\\end{array}$
$\begin{array}{rr}&625\\\times&405\\\hline\\\end{array}$
$\begin{array}{rr}&275\\\times&306\\\hline\\\end{array}$
$\begin{array}{rr}&765\\\times&506\\\hline\\\end{array}$
$\begin{array}{rr}&1576\\\times&702\\\hline\\\end{array}$
$\begin{array}{rr}&432\\\times&405\\\hline\\\end{array}$
$\begin{array}{rr}&625\\\times&409\\\hline\\\end{array}$
$\begin{array}{rr}&175\\\times&306\\\hline\\\end{array}$
$\begin{array}{rr}&5874\\\times&309\\\hline\\\end{array}$
$\begin{array}{rr}&7384\\\times&104\\\hline\\\end{array}$
$\begin{array}{rr}&6538\\\times&603\\\hline\\\end{array}$

Answers to Exercise 4

371969
89640
99292
88560
253125
84150
387090
1106352
174960
255625
53550
1815066
767936
3942414

Multiplying by 10, 100, and 1000

Exercise 5

Do the following questions and see if you can find the pattern. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&83\\\times&10\\\hline&830\end{array}$
$\begin{array}{rr}&46\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&97\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&123\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&70\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&129\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&1852\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&29871\\\times&10\\\hline\\\end{array}$
$\begin{array}{rr}&45\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&26\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&432\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&679\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&2482\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&9037\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&46207\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&97512\\\times&100\\\hline\\\end{array}$
$\begin{array}{rr}&23\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&452\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&207\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&348\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&2118\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&2431\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&23681\\\times&1000\\\hline\\\end{array}$
$\begin{array}{rr}&48203\\\times&1000\\\hline\\\end{array}$

Answers to Exercise 5

830
460
970
1230
700
1290
18520
298710
4500
2600
43200
67900
248200
903700
4620700
9751200
23000
452000
207000
348000
2118000
2431000
23681000
48203000

And the pattern is …

When multiplying by 10, 100, 1000, 10000, etc., place as many zeros to the right of the number as there are zeros in the 10, 100, 1000, etc.

To multiply by 10 put one zero after the number.
To multiply by 100 put two zeros after the number.
To multiply by 1000 put three zeros after the number.

Exercise 6

Find the products using the short method. Do not rewrite the questions. Check your work using the answer key at the end of the exercise.

12 × 10 = 120
10 × 3175 =
162 × 10 =
10 × 53821 =
10 × 123 =
27342 × 10 =
10 × 98 =
1134 × 10 =
15 × 100 =
100 × 278 =
9134 × 100 =
651 × 100 =
100 × 5169 =
100 × 24815 =
10 × 905 =
45683 × 10 =
1000 × 87 =
521 × 1000 =
1000 × 68935 =
1000 × 8902 =
1576 × 1000 =
31584 × 1000 =
1000 × 426 =
72 × 1000 =

Answers for Exercise 6

120
31750
1620
538210
1230
273420
980
11340
1500
27800
913400
65100
516900
2481500
9050
456830
87000
521000
68935000
8902000
1576000
31584000
426000
72000

Topic B: Self-Test

Mark /12 Aim 10/12

Multiply these numbers.
1. $\begin{array}{rr}&47\\\times&39\\\hline\\\end{array}$
2. $\begin{array}{rr}&58\\\times&93\\\hline\\\end{array}$
3. $\begin{array}{rr}&48\\\times&100\\\hline\\\end{array}$
4. $\begin{array}{rr}&982\\\times&1000\\\hline\\\end{array}$
5. $\begin{array}{rr}&678\\\times&39\\\hline\\\end{array}$
6. $\begin{array}{rr}&4579\\\times&86\\\hline\\\end{array}$
7. $\begin{array}{rr}&8703\\\times&93\\\hline\\\end{array}$
8. $\begin{array}{rr}&7390\\\times&85\\\hline\\\end{array}$
9. $\begin{array}{rr}&8047\\\times&236\\\hline\\\end{array}$
10. $\begin{array}{rr}&4238\\\times&197\\\hline\\\end{array}$
11. $\begin{array}{rr}&8200\\\times&444\\\hline\\\end{array}$
12. $\begin{array}{rr}&7265\\\times&409\\\hline\\\end{array}$

Answers to Topic B Self-Test

Multiply these numbers.
1. 1833
2. 5394
3. 4800
4. 982000
5. 26442
6. 393794
7. 809379
8. 628150
9. 1899092
10. 834886
11. 3640800
12. 2971385

Topic C: Estimating Products

Before you practice estimating products, look at the easy multiplication that can be done when the factors end in zeros.

You already know how to quickly multiply by 10, 100, 1000, etc…

You can find the products in questions like these using regular multiplication:

But here is a shortcut:

Count all the zeros at the end of the numbers in both factors.
Write down that many zeros at the end of the product.
Multiply the other digits in the usual way and put them before the zeros.

Example A

30 × 500 =

How many zeros at the end of the factors? 3
Write them down. 000
Multiply the other digits and put them before the zeros. 3 × 5 = 15

30 × 500 = 15000

Example B

400 × 3000 =

How many zeros at the end of the factors? 5
Write them down. 00000
Multiply the other digits and put them before the zeros. 4 × 3 = 12

400 × 3000 = 1200000

Exercise 1

Find the products using the shortcut. Check your work using the answer key at the end of the exercise.

$\begin{array}{rr}&300\\\times&20\\\hline&6000\end{array}$
$\begin{array}{rr}&6000\\\times&200\\\hline\\\end{array}$
$\begin{array}{rr}&210\\\times&20\\\hline\\\end{array}$
$\begin{array}{rr}&800\\\times&600\\\hline\\\end{array}$
$\begin{array}{rr}&400\\\times&500\\\hline\\\end{array}$
$\begin{array}{rr}&6000\\\times&90\\\hline\\\end{array}$
$\begin{array}{rr}&50000\\\times&6000\\\hline\\\end{array}$
$\begin{array}{rr}&80000\\\times&30\\\hline\\\end{array}$
$\begin{array}{rr}&5000\\\times&50\\\hline\\\end{array}$
$\begin{array}{rr}&70\\\times&80\\\hline\\\end{array}$
$\begin{array}{rr}&3000\\\times&700\\\hline\\\end{array}$
$\begin{array}{rr}&50000\\\times&900\\\hline\\\end{array}$
$\begin{array}{rr}&9000\\\times&8000\\\hline\\\end{array}$
$\begin{array}{rr}&60000\\\times&90\\\hline\\\end{array}$
$\begin{array}{rr}&90000\\\times&2000\\\hline\\\end{array}$
$\begin{array}{rr}&600\\\times&600\\\hline\\\end{array}$
$\begin{array}{rr}&40000\\\times&800\\\hline\\\end{array}$
$\begin{array}{rr}&2400\\\times&70\\\hline\\\end{array}$
$\begin{array}{rr}&390\\\times&40\\\hline\\\end{array}$
$\begin{array}{rr}&7200\\\times&5000\\\hline\\\end{array}$
$\begin{array}{rr}&7000\\\times&7000\\\hline\\\end{array}$
$\begin{array}{rr}&61000\\\times&400\\\hline\\\end{array}$
$\begin{array}{rr}&5200\\\times&300\\\hline\\\end{array}$
$\begin{array}{rr}&40\\\times&60\\\hline\\\end{array}$

Answers to Exercise 1

6 000
1 200 000
4 200
480 000
200 000
540 000
300 000 000
2 400 000
250 000
5 600
2 100 000
45 000 000
72 000 000
5 400 000
180 000 000
360 000
32 000 000
168 000
15 600
36 000 000
49 000 000
24 400 000
1 560 000
2 400

To find an estimated product, round the factors before you multiply. If a factor has only one digit, do not round it.

Example C

$\begin{array}{rr}&6258\\\times&3\\\hline\end{array}$

6258 rounds to 6000. Leave 3 as is.

$\begin{array}{rr}&6000\\\times&3\\\hline&18000\end{array}$

18000 is the estimated product.

Example D

$\begin{array}{rr}&491\\\times&24\\\hline\end{array}$

491 rounds to 500. 24 rounds to 20.

$\begin{array}{rr}&500\\\times&20\\\hline&10000\end{array}$

10000 is the estimated product.

Topic C: Self-Test

Mark /18 Aim 15/18

Multiply these numbers.
1. $\begin{array}{rr}&600\\\times&70\\\hline\end{array}$
2. $\begin{array}{rr}&9000\\\times&30\\\hline\end{array}$
3. $\begin{array}{rr}&5000\\\times&600\\\hline\end{array}$
4. $\begin{array}{rr}&3000\\\times&500\\\hline\end{array}$
5. $\begin{array}{rr}&400\\\times&50\\\hline\end{array}$
6. $\begin{array}{rr}&8000\\\times&1000\\\hline\end{array}$
Find an estimated product.
1. $\begin{array}{rr}&87\\\times&23\\\hline\end{array}$
2. $\begin{array}{rr}&268\\\times&25\\\hline\end{array}$
3. $\begin{array}{rr}&9421\\\times&75\\\hline\end{array}$
4. $\begin{array}{rr}&2632\\\times&49\\\hline\end{array}$
5. $\begin{array}{rr}&365\\\times&455\\\hline\end{array}$
6. $\begin{array}{rr}&7264\\\times&590\\\hline\end{array}$

Answers to Topic C Self-Test

Multiply these numbers.
1. 42000
2. 270000
3. 3000000
4. 1500000
5. 20000
6. 8000000
Find an estimated product.
1. 90 × 20 = 1800
2. 300 × 30 = 9000
3. 9000 × 80 = 720000
4. 3000 × 50 = 150000
5. 400 × 500 = 200000
6. 7000 × 600 = 4200000

Topic D: Multiplication Problems

Multiplication problems usually give information for one unit and ask information for many units. For example, you may be given an average distance for one hour and asked for a distance for six hours. You may be given the price for one item and asked the price for 96 items.

Remember that multiplication is fast addition of equal amounts. Key words for addition may also point to multiplication.

Key words that point to multiplication:

altogether
combined
in all
total
the average is
product
how many?
how much?
each

Exercise 1

Do these problems by following the five problem-solving steps. Remember to circle the information and underline what is being asked. Be sure to write down your estimation before you find the actual solution. Check your work using the answer key at the end of the exercise.

Maria spent $151 a month to ride to work. How much did it cost her to ride to work for the whole year? (1 year = 12 months)
Fred travels 121 km a day on his delivery route. How far does he travel in 5 working days?
Manuel buys 340 L of gas a month. In 6 months, how many litres of gas does Manual buy?
An apartment building has 16 apartments, each rented for $870 a month. What is the total monthly rental income from this building?
A farmer sells potatoes at $53 per 50 kg sack. How much will he get for 75 sacks?
Bob is paid $12 per hour. If he worked 39 hours last week and 24 hours this week, how much did he earn for the two weeks? (two operations)
It costs $35260 per working day to run the factory. How much does it cost to run this factory for a month of 23 working days?
The train has an average speed of 75 km an hour. How far does this train travel in 14 hours?
Lee’s sports car averages 18 km per litre. How far can she drive on 12 L of gasoline?
Frank bought a used car and paid for it over 15 months. He made 15 monthly payments of $325 each. How much did he pay?

Answers to Exercise 1

$1812
605 km
2040 L
$13920
$3975
63hours, $756
$810, 980
1050 km
216 km
$4875

Topic D: Self-Test

Mark /8 Aim 6/8

Solve these problems. Show all your work. Give yourself one mark for the correct method and one mark for the correct answer.
1. A freight train has 70 cars. Each car can hold 22680 kilograms of cargo. How much cargo can the train hold in all?
2. The highway distance between Fernie and Edmonton is 621 kilometres. How many kilometres will a bus travel in 68 trips from Fernie to Edmonton?
3. A tanker truck made 275 trips in one year. The truck hauled 23 800 litres each time. How many litres did the truck haul during the year?
4. The college cafeteria hopes to serve 425 people each day. Estimate how many meals will be served if the cafeteria is open 175 days.

Answers to Topic D Self-Test

Solve these problems. Show all your work. Give yourself one mark for the correct method and one mark for the correct answer.
1. 1587600 kilograms
2. 42228 kilometres
3. 6545000 litres
4. 400 × 200 = 80000 meals

Unit 1 Review: Multiplication

You will now practice all the skills you learned in Unit 1. Check your work using the answer key at the end of the review.

Find the products.
1. $\begin{array}{rr}&81\\\times&5\\\hline\\\end{array}$
2. $\begin{array}{rr}&73\\\times&3\\\hline\\\end{array}$
3. $\begin{array}{rr}&441\\\times&2\\\hline\\\end{array}$
4. $\begin{array}{rr}&512\\\times&4\\\hline\\\end{array}$
5. $\begin{array}{rr}&9342\\\times&2\\\hline\\\end{array}$
6. $\begin{array}{rr}&8132\\\times&3\\\hline\\\end{array}$
Find the products.
1. $\begin{array}{rr}&48\\\times&7\\\hline\\\end{array}$
2. $\begin{array}{rr}&78\\\times&9\\\hline\\\end{array}$
3. $\begin{array}{rr}&892\\\times&8\\\hline\\\end{array}$
4. $\begin{array}{rr}&536\\\times&6\\\hline\\\end{array}$
5. $\begin{array}{rr}&2375\\\times&4\\\hline\\\end{array}$
6. $\begin{array}{rr}&5649\\\times&3\\\hline\\\end{array}$
Find the products.
1. $\begin{array}{rr}&67\\\times&19\\\hline\\\end{array}$
2. $\begin{array}{rr}&581\\\times&34\\\hline\\\end{array}$
3. $\begin{array}{rr}&7310\\\times&46\\\hline\\\end{array}$
4. $\begin{array}{rr}&754\\\times&692\\\hline\\\end{array}$
5. $\begin{array}{rr}&2735\\\times&846\\\hline\\\end{array}$
6. $\begin{array}{rr}&857\\\times&308\\\hline\\\end{array}$
7. $\begin{array}{rr}&629\\\times&407\\\hline\\\end{array}$
8. $\begin{array}{rr}&2805\\\times&15\\\hline\\\end{array}$
9. $\begin{array}{rr}&5102\\\times&743\\\hline\\\end{array}$
Find the products. Use the shortcut.
1. $\begin{array}{rr}&1000\\\times&82\\\hline\\\end{array}$
2. $\begin{array}{rr}&100\\\times&26\\\hline\\\end{array}$
3. $\begin{array}{rr}&6263\\\times&1000\\\hline\\\end{array}$
4. 407 × 100 =
5. 100 × 9482 =
6. 3614 × 10 =
7. 1000 × 1795 =
Find the products. Use the shortcut.
1. $\begin{array}{rr}&50\\\times&40\\\hline\\\end{array}$
2. $\begin{array}{rr}&600\\\times&800\\\hline\\\end{array}$
3. $\begin{array}{rr}&9000\\\times&500\\\hline\\\end{array}$
Find an estimated product.
1. $\begin{array}{rr}&68\\\times&39\\\hline\\\end{array}$
2. $\begin{array}{rr}&185\\\times&94\\\hline\\\end{array}$
3. $\begin{array}{rr}&6763\\\times&69\\\hline\\\end{array}$
4. $\begin{array}{rr}&853\\\times&399\\\hline\\\end{array}$
5. $\begin{array}{rr}&2735\\\times&846\\\hline\\\end{array}$
6. $\begin{array}{rr}&68924\\\times&268\\\hline\\\end{array}$
Word problems.
1. The Yellow River in China is 5464 kilometres long. How many kilometres will 75 boats cover if they travel the length of the river?
2. Mount Logan in Canada is 5959 metres high. How many metres would 24 climbers cover if they were able to climb to the top of Mt. Logan?
3. How many pieces of candy are needed to pack 500 boxes, if each box has 8 rows and each row has 15 pieces of candy? (2-step question)
4. The distance between Prince Rupert and Kelowna is 1409 km. Estimate how many kilometres 42 trucks will travel if each truck makes one trip.

Answers to Unit 1 Review

Find the products.
1. 405
2. 219
3. 882
4. 2048
5. 18684
6. 24396
Find the products.
1. 336
2. 702
3. 7136
4. 3216
5. 9500
6. 16947
Find the products.
1. 1273
2. 19754
3. 336260
4. 521768
5. 2313810
6. 263956
7. 256003
8. 42075
9. 3790786
Find the products. Use the shortcut.
1. 82000
2. 2600
3. 6263000
4. 40700
5. 948200
6. 36140
7. 1795000
Find the products. Use the shortcut.
1. 2000
2. 480000
3. 4500000
Find an estimated product.
1. 70 × 40 = 2800
2. 200 × 90 = 18000
3. 7000 × 70 = 490000
4. 900 × 400 = 360000
5. 3000 × 800 = 2400000
6. 70000 × 300 = 21000000
Word problems.
1. 409800 kilometres
2. 143016 metres or 286032 there and back
3. 60000 pieces of candy
4. 1000 × 40 = 40000 kilometres

CONGRATULATIONS!!

Now you have finished Unit 1.

TEST TIME!

Ask your instructor for the practice test for this unit.

Once you’ve done the practice test, you need to do the unit 1 test. Again, ask your instructor for this.

Good luck!

Unit 2: Division

Topic A: Introduction and Division Facts

Division is an interesting operation. Both these signs tell you to divide:

$8\div4$ $\require{enclose}4\enclose{longdiv}{8}$

Division is the opposite of multiplication.

Multiplication takes equal-sized groups and puts the groups together to find the total.

3 × 4 = 12

Division takes the total and separates that amount into equal groups. You can find the number of equal groups or the size of each group.

12 ÷ 4 = 3

$4 × 3 = 12$	$12 ÷ 3 = 4$	$\begin{array}{r}4\\3 \enclose{longdiv}{12} \end{array}$
$3 × 4 = 12$	$12 ÷ 4 = 3$	$\begin{array}{r}3\\4 \enclose{longdiv}{12} \end{array}$

Learn this vocabulary for division:

Dividend – The number or quantity to be divided; the amount altogether.
Divisor – The number we divide by. The divisor tells us the number of groups or the quantity in each group that the dividend is to be separated into.
Quotient – (“kw ō shent”) The answer to a division question.

$\begin{array}{r}\text{quotient}\\\text{divisor} \enclose{longdiv}{\text{dividend}}\end{array}$ $\text{dividend}\div\text{divisor}=\text{quotient}$

If you have learned your times tables well, the division facts will be easier. The times table chart can be used to find division facts.

To use the times table chart for division facts:

Find the divisor in the column on the left of the times table chart.
Run your finger in a straight line across the divisor row until you come to the dividend.
Go straight up that column and the quotient should be the top number.

Try a few while you are doing Exercise 1.

Exercise 1

Complete this chart to help yourself understand the connection between multiplication and division. The first one is done for you. Check your work using the answer key at the end of the exercise.

#	Multiplication	Division	Division	“Say”
a	5 × 3 = 15 3 × 5 = 15	15 ÷ 3 = 5 15 ÷ 5 = 3	$\begin{array}{r}5\\3 \enclose{longdiv}{15} \end{array}$ $\begin{array}{r}3\\5 \enclose{longdiv}{15} \end{array}$	15 divided by 3 is 5 15 divided by 5 is 3
b	8 × 6 = 48 6 × 8 = 48	48 ÷ 6 = 8 48 ÷ 8 = 6	$\begin{array}{r}8\\6 \enclose{longdiv}{48} \end{array}$ $\begin{array}{r}6\\8 \enclose{longdiv}{48} \end{array}$	48 divided by 6 is 8.
c	3 × 7 = 21
d	5 × 9 = 45
e	4 × 6 = 24
f	2 × 8 = 16
g	7 × 10 = 70
h	6 × 9 = 54
i	9 × 4 = 36
j	6 × 7 = 42

Answers to Exercise 1

#	Multiplication	Division	Division	“Say”
a	5 × 3 = 15 3 × 5 = 15	15 ÷ 3 = 5 15 ÷ 5 = 3	$\begin{array}{r}5\\3 \enclose{longdiv}{15} \end{array}$ $\begin{array}{r}3\\5 \enclose{longdiv}{15} \end{array}$	15 divided by 3 is 5 15 divided by 5 is 3
b	8 × 6 = 48 6 × 8 = 48	48 ÷ 6 = 8 48 ÷ 8 = 6	$\begin{array}{r}8\\6 \enclose{longdiv}{48} \end{array}$ $\begin{array}{r}6\\8 \enclose{longdiv}{48} \end{array}$	48 divided by 6 is 8. 48 divided by 8 is 6.
c	3 × 7 = 21 7 × 3 = 21	21 ÷ 7 = 3 21 ÷ 3 = 7	$\begin{array}{r}3\\7 \enclose{longdiv}{21} \end{array}$ $\begin{array}{r}7\\3 \enclose{longdiv}{21} \end{array}$	21 divided by 7 is 3. 21 divided by 3 is 7.
d	5 × 9 = 45 9 × 5 = 45	45 ÷ 9 = 5 45 ÷ 5 = 9	$\begin{array}{r}5\\9 \enclose{longdiv}{45} \end{array}$ $\begin{array}{r}9\\5 \enclose{longdiv}{45} \end{array}$	45 divided by 9 is 5. 45 divided by 5 is 9.
e	4 × 6 = 24 6 × 4= 24	24 ÷ 6 = 4 24 ÷ 4 = 6	$\begin{array}{r}4\\6 \enclose{longdiv}{24} \end{array}$ $\begin{array}{r}6\\4 \enclose{longdiv}{24} \end{array}$	24 divided by 6 is 4. 24 divided by 4 is 6.
f	2 × 8 = 16 8 × 2 = 16	16 ÷ 8 = 2 16 ÷ 2 = 8	$\begin{array}{r}2\\8 \enclose{longdiv}{16} \end{array}$ $\begin{array}{r}8\\2 \enclose{longdiv}{16} \end{array}$	16 divided by 8 is 2. 16 divided by 2 is 8.
g	7 × 10 = 70 10 × 7 = 70	70 ÷ 10 = 7 70 ÷ 7 = 10	$\begin{array}{r}7\\10 \enclose{longdiv}{70} \end{array}$ $\begin{array}{r}10\\7 \enclose{longdiv}{70} \end{array}$	70 divided by 10 is 7. 70 divided by 7 is 10.
h	6 × 9 = 54 9 × 6 = 54	54 ÷ 9 = 6 54 ÷ 6 = 9	$\begin{array}{r}6\\9 \enclose{longdiv}{54} \end{array}$ $\begin{array}{r}9\\6 \enclose{longdiv}{54} \end{array}$	54 divided by 9 is 6. 54 divided by 6 is 9.
i	9 × 4 = 36 4 × 9 = 36	36 ÷ 4 = 9 36 ÷ 9 = 4	$\begin{array}{r}9\\4 \enclose{longdiv}{36} \end{array}$ $\begin{array}{r}4\\9 \enclose{longdiv}{36} \end{array}$	36 divided by 4 is 9. 36 divided by 9 is 4.
j	6 × 7 = 42 7 × 6 = 42	42 ÷ 7 = 6 42 ÷ 6 = 7	$\begin{array}{r}6\\7 \enclose{longdiv}{42} \end{array}$ $\begin{array}{r}7\\6 \enclose{longdiv}{42} \end{array}$	42 divided by 7 is 6. 42 divided by 6 is 7.

Exercise 2

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

72 ÷ 6 =
12 ÷ 2 =
3 ÷ 1 =
80 ÷ 10 =
18 ÷ 6 =
40 ÷ 4 =
21 ÷ 7 =
50 ÷ 5 =
54 ÷ 9 =
8 ÷ 2 =
22 ÷ 11 =
45 ÷ 9 =
4 ÷ 4 =
24 ÷ 6
81 ÷ 9 =
88 ÷ 8 =
30 ÷ 3 =
12 ÷ 4 =
33 ÷ 3 =
66 ÷ 11 =
20 ÷ 5 =

Answers to Exercise 2

Exercise 3

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

90 ÷ 10 =
70 ÷ 7 =
28 ÷ 7 =
32 ÷ 8 =
24 ÷ 3 =
36 ÷ 12 =
84 ÷ 7 =
10 ÷ 2 =
64 ÷ 8 =
6 ÷ 6 =
60 ÷ 12 =
48 ÷ 4 =
72 ÷ 9 =
20 ÷ 10 =
49 ÷ 7 =
48 ÷ 6 =
36 ÷ 9 =
21 ÷ 3 =

Answers to Exercise 3

Exercise 4

Check your division facts by quickly doing this exercise. Check your work using the answer key at the end of the exercise.

12 ÷ 6 =
27 ÷ 9 =
56 ÷ 7 =
3 ÷ 1 =
20 ÷ 2 =
9 ÷ 3 =
55 ÷ 5 =
14 ÷ 7 =
42 ÷ 6 =
18 ÷ 3 =
88 ÷ 11 =
63 ÷ 9 =
28 ÷ 4 =
6 ÷ 1 =
30 ÷ 5 =
4 ÷ 2 =
7 ÷ 7 =
48 ÷ 12 =

Answers to Exercise 4

Tip: Make a list of any errors that you made and the facts that you had to really think about. If you have any more than five facts on your list, ask your instructor for suggestions on learning and drilling the division facts.

Remainders

You have been practicing the division facts that always work out evenly – nothing is left over.

Well, in the real world things are not usually so perfect!

You have 7 candies to share among your 3 children.

Seven candies. There are three groups of two candies with one left over.

We call the left-over the remainder. For now, put R and the left over number after your quotient.

$7\div3=2\text{ R}1$ $\begin{array}{r}2\text{ R}1\\3 \enclose{longdiv}{7}\hspace{1.3em}\end{array}$

Example A

Here are 22 cookies. Circle groups of 5. How many groups of 5 in 22?

Twenty-two ginger-person cookies.

You should have 4 groups with 2 left over.

$22\div5=4\text{ R}2$ $\begin{array}{r}4\text{ R}2\\5 \enclose{longdiv}{22}\hspace{1.3em}\end{array}$

The remainder must not be the same size or bigger than the divisor. If it is bigger, it means another group could be made.

Example B

Here are 66 suns. Make groups of 9. How many groups of 9 in 66?

Sixty-six suns.

How many groups? 7

How many left over? 3

$66\div9=7\text{ R}3$ $\begin{array}{r}7\text{ R}3\\9 \enclose{longdiv}{66}\hspace{1.3em}\end{array}$

Tip: The remainder must not be the same size or bigger than the divisor. Why not? It would make another group.

Exercise 5

Draw pictures to find the quotient and the remainder. Write each question using the other division sign as well. Have your instructor check your work.

$23\div10=2\text{ R}3$ $\begin{array}{r}2\text{ R}3\\10 \enclose{longdiv}{23}\hspace{1.3em}\end{array}$

24 ÷ 7 =
19 ÷ 3 =
39 ÷ 12 =

You cannot always draw pictures, so how should you find the quotients?

Example C

29 ÷ 3 =

Use multiplication tables or the division facts to find a trial quotient. What can you multiply by 3 to find a number close to 29?

3 × 9 = 27
3 × 10 = 30

Use 9 as the trial quotient. Do not use 10 because 3 × 10 = 30, which is more than the dividend 29.

Divide.
$\begin{array}{r}9\\3 \enclose{longdiv}{29}\end{array}$
Multiply. 9 × 3 = 27. Write the product under the 29.
$\begin{array}{r}9\\3 \enclose{longdiv}{29}\\27\end{array}$
Subtract 27 from 29 to find the remainder.
$\begin{array}{r}9\\3 \enclose{longdiv}{29}\\-27\\\hline2\end{array}$
Check (compare) to be sure the remainder is less then (<) the divisor.
2 < 3

29 ÷ 3 = 9 R2

Example D

60 ÷ 7 =

Think what can be multiplied by 7 to find a number close to 60.

7 × 8 = 56 (good)
7 × 9 = 63 (too big)

Use 8 as the trial quotient.

Divide.
$\begin{array}{r}8\\7 \enclose{longdiv}{60}\end{array}$
Multiply.
$\begin{array}{r}8\\7 \enclose{longdiv}{60}\\56\end{array}$
Subtract.
$\begin{array}{r}8\\7 \enclose{longdiv}{60}\\-56\\\hline4\end{array}$
Compare to be sure the remainder is less than the divisor.
4 < 7

60 ÷ 7 = 8 R4

Exercise 6

Find the quotients and remainders (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.

$5\enclose{longdiv}{28}$
$4\enclose{longdiv}{15}$
$6\enclose{longdiv}{47}$
$9\enclose{longdiv}{37}$
$2\enclose{longdiv}{13}$
$6\enclose{longdiv}{25}$
$8\enclose{longdiv}{75}$
$3\enclose{longdiv}{19}$
$7\enclose{longdiv}{32}$
$4\enclose{longdiv}{9}$
$9\enclose{longdiv}{55}$
$10\enclose{longdiv}{98}$
$3\enclose{longdiv}{26}$
$8\enclose{longdiv}{47}$
$9\enclose{longdiv}{46}$
$6\enclose{longdiv}{43}$
$5\enclose{longdiv}{49}$
$4\enclose{longdiv}{38}$
$2\enclose{longdiv}{19}$
$7\enclose{longdiv}{61}$
$3\enclose{longdiv}{23}$
$8\enclose{longdiv}{78}$
$9\enclose{longdiv}{67}$
$6\enclose{longdiv}{45}$

Answers to Exercise 6

5 R3
3 R3
7 R5
4 R1
6 R1
4 R1
9 R3
6 R1
4 R4
2 R1
6 R1
9 R8
8 R2
5 R7
5 R1
7 R1
9 R4
9 R2
9 R1
8 R5
7 R2
9 R6
7 R4
7 R3

Exercise 7

Find the quotients and remainders (divide, multiply, subtract, compare.) Check your work using the answer key at the end of the exercise.

Example

59 ÷ 7 =

Rewrite: $7\enclose{longdiv}{59}$

Then solve: $\begin{array}{r}8\\7 \enclose{longdiv}{59}\\-56\\\hline3\end{array}$

Answer: 8 R3

27 ÷ 5 =
13 ÷ 2 =
46 ÷ 9 =
38 ÷ 6 =
61 ÷ 7 =
14 ÷ 5 =
49 ÷ 8 =
28 ÷ 3 =
78 ÷ 8 =
37 ÷ 4 =
67 ÷ 9 =
52 ÷ 6 =
45 ÷ 8 =
25 ÷ 7 =
11 ÷ 3 =
53 ÷ 9 =
19 ÷ 4 =
77 ÷ 8 =
20 ÷ 3 =
11 ÷ 2 =
23 ÷ 5 =
54 ÷ 7 =
87 ÷ 9 =
9 ÷ 4 =

Answers for Exercise 7

5 R2
6 R1
5 R1
6 R2
8 R5
2 R4
6 R1
9 R1
9 R6
9 R1
7 R4
8 R4
5 R5
3 R4
3 R2
5 R8
4 R3
9 R5
6 R2
5 R1
4 R3
7 R5
9 R6
2 R1

Topic A: Self-Test

Mark /12 Aim 9/12

Give the answer. (6 marks)
1. 63 ÷ 9 =
2. 21 ÷ 7 =
3. 72 ÷ 8 =
4. 36 ÷ 4 =
5. 32 ÷ 4 =
6. 48 ÷ 6 =
Find the quotient. (6 marks)
1. $6\enclose{longdiv}{59}$
2. $9\enclose{longdiv}{87}$
3. $7\enclose{longdiv}{51}$
4. $8\enclose{longdiv}{76}$
5. $5\enclose{longdiv}{49}$
6. $3\enclose{longdiv}{26}$

Answers to Topic A Self-Test

Give the answer.
1. 7
2. 3
3. 9
4. 9
5. 8
6. 8
Find the quotient.
1. 9 R5
2. 9 R6
3. 7 R2
4. 9 R4
5. 9 R4
6. 8 R2

Topic B: Divisibility

Divisibility is when two numbers can be divided without a remainder.

For example, 18 is divisible by 3 because 18 ÷ 3 = 6.

18 is not divisible by 5 because 18 ÷ 5 = 3 with a remainder of 3.

Divisibility by 2

A number is divisible by 2 if it ends in 0, 2, 4, 6 or 8.

Example A

438, 540, 256, 12, and 154 are all divisible by 2 because each number ends in a 0, 2, 4, 6, or 8.

Example B

351, 945, 849, 347, and 193 are not divisible by 2 because each number does not end in a 0, 2, 4, 6 or 8.

Exercise 1

Circle the numbers that are divisible by 2. Check your work using the answer key at the end of the exercise.

22
35
17
10
274
345
639
456
2437
7548
6754
5543

Answers to Exercise 1

The following numbers are divisible by 2: a, d, e, h, j, k.

Divisibility by 3

A number is divisible by 3 if the sum (total) of the digits is divisible by 3.

Example C

6 + 3 = 9

9 is divisible by three, so 63 is divisible by three.

Example D

148

1 + 4 + 8 = 13

13 is not divisible by three, so 148 is not divisible by 3.

Example E

5892

5 + 8 + 9 + 2 = 24

Add again: 2 + 4 = 6

6 is divisible by 3, so 5892 is divisible by 3.

Exercise 2

Circle the numbers that are divisible by 3. Check your work using the answer key at the end of the exercise.

27
35
81
94
274
581
564
316
3175
1458
1890
3934

Answers to Exercise 2

The following numbers are divisible by 3: a, c, g, j, k.

Divisibility by 5

A number is divisible by 5 if the number ends in 0 or 5.

Example F

290 is divisible by 5 because it ends in 0.

Example G

132 is not divisible by 5 because it does not end in 0 or 5.

Exercise 3

Circle the numbers that are divisible by 5. Check your work using the answer key at the end of the exercise.

45
84
72
90
800
753
672
355
6009
6375
7020
1704

Answers to Exercise 3

The following numbers are divisible by 5: a, d, e, h, j, k.

Exercise 4

For each number, identify if it is divisible by 2, 3, and 5. Check your work using the answer key at the end of the exercise.

Number	Divisible by 2	Divisible by 3	Divisible by 5
474	Yes	Yes	No
615
412
865
300
831
525
350
710
429
906
634
430
275

Answers to Exercise 4

Number	Divisible by 2	Divisible by 3	Divisible by 5
474	Yes	Yes	No
615	No	Yes	Yes
412	Yes	No	No
865	No	No	Yes
300	Yes	Yes	Yes
831	No	Yes	No
525	No	Yes	Yes
350	Yes	No	Yes
710	Yes	No	Yes
429	No	Yes	No
906	Yes	Yes	No
634	Yes	No	No
430	Yes	No	Yes
275	No	No	Yes

Exercise 5

For each number, identify if it is divisible by 2, 3, and 5. Check your work using the answer key at the end of the exercise.

Number	Divisible by 2	Divisible by 3	Divisible by 5
3585
7548
5890
6318
3905
5280
1760
8007
6752
7375
5523
2625
8956
9150

Answers to Exercise 5

Number	Divisible by 2	Divisible by 3	Divisible by 5
3585	No	Yes	Yes
7548	Yes	Yes	No
5890	Yes	No	Yes
6318	Yes	Yes	No
3905	No	No	Yes
5280	Yes	Yes	Yes
1760	Yes	No	Yes
8007	No	Yes	No
6752	Yes	No	No
7375	No	No	Yes
5523	No	Yes	No
2625	No	Yes	Yes
8956	Yes	No	No
9150	Yes	Yes	Yes

Divisibility by 9

A number is divisible by 9 if the sum (total) of the digits is divisible by 9.

Example H

135

1 + 3 + 5 = 9

9 is divisible by 9, so 135 is divisible by 9.

Example I

7578

7 + 5 + 7 + 8 = 27

27 is divisible by 9, so 7578 is divisible by 9.

Example J

57896

5 + 7 + 8 + 9 + 6 = 35

35 is not divisible by 9, so 57896 is not divisible by 9.

Exercise 6

Circle the numbers that are divisible by 9. Check your work using the answer key at the end of the exercise.

538
783
954
762
6213
5742
7083
5738
34937
39402
74124
45683

Answers to Exercise 6

The following numbers are divisible by 9: b, c, f, g, j, k

Exercise 7

For each number, identify if it is divisible by 2, 3, 5, and 9. Check your work using the answer key at the end of the exercise.

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
453
320
216
726
712
425
630
375
990
210

Answers to Exercise 7

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
453	No	Yes	No	No
320	Yes	No	Yes	No
216	Yes	Yes	No	Yes
726	Yes	Yes	No	No
712	Yes	No	No	No
425	No	No	Yes	No
630	Yes	Yes	Yes	Yes
375	No	Yes	Yes	No
990	Yes	Yes	Yes	Yes
210	Yes	Yes	Yes	No

Exercise 8

For each number, identify if it is divisible by 2, 3, 5, and 9. Check your work using the answer key at the end of the exercise.

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
837
360
648
981
465
1002
3520
6435
8022
7425

Answers to Exercise 8

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
837	No	Yes	No	Yes
360	Yes	Yes	Yes	Yes
648	Yes	Yes	No	Yes
981	No	Yes	No	Yes
465	No	Yes	Yes	No
1002	Yes	Yes	No	No
3520	Yes	No	Yes	No
6435	No	Yes	Yes	Yes
8022	Yes	Yes	No	No
7425	No	Yes	Yes	Yes

Exercise 9

For each number, identify if it is divisible by 2, 3, 5, and 9. Check your work using the answer key at the end of the exercise

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
1200
7164
3681
8205
2745
4320
7350
4000
1368
6720

Answers to Exercise 9

Number	Divisible by 2	Divisible by3	Divisible by 5	Divisible by 9
1200	Yes	Yes	Yes	No
7164	Yes	Yes	No	Yes
3681	No	Yes	No	Yes
8205	No	Yes	Yes	No
2745	No	Yes	Yes	Yes
4320	Yes	Yes	Yes	Yes
7350	Yes	Yes	Yes	No
4000	Yes	No	Yes	No
1368	Yes	Yes	No	Yes
6720	Yes	Yes	Yes	No

Topic B: Self-Test

Mark /12 Aim 9/12

From the list of numbers, write the numbers that are divisible by the specified number.
48, 925, 1467, 2645, 5534, 7512, 31183, 52361
1. Which numbers are divisible by 2?
2. Which numbers are divisible by 3?
3. Which numbers are divisible by 5?
4. Which numbers are divisible by 9?
For each number, identify if it is divisible by 2, 3, 5, and 9.

Number Divisible by 2 Divisible by 3 Divisible by 5 Divisible by 9

1200

7164

3681

8205

2745

Answers to Topic B Self-Test

From the list of numbers, write the numbers that are divisible by the specified number.
1. 48, 7512, 5534
2. 48, 1467, 7512
3. 925, 2645
4. 1467
For each number, identify if it is divisible by 2, 3, 5, and 9.

Number Divisible by 2 Divisible by 3 Divisible by 5 Divisible by 9

1200 Yes Yes Yes No

7164 Yes Yes No Yes

3681 No Yes No Yes

8205 No Yes Yes No

2745 No Yes Yes Yes

Topic C: Dividing Larger Numbers by One-Digit Divisors

Several methods are used to divide larger numbers. This book will only teach one method. If you have learned a different method for dividing, ask your instructor to review it with you. You can use the practice exercises in this workbook using whichever method you prefer.

One-Digit Divisors without Remainders

Division has four steps which are repeated until the dividend is completely divided. Work through the three examples which show these steps.

Divide.
Multiply
Subtract and compare the remainder to the divisor.
Bring down the next digit in the dividend and repeat.

Example A

294 ÷ 7 =

Rewrite as $7\enclose{longdiv}{294}$

Step 1: Divide.

Find a trial quotient using multiplication tables or division facts.
Look at the dividend one digit at a time.
1. The first digit is a 2, which is really 2 hundreds.
2. Will 7 go into 2? Can you divide 2 by 7? No.
3. Look at the first 2 digits, 29, which is really 29 tens.
4. Will 7 go into 29? Yes. 4 × 7 = 28.
The first number in the quotient is 4. Place the 4 in the quotient directly above the 9 tens. The 4 is 4 tens in the quotient.

$\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\end{array}$

Step 2: Multiply.

Multiply 4 × 7 = 28.
Write the 28 under the 29. Draw a line.

$\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\28\hspace{0.8em}\\\hline\end{array}$

Step 3: Subtract.

Subtract 29 − 28 = 1 (ten).
Check 1 < 7.

$\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline1\hspace{0.8em}\end{array}$

Step 4: Bring down the next number in the dividend.

Bring down the next number in the dividend (4), and put it beside the result of the subtraction you just completed (1).
Together, they make 14. This 14 is the number that you must now divide.

$\begin{array}{r}4\hspace{0.8em}\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}$

Repeat Step 1: Divide.

Divide 14 ÷ 7 = 2
Put the 2 in the quotient right after the 4 in the ones place.

$\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\end{array}$

Repeat Step 2: Multiply.

Multiply 2 × 7 = 14.
Write the 14 under the 14. Draw a line.

$\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\14\\\hline\end{array}$

Repeat Step 3: Subtract.

Subtract 14 − 14 = 0. There is 0 remainder.
Check 0 < 7.

$\begin{array}{r}42\\7 \enclose{longdiv}{294}\\-28\hspace{0.8em}\\\hline14\\-14\\\hline0\end{array}$

There are no more numbers in the dividend to bring down, so you are done.

294 ÷ 7 = 42

Example B

128 ÷ 2 =

$\begin{array}{r}64\\2 \enclose{longdiv}{128}\\-12\hspace{0.8em}\\\hline08\\-8\\\hline0\end{array}$

Step 1: Divide.

Can 2 go into 1? No.
Can 2 go into 12? Yes.
How many times?
- 2 × 6 = 12; 12 ÷ 2 = 6
The first number in the trial quotient is 6. Put the 6 in the quotient directly above the 2 tens dividend.

Step 2: Multiply.

6 × 2 = 12.
Write the 12 under the 12. Draw a line.

Step 3: Subtract.

12 − 12 = 0
Check 0 < 12

Step 4: Bring down the next number in the dividend.

Bring down the next number in the dividend (8), and put it beside the result of the subtraction you just completed (0).
8 is now the number to be divided.

Repeat

Divide: 8 ÷ 2 = 4
Multiple: 4 × 2 = 8
Subtract: 8 − 8 = 0
1. Check 0 < 2
Bring down the next digit. There are no more digits in the dividend.

128 ÷ 2 = 64

Exercise 1

Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.

$4\enclose{longdiv}{364}$
$2\enclose{longdiv}{144}$
$5\enclose{longdiv}{455}$
$7\enclose{longdiv}{651}$
$8\enclose{longdiv}{144}$
$2\enclose{longdiv}{166}$
$7\enclose{longdiv}{588}$
$2\enclose{longdiv}{196}$
$5\enclose{longdiv}{230}$
$8\enclose{longdiv}{584}$
$6\enclose{longdiv}{366}$
$4\enclose{longdiv}{244}$
$5\enclose{longdiv}{375}$
$8\enclose{longdiv}{200}$
$2\enclose{longdiv}{628}$
$7\enclose{longdiv}{357}$
$9\enclose{longdiv}{837}$
$8\enclose{longdiv}{248}$
$8\enclose{longdiv}{312}$
$7\enclose{longdiv}{462}$
$5\enclose{longdiv}{295}$
$6\enclose{longdiv}{384}$
$2\enclose{longdiv}{276}$
$4\enclose{longdiv}{372}$

Answers to Exercise 1

Example C

856 ÷ 8 =

$8\enclose{longdiv}{856}$

Divide

Does 8 go into 8? Yes. 8 ÷ 8 = 1.

$\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\end{array}$

Multiply

1 × 8 = 8

$\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\8\hspace{1.2em}\end{array}$

Subtract

8 − 8 = 0. Check 0 < 8.

$\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline0\hspace{1.2em}\end{array}$

Bring down the next digit

5 is now the number to be divided.

$\begin{array}{r}1\hspace{1.2em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}$

Now repeat.

Divide

Does 8 go into 5? No.

You must put a zero to hold the place in the quotient. If a digit is brought down, a digit must be placed in the quotient.

Multiply

0 × 8 = 0

$\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\end{array}$

Subtract

5 − 0 = 5. Check 5 < 8.

$\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline5\hspace{0.8em}\end{array}$

Bring down the next digit

56 is now the number to be divided.

$\begin{array}{r}10\hspace{0.8em}\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}$

Now repeat.

Divide

Does 8 go into 56? Yes. 56 ÷ 8 = 7.

Write 7 in the quotient in the ones place above the 6 in the dividend.

$\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\end{array}$

Multiply

7 × 8 = 56

$\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\56\end{array}$

Subtract

56 − 56 = 0. Check 0 < 8.

$\begin{array}{r}107\\8 \enclose{longdiv}{856}\\-8\hspace{1.2em}\\\hline05\hspace{0.8em}\\-0\hspace{0.8em}\\\hline56\\-56\\\hline0\end{array}$

Bring down the next digit.

No more digits.

856 ÷ 8 = 107

Exercise 2

Find the quotients (divide, multiply, subtract, compare). Check your work using the answer key at the end of the exercise.

$7\enclose{longdiv}{721}$
$9\enclose{longdiv}{954}$
$3\enclose{longdiv}{927}$
$3\enclose{longdiv}{621}$
$4\enclose{longdiv}{824}$
$9\enclose{longdiv}{972}$
$7\enclose{longdiv}{714}$
$2\enclose{longdiv}{416}$
$5\enclose{longdiv}{540}$
$6\enclose{longdiv}{654}$
$8\enclose{longdiv}{832}$
$4\enclose{longdiv}{436}$
$5\enclose{longdiv}{515}$
$2\enclose{longdiv}{814}$
$6\enclose{longdiv}{648}$
$8\enclose{longdiv}{856}$
$5\enclose{longdiv}{525}$
$7\enclose{longdiv}{763}$
$9\enclose{longdiv}{945}$
$3\enclose{longdiv}{315}$
$8\enclose{longdiv}{872}$
$4\enclose{longdiv}{416}$
$6\enclose{longdiv}{618}$
$2\enclose{longdiv}{612}$

Answers to Exercise 2

One-Digit Divisors with Remainders

Do the division exactly the same way that you have been learning. Often there is a remainder after the last subtraction. Write it with the quotient as you already know how to do.

Example D

259 ÷ 8 =

$\begin{array}{l}\hspace{1.5em}32\text{ R}3\\8 \enclose{longdiv}{259}\\\hspace{0.3em}-24\\\hline\hspace{1.7em}19\\\hspace{1em}-16\\\hline\hspace{2.3em}3\end{array}$

259 ÷ 8 = 32 R3

Exercise 3

Divide and show any remainders. Check your work using the answer key at the end of the exercise.

$20\enclose{longdiv}{93}$
$3\enclose{longdiv}{52}$
$5\enclose{longdiv}{94}$
$7\enclose{longdiv}{74}$
$4\enclose{longdiv}{95}$
$9\enclose{longdiv}{96}$
$6\enclose{longdiv}{97}$
$8\enclose{longdiv}{99}$
$9\enclose{longdiv}{98}$
$4\enclose{longdiv}{59}$
$6\enclose{longdiv}{76}$
$3\enclose{longdiv}{79}$
$7\enclose{longdiv}{96}$
$5\enclose{longdiv}{57}$
$2\enclose{longdiv}{47}$
$8\enclose{longdiv}{91}$
$7\enclose{longdiv}{89}$
$6\enclose{longdiv}{82}$
$5\enclose{longdiv}{67}$
$2\enclose{longdiv}{85}$
$4\enclose{longdiv}{71}$
$3\enclose{longdiv}{65}$
$9\enclose{longdiv}{92}$
$8\enclose{longdiv}{94}$

Answers to Exercise 3

46 R1
17 R1
18 R4
10 R4
23 R3
10 R6
16 R1
12 R3
10 R8
14 R3
12 R4
26 R1
13 R5
11 R2
23 R1
11 R3
12 R5
13 R4
13 R2
42 R1
17 R3
21 R2
10 R2
11 R6

Check Division by Multiplying

To check your division, do this:

Multiply the quotient by the divisor.
Add on any remainder.

The product will equal the dividend if your arithmetic is correct.

Example E

$\begin{array}{l}\hspace{1.5em}52\text{ R}1\\9 \enclose{longdiv}{469}\\\hspace{0.3em}-45\\\hline\hspace{1.7em}19\\\hspace{1em}-18\\\hline\hspace{2.3em}1\end{array}$ $\begin{array}{rr}&52\\\times&9\\\hline&468\end{array}$ $468+1=469$

Exercise 4

Divide and check your answer by multiplying. Check your work using the answer key at the end of the exercise.

$7\enclose{longdiv}{709}$
$2\enclose{longdiv}{423}$
$5\enclose{longdiv}{538}$
$4\enclose{longdiv}{609}$
$9\enclose{longdiv}{406}$
$6\enclose{longdiv}{125}$
$3\enclose{longdiv}{605}$
$9\enclose{longdiv}{928}$
$3\enclose{longdiv}{962}$
$4\enclose{longdiv}{805}$
$8\enclose{longdiv}{301}$
$2\enclose{longdiv}{807}$
$6\enclose{longdiv}{725}$
$7\enclose{longdiv}{320}$
$9\enclose{longdiv}{140}$
$8\enclose{longdiv}{483}$
$2\enclose{longdiv}{197}$
$6\enclose{longdiv}{307}$
$5\enclose{longdiv}{504}$
$8\enclose{longdiv}{709}$
$7\enclose{longdiv}{876}$
$4\enclose{longdiv}{101}$
$3\enclose{longdiv}{269}$
$5\enclose{longdiv}{473}$

Answers to Exercise 4

101 R2
211 R1
107 R3
152 R1
45 R1
20 R5
201 R2
103 R1
320 R2
201 R1
37 R5
403 R1
120 R5
45 R5
15 R5
60 R3
98 R1
51 R1
100 R4
88 R5
125 R1
25 R1
89 R2
94 R3

Topic C: Self-Test

Mark /24 Aim 19/24

Find the quotient. (6 marks)
1. $6\enclose{longdiv}{96}$
2. $4\enclose{longdiv}{92}$
3. $7\enclose{longdiv}{91}$
4. $2\enclose{longdiv}{93}$
5. $5\enclose{longdiv}{94}$
6. $3\enclose{longdiv}{52}$
Divide. (6 marks)
1. $7\enclose{longdiv}{182}$
2. $8\enclose{longdiv}{736}$
3. $6\enclose{longdiv}{162}$
4. $5\enclose{longdiv}{295}$
5. $4\enclose{longdiv}{184}$
6. $9\enclose{longdiv}{576}$
Divide and show your check for each answer (12 marks – 1 mark for question, 1 mark for check).
1. $9\enclose{longdiv}{705}$
2. $4\enclose{longdiv}{257}$
3. $7\enclose{longdiv}{899}$
4. $5\enclose{longdiv}{538}$
5. $8\enclose{longdiv}{876}$
6. $6\enclose{longdiv}{628}$

Answers to Topic C Self-Test

Find the quotient.
1. 16
2. 23
3. 13
4. 46 R1
5. 18 R4
6. 17 R1
Divide.
1. 26
2. 92
3. 27
4. 59
5. 46
6. 64
Divide and show your check for each answer.
1. 78 R3
2. 64 R1
3. 128 R3
4. 107 R3
5. 109 R4
6. 104 R4

Topic D: Dividing by Two- and Three- Digit Divisors

Finding Trial Quotients

When dividing by 2-digit numbers, you will need to estimate the quotient. This guess is called a trial quotient.

Example A

$624 \div 24$

Rewrite as $24\enclose{longdiv}{624}$

Step 1: Divide

Think: $2\enclose{longdiv}{6}$ is $3$ . So $24\enclose{longdiv}{62}$ is about $3$ .

Step 2: Multiply and subtract.

$\begin{array}{r}3 \ \ \ \\ 24\enclose{longdiv}{624}\\72 \ \ \ \\ \hline \end{array}$
Since 72 > 62, 3 is too large.

Step 3: Try a smaller number, multiply and subtract.

$\begin{array}{r}2\ \ \ \\ 24\enclose{longdiv}{624}\\48 \ \ \ \\ \hline 14 \ \ \ \end{array}$
Since 14 < 24, 2 is correct.

Step 4: Finish the problem.

Example B

$630 \div 15$

Rewrite as $15\enclose{longdiv}{630}$

Step 1: Divide

15 rounds to 20. Think: $2\enclose{longdiv}{6}$ is $3$ . So $15\enclose{longdiv}{63}$ is about $3$ .

Step 2: Multiply and subtract.

$\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{630}\\45 \ \ \ \\ \hline 18 \ \ \ \end{array}$
Since 18 > 15, 3 is too small.

Step 3: Try a larger number, multiply and subtract.

$\begin{array}{r}4 \ \ \ \\ 24\enclose{longdiv}{630}\\60 \ \ \\ \hline 3 \ \ \end{array}$
Since 3 < 15, 4 is correct.

Step 4: Finish the problem.

Exercise 1

In each question, the trial quotient is either too large or too small. Multiply. Write too large or too small on the line. Then, write the correct trial quotient beside. Check your work using the answer key at the end of the exercise.

$\begin{array}{r}2 \ \ \ \\ 25\enclose{longdiv}{475}\end{array}$
$\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{682}\end{array}$
$\begin{array}{r}5 \ \ \ \\ 18\enclose{longdiv}{813}\end{array}$
$\begin{array}{r}2\ \ \ \\ 25\enclose{longdiv}{810}\end{array}$
$\begin{array}{r}3\ \ \ \\ 33\enclose{longdiv}{891}\end{array}$
$\begin{array}{r}3\ \ \ \\ 18\enclose{longdiv}{819}\end{array}$
$\begin{array}{r}3\ \ \ \\ 27\enclose{longdiv}{727}\end{array}$
$\begin{array}{r}2 \ \ \ \\ 35\enclose{longdiv}{652}\end{array}$
$\begin{array}{r}3\ \ \ \\ 25\enclose{longdiv}{650}\end{array}$
$\begin{array}{r}4\ \ \ \\34\enclose{longdiv}{176}\end{array}$
$\begin{array}{r}4\ \ \ \\ 12\enclose{longdiv}{420}\end{array}$
$\begin{array}{r}2\ \ \ \\ 43\enclose{longdiv}{801}\end{array}$
$\begin{array}{r}3\ \ \ \\ 31\enclose{longdiv}{899}\end{array}$
$\begin{array}{r}4\ \ \ \\ 18\enclose{longdiv}{648}\end{array}$
$\begin{array}{r}4\ \ \ \\ 27\enclose{longdiv}{946}\end{array}$
$\begin{array}{r}3\ \ \ \\ 23\enclose{longdiv}{943}\end{array}$
$\begin{array}{r}3\ \ \ \\ 24\enclose{longdiv}{578}\end{array}$
$\begin{array}{r}2\ \ \ \\ 29\enclose{longdiv}{406}\end{array}$
$\begin{array}{r}2\ \ \ \\ 48\enclose{longdiv}{892}\end{array}$
$\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{534}\end{array}$
$\begin{array}{r}3\ \ \ \\ 37\enclose{longdiv}{939}\end{array}$
$\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{854}\end{array}$

Answers to Exercise 1

too large, 1
too small, 4
too large, 4
too small, 3
too large, 2
too small, 4
too large, 2
too large 1
too large, 2
too small, 5
too large, 3
too large, 1
too. large 2
too large, 3
too large, 3
too small, 4
too large, 2
too large, 1
too large, 1
too large 1
too large, 2
too small, 3

Example C

$78\enclose{longdiv}{2706}$

Since 78 rounds to 80, think $8\enclose{longdiv}{27}$ . 8 goes into 27 ≈ 3. 3 would be a good trial quotient.

$\begin{array}{r}3\ \ \ \\ 78\enclose{longdiv}{2706}\\234\ \ \ \\ \hline 36\ \ \ \end{array}$

Since 36 < 78, 3 is a good trial quotient.

Example D

$27\enclose{longdiv}{2205}$

Since 27 rounds to 30, think $3\enclose{longdiv}{22}$ . 3 goes into 22 ≈ 7. 7 would be a good trial quotient.

$\begin{array}{r}7\ \ \ \\ 27\enclose{longdiv}{2205}\\189\ \ \ \\ \hline 31\ \ \ \end{array}$

Since 31 > 27, so 7 is too small. A better trial quotient would be 8.

Exercise 2

Find the first digit in the trial quotient. Check your work using the answer key at the end of the exercise.

$43\enclose{longdiv}{1772}$ $\begin{array}{r}4\\ 4\enclose{longdiv}{17}\\16\\ \hline 1\end{array}$ . 1 < 4
$64\enclose{longdiv}{3276}$
$28\enclose{longdiv}{6008}$
$33\enclose{longdiv}{2731}$
$59\enclose{longdiv}{4164}$
$75\enclose{longdiv}{2420}$
$54\enclose{longdiv}{3316}$
$38\enclose{longdiv}{2759}$
$46\enclose{longdiv}{387}$
$35\enclose{longdiv}{3316}$
$83\enclose{longdiv}{7237}$
$77\enclose{longdiv}{6763}$
$93\enclose{longdiv}{3724}$
$52\enclose{longdiv}{4690}$
$86\enclose{longdiv}{2089}$
$26\enclose{longdiv}{1417}$
$72\enclose{longdiv}{1462}$
$27\enclose{longdiv}{6939}$
$32\enclose{longdiv}{7840}$
$24\enclose{longdiv}{7605}$
$16\enclose{longdiv}{8640}$
$45\enclose{longdiv}{3060}$
$38\enclose{longdiv}{2158}$
$42\enclose{longdiv}{1491}$

Answers to Exercise 2

Two Digit Divisors

Dividing by large divisors is a challenge!

You must estimate how many times one number will divide into another. Use pencil and have an eraser close by when you do these questions. You will use the same steps that you already know.

Example E

$964 \div 75 =$

Step 1: Divide

Does 75 go into 9? NO
Does 75 go into 96? YES
Estimate
1. Round 75 to 80 – think “8”
2. Round 96 to 100 – think “10”
3. How many 8’s in 10? (8 1 = 8, 10 8 = 1)
4. The estimate for the first digit in the trial quotient is 1.
Write 1 in the quotient above the 6 tens.

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\end{array}$

Step 2: Multiply

$1 \times 75 = 75$

Write 75 under 96.

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \ \ \\ \hline \end{array}$

Step 3: Subtract

$96 - 75 = 21$

Check 21 < 75? YES!

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75\ \ \\ \hline 21 \ \ \end{array}$

Step 4: Bring down the next digit in the dividend. 214 is now the number to be divided.

$\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}$

REPEAT

Step 1: Divide

Does 75 go into 214? YES
Estimate 75 as 80 – think “8”
1. Estimate 214 as 200 – think “20”
2. 8 goes into 20 2 times ( 8 2 = 16, so 20 8 2)
The estimate for the second digit in the trial quotient is 2. Write 2 in the quotient above the 4 in the dividend.

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}$

Step 2: Multiply

$2 \times 75 = 150$

Write 150 under the 214.

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline\end{array}$

Step 3: Subtract and check that the remainder is less than the divisor.

$\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline 64\end{array}$

Step 4: Bring down – no more digits in dividend.

To check your answer

$\begin{array}{rrr}&75\\\times&12\\\hline&150 \\ &750 \\ \hline &900 \\ + &64* \\ \hline &964\end{array}$

*Don’t forget to add the remainder when you check.

Exercise 3

Carefully divide these questions. Be careful to keep the hundreds in line with the hundreds, the tens with the tens, and so on. You might want to use squared paper for long division. Check your work using the answer key at the end of the exercise.

If you are having any difficulty, ask your instructor to watch you doing a few questions to be sure you are using a correct method.

$10\enclose{longdiv}{720}$
$12\enclose{longdiv}{564}$
$21\enclose{longdiv}{882}$
$22\enclose{longdiv}{946}$
$32\enclose{longdiv}{1632}$
$23\enclose{longdiv}{943}$
$62\enclose{longdiv}{2528}$
$71\enclose{longdiv}{2414}$
$24\enclose{longdiv}{7578}$
$82\enclose{longdiv}{2958}$
$18\enclose{longdiv}{6250}$
$25\enclose{longdiv}{1550}$
$19\enclose{longdiv}{9595}$
$47\enclose{longdiv}{3854}$
$58\enclose{longdiv}{6500}$
$24\enclose{longdiv}{9648}$
$49\enclose{longdiv}{1312}$
$67\enclose{longdiv}{7683}$

Answers to Exercise 3

72
47
42
43
51
41
40 R48
34
315 R18
36 R6
347 R4
62
505
82
112 R4
402
26 R38
114 R45

If the estimate for your trial quotient is too large, the result of the multiplication will be larger than the numbers in the dividend.

Take, for example, the following division question:

$23\enclose{longdiv}{784}$

When you divide, the trial estimate is 4:

$\begin{array}{r}4\ \ \ \\ 23\enclose{longdiv}{784}\end{array}$

When you multiply:

$4 \times 23 = 92$
92 is larger than 78, so 4 is too large an estimate. Erase it. Try 3. $3 \times 23 = 69$
3 is the correct estimate and you can complete the division, as seen below:

$\begin{array}{l}\hspace{2.3em}34\text{ R}2\\23 \enclose{longdiv}{784}\\\hspace{1.5em}69 \downarrow\\\hline\hspace{2.3em}94\\\hspace{2.3em}92\\\hline\hspace{2.8em}2\end{array}$

If the estimate is too small, the result of the subtraction will be larger than the divisor.

Take this division question as an example:

$72\enclose{longdiv}{4487}$

Say your trial quotient is 5. When you multiply, you get $5 \times 72 = 360$ . Since 360 is smaller than 448, you can go on to subtract:

$448 - 360=88$

Is 88 less than 72? NO, 88 is greater than 72. This means that 5 is too small for the trial quotient. Use the larger number 6.

Multiply $6 \times 72 = 432$ . Subtract $448 - 432 = 16$ . Since 16 is less than 72, you can continue the division.

See the full division process below:

$\begin{array}{l}\hspace{1.1cm}62\text{ R}23\\72 \enclose{longdiv}{4487}\\\hspace{1.45em}432 \downarrow\\\hline\hspace{2.1em}167\\\hspace{2.1em}144\\\hline\hspace{2.55em}23\end{array}$

Exercise 4

Divide and check your work by multiplying. Check your work using the answer key at the end of the exercise.

$18\enclose{longdiv}{648}$
$26\enclose{longdiv}{6766}$
$52\enclose{longdiv}{1968}$
$84\enclose{longdiv}{8640}$
$72\enclose{longdiv}{2883}$
$94\enclose{longdiv}{8126}$
$20\enclose{longdiv}{4060}$
$47\enclose{longdiv}{1728}$
$33\enclose{longdiv}{1886}$
$25\enclose{longdiv}{5750}$
$79\enclose{longdiv}{2765}$
$42\enclose{longdiv}{8442}$
$57\enclose{longdiv}{9144}$
$96\enclose{longdiv}{20160}$
$75\enclose{longdiv}{23550}$

Answers to Exercise 4

36
260 R6
37 R44
102 R72
40 R3
86 R42
203
36 R36
57 R5
230
35
201
160 R24
210
314

Dividing by 10, 100, 1000…

What is the pattern? When you divide by 10

The ones digit in the dividend becomes the remainder.

$10\enclose{longdiv}{324} = 32 R4$

The other numbers in the dividend stay the same, but each digit is one place value less.

- the hundreds become tens
- the tens become ones
- the ones become the remainder

Exercise 5

Find the quotients. Look for the pattern. Check your work using the answer key at the end of the exercise.

$10\enclose{longdiv}{46}$
$10\enclose{longdiv}{75}$
$10\enclose{longdiv}{136}$
$10\enclose{longdiv}{832}$
$10\enclose{longdiv}{674}$
$10\enclose{longdiv}{952}$
$10\enclose{longdiv}{2457}$
$10\enclose{longdiv}{3685}$

Answers to Exercise 5

4 R6
7 R5
13 R6
83 R2
67 R4
95 R2
245 R7
368 R5

When you divide by 100

The ones and tens digits in the dividend become the remainder.
The other digits in the dividend stay the same but each digit is two places less.
- the thousands become tens

- the hundreds become ones
- the tens and ones become the remainder

Exercise 6

Find these quotients. Look for the pattern when you divide. Check your work using the answer key at the end of the exercise

$100\enclose{longdiv}{386}$
$100\enclose{longdiv}{995}$
$100\enclose{longdiv}{269}$
$100\enclose{longdiv}{175}$
$100\enclose{longdiv}{2948}$
$100\enclose{longdiv}{4671}$
$100\enclose{longdiv}{92045}$
$100\enclose{longdiv}{43821}$

Answers to Exercise 6

3 R86
9 R95
2 R69
1 R75
29 R48
46 R71
920 R45
438 R21

When you divide by 1000

The ones, tens, and hundreds digits become the remainder.
The other digits stay the same but are three place values less.
- thousands become ones
- ten thousands become tens
- hundred thousands become hundreds

Exercise 7

Try these. Check your work using the answer key at the end of the exercise.

$1000\enclose{longdiv}{2398}$
$1000\enclose{longdiv}{6475}$
$1000\enclose{longdiv}{4835}$
$1000\enclose{longdiv}{63291}$
$1000\enclose{longdiv}{82405}$
$1000\enclose{longdiv}{293591}$

Answers to Exercise 7

2 R398
6 R475
4 R835
63 R291
82 R405
293 R591

Three-Digit Divisors

If the divisor has three digits, use the method you know for two-digit divisors, but estimate the divisor to the nearest hundred to find the trial quotient. Be very careful to put the first digit in the quotient in the correct place.

Example F

17902 ÷ 381 =

Step 1: Divide

Does 381 go into 1? NO
Does 381 go into 17? NO
Does 381 go into 179? NO
Does 381 go into 1790? YES
Estimate 381 as 400. Think 4.
Estimate 1790 as 1800. Think 18.
4 goes into 18 ≈ 4 times ( $4\times 4 = 16$ )
Your estimate is 4. Write 4 in the quotient above the 0 in the dividend.

$\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \end{array}$

Step 2: Multiply

4 × 381 = 1524

$\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \ \ \ \\ \hline \end{array}$

Step 3: Subtract

1790 − 1524 = 266
Check 266 < 381? YES!

$\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \ \ \ \\ \hline 266 \ \ \ \end{array}$

Step 4: Bring down the 2. 2662 is now the number to be divided

$\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \end{array}$

REPEAT

Step 1: Divide $2662 \div 381 =$

Estimate 381 as 400. Think of 4.
Estimate 2662 as 2700. Think 27.
4 goes into 27 ≈ 6 times ( $4 \times 6 = 24$ ).
Place this estimate in the quotient above the 2

$\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \end{array}$

Step 2: Multiply

$6\times381 = 2286$

$\begin{array}{r}46 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \\ 2286 \\ \hline \end{array}$

Step 3: Subtract

2662 − 2286 = 376
Check 376 < 381? YES!

$\begin{array}{r}46 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \\ 2286 \\ \hline 376 \end{array}$

Step 4: No more digits to bring down.

17902 ÷ 381 = 46 R376

Exercise 8

Divide and check your answers. These questions are hard work! Check your work using the answer key at the end of the exercise.

$115\enclose{longdiv}{8682}$
$205\enclose{longdiv}{2384}$
$325\enclose{longdiv}{66321}$
$241\enclose{longdiv}{13284}$
$860\enclose{longdiv}{262412}$
$659\enclose{longdiv}{270190}$

Answers to Exercise 8

75 R57
11 R129
204 R21
55 R29
305 R112
410

Topic D: Self-Test

Mark /12 Aim 10/12

Divide and check your work for questions b and f using multiplication. (12 marks)
1. $185 \div 10 =$
2. $4\enclose{longdiv}{92}$
3. $408 \div 50 =$
4. $72\enclose{longdiv}{6768}$
5. $67\enclose{longdiv}{5963}$
6. $53\enclose{longdiv}{4856}$
7. $91\enclose{longdiv}{8736}$
8. $265\enclose{longdiv}{133624}$
9. $606\enclose{longdiv}{26094}$
10. $1000\enclose{longdiv}{83652}$

Answers to Topic D Self-Test

Divide and check your work for questions b and f using multiplication.
1. 18 R5
2. 23
3. 8 R8
4. 94
5. 89
6. 91 R33
7. 96
8. 504 R64
9. 43 R36
10. 83 R652

Topic E: Estimating Quotients

Divide Numbers That Both End With Zeros

In Unit 1 you learned a shortcut for multiplying numbers that end with zeros.

Now you will learn a short way to divide numbers that both end with zeros. First do this exercise and notice the pattern in the quotients.

Exercise 1

Divide. Check your work using the answer key at the end of the exercise.

$2 \enclose{longdiv}{6}$
$20 \enclose{longdiv}{60}$
$200 \enclose{longdiv}{600}$
$2000 \enclose{longdiv}{6000}$
$5 \enclose{longdiv}{25}$
$50 \enclose{longdiv}{250}$
$500 \enclose{longdiv}{2500}$
$5000 \enclose{longdiv}{25000}$
$14 \enclose{longdiv}{28}$
$140 \enclose{longdiv}{280}$
$1400 \enclose{longdiv}{2800}$
$14000 \enclose{longdiv}{28000}$

Answers to Exercise 1

Here is the shortcut:

When dividing numbers that both end with zeros, cross off the same number of zeros from the end of the divisor and the dividend. This is sometimes called cancelling zeros.

Example A

$4800 \div 60 = 480\cancel{0} \div 6\cancel{0}$

$\begin{array}{r}80\\ 6\enclose{longdiv}{480}\end{array}$

Example B

$23000 \div 500 = 230\cancel{00} \div 5\cancel{00}$

$\begin{array}{r}46\\ 5\enclose{longdiv}{230} \\ 20 \downarrow \\ \hline 30 \\ 30 \\ \hline 0\end{array}$

Example C

$2\cancel{000}\enclose{longdiv}{680\cancel{000}}$

$\begin{array}{r}340 \\ 2\enclose{longdiv}{680}\\ 6\downarrow \ \\\hline 08 \ \ \ \\ 8 \downarrow \\ \hline 00 \\ 0 \\ \hline 0\end{array}$

If you are interested in the facts of arithmetic that make this shortcut work, ask your instructor for an explanation.

Exercise 2

Quickly find the quotients. Remember to cancel the same number of zeros in both the divisor and dividend in each question. Check your work using the answer key at the end of the exercise.

$30\enclose{longdiv}{90}$
$40\enclose{longdiv}{1600}$
$300\enclose{longdiv}{1200}$
$400\enclose{longdiv}{20000}$
$500\enclose{longdiv}{35000}$
$700\enclose{longdiv}{42000}$
$60000\enclose{longdiv}{2400000}$
$800000\enclose{longdiv}{400000000}$

Answers to Exercise 2

Rounding Division Questions to Estimate

We round numbers and estimate to get a quick answer.

In division, round the divisor and dividend before you divide.

If the divisor only has one digit, do not round it,
Round the dividend to make the arithmetic easier for yourself.

Example D

Look at the two ways of rounding this question.

$1796 \div 32 =$

The divisor (32) will round to 30. This dividend (1796) can be rounded to 1800 or to 2000. Let’s try each:

Round to 1800.

$\begin{array}{r}60\ \ \ \ \\ 3\cancel{0}\enclose{longdiv}{180\cancel{0}}\end{array}$

Round to 2000.

$\begin{array}{l}\hspace{2.5em}66\text{ R}2\\3\cancel{0}\enclose{longdiv}{200\cancel{0}} \\\hspace{2em}18\downarrow \\\hline\hspace{2.5em}20\\\hspace{2.5em}18\\\hline\hspace{1.5cm}2\end{array}$

Rounding 1796 to 1800 is easier arithmetic because $18 \div 3$ works out evenly, so $180 \div 3$ works out evenly. Both estimates are correct.

Example E

2688 ÷ 28 =

Round the divisor (28) to 30.
Round the dividend (2688) to 2700 or to 3000.

Round to 2700

$\begin{array}{r}90\ \ \ \ \\\ 3\cancel{0}\enclose{longdiv}{270\cancel{0}}\end{array}$

Round to 3000

$\begin{array}{r}100\ \ \ \ \ \\ 3\cancel{0}\enclose{longdiv}{300\cancel{0}}\end{array}$

Both estimates are correct and both are easy to do.

Example F

2893 ÷ 47 =

Round the divisor (47) to 50.
Round the dividend (2893) to 2900 or 3000.
Which rounded dividend will be easier to divide by 50?
1. Answer: The 3000 because 5 goes evenly into 30.

$\begin{array}{r}60\ \ \ \ \ \\5\cancel{0} \enclose{longdiv}{300\cancel{0}}\end{array}$

Exercise 3

Give an estimated quotient. Show your rounding. Check your work using the answer key at the end of the exercise.

$365\enclose{longdiv}{27692}$
$23\enclose{longdiv}{34459}$
$45\enclose{longdiv}{4590}$
$16\enclose{longdiv}{6729}$
$56\enclose{longdiv}{4792}$
$75\enclose{longdiv}{7648}$
$81\enclose{longdiv}{4049}$
$68\enclose{longdiv}{5636}$
$19\enclose{longdiv}{1672}$
$200\enclose{longdiv}{20000}$
$557\enclose{longdiv}{41680}$

Answers to Exercise 3

28000 ÷ 400 = 70
34000 ÷ 20 =1700
5000 ÷ 50 =100
7000 ÷ 20 = 350
4800 ÷ 60 = 80
8000 ÷ 80 = 100
4000 ÷ 80 = 50
5600 ÷ 70 = 80
2000 ÷ 20 = 100
20000 ÷ 200 = 100
42000 ÷ 600 = 70

Topic E: Self-Test

Mark /6 Aim 5/6

Give an estimated quotient. Show your work. (6 marks)
1. $98\enclose{longdiv}{8541}$
2. $27\enclose{longdiv}{2963}$
3. $241\enclose{longdiv}{26348}$
4. $55\enclose{longdiv}{3276}$
5. $24\enclose{longdiv}{1776}$
6. $59\enclose{longdiv}{11830}$

Answers to Topic E Self-Test

Give an estimated quotient. Show your work.
1. 8500 ÷ 100 = 85
2. 3000 ÷ 30 = 100
3. 26000 ÷ 200 = 130
4. 3000 ÷ 60 = 50
5. 2000 ÷ 20 = 100
6. 12000 ÷ 60 = 200

Topic F: Division Problems

Review Topic F: Problem Solving in Adult Literacy Fundamentals Mathematics: Book 2.

One common type of division problem gives a total amount for several things and asks you to find what the amount would be for one.

Problems may tell you…	and ask you to find…
kilometres driven in 8 hours (h)	km driven in 1 h
cost for 15 kg (or litres, etc.)	cost for one kg
pay for 40 hours	pay for one hour
rent for one year (12 months)	rent for one month
work done in eight hours	work done in one hour
kilometres driven on 55 L of gas	km driven on 1 L of gas

Problems may tell you…

and ask you to find…

kilometres driven in 8 hours (h)

km driven in 1 h

cost for 15 kg (or litres, etc.)

cost for one kg

pay for 40 hours

pay for one hour

rent for one year (12 months)

rent for one month

work done in eight hours

work done in one hour

kilometres driven on 55 L of gas

km driven on 1 L of gas

The word per is a Latin word meaning “for each”. For example, “Find the kilometres per hour” may be used to mean, “Find the kilometres driven in one hour.” A slash (/ ) also means per e.g. km/h.

“Find the average” is another way of asking you to find the amount for one.

It may be difficult to decide which number is the dividend and which is the divisor. These suggestions should help:

Look at the question in the problem. What do you have to find out? Look for the words “per” and “for one.”
How will the answer be written? That is your clue. If the answer is km/h then the division equation will be total km ÷ h. Study these examples:
- $\text{total of kilometres} \div \text{number of hours} = \text{km/h}$
- $\text{total of kilometres} \div \text{number of litres} = \text{km/L}$
- $\text{total cost} \div \text{unit} = \text{cost per unit}$
- $\text{total pay} \div \text{hours (or days, etc.)} = \text{pay per hour}$
- $\text{total rent}\div \text{number of months} = \text{rent/month}$
- $\text{total things done} \div \text{total time} = \text{number done/unit of time}$
Do a quick estimate.
Look at your estimate and re-read the problem. Does your answer make sense?

To find the average, divide the total amount by the number of items that make up the total. You may first have to add the different items together to find the total.

Average = The sum of all values given ÷ the number of values added up

Example A

You bowled 5 games with scores of 124, 187, 164, 205, 130. What was your average score?

Find the total by adding $124 + 187 + 164 + 205 + 130 = 810$
Divide the total by number of items $810 \div 5 \text{ games} = 162 \text{ per game}$

Example B

Joan and Rick have been keeping track of their household costs. They want to plan a monthly budget. Their grocery bills for six months were $428, $605, $397, $530, $590, and $474. What is their average monthly grocery cost?

Find the total amount.

$428 + $605 + $397 + $530 + $590 + $474 = $3024

Divide total amount by number of items.

$3024 ÷ 6 = $504 average cost per month

Some details to remember:

1 minute = 60 seconds
1 hour = 60 minutes
1 year = 365 days
1 year = 12 months
1 year = 52 weeks

Key words that point to division:

Find the average
Separated
Find the ____ per ____
Split
Find the unit price
Shared

Exercise 1

Solve these problems. Use the five problem solving steps. Be sure to write down an estimate and check that the estimate makes sense before you find the actual solution. Check your work using the answer key at the end of the exercise.

A machine shop can stamp out 360 car parts in an 8-hour working day. How many parts is that per hour?
Izyan paid $560 for 4 tires. How much did each tire cost?
Bjork earned $8840 in 4 months.
1. How much did he earn each month?
2. How much did he earn per week? (4 months is 17 weeks)
Theron used 9 L of gasoline to drive 207 km. How many kilometres did he drive per litre?
The total cost of the car Elena bought is $14880 including taxes and interest. She will pay for it in 24 equal payments. How much will each payment be?
Diego worked 8 hours a day for five days and earned $360. How much was he paid per hour? (This is a 2-step problem — you must first find the total number of hours Diego worked.)
Dae-Hyun and Mi-Ok can afford no more than a total of $14940 per year for rent, electricity, and phone. How much can they pay per month?

Answers to Exercise 1

45 parts per hour
$140 per tire
1. $2210 per month
2. $520 per week
23 km/L
$620 per payment
$9 per hour
$1245 per month

A second type of division problem gives the total amount and the size of each group. You will find the number of groups. You will notice that both numbers have the same units. The answer to the problem will give another unit. This other unit will be asked for in the problem.

Example C

One necklace uses 125 beads. How many necklaces can Susan make for the craft fair if she has 6250 beads?

Find how many groups of 125 there are in 6520.

6250 ÷ 125 =

$\begin{array}{r}50\\125\enclose{longdiv}{6250}\\625 \\ \hline 0 \end{array}$

She can make 50 necklaces.

Example D

If you drive an average speed of 80 km an hour, how many hours will it take you to drive 560 km?

Find how many groups of 80 km there are in 560 km.

$56\cancel{0} \div 8\cancel{0} = 7$

The 560 km trip will take 7 hours.

Exercise 2

Pay attention to wording and situations as you solve these problems. Use the five problem solving steps. Be sure to write down an estimate and check that the estimate makes sense before you find the actual solution. Check your work using the answer key at the end of the exercise.

A train travels 90 km per hour. How many hours will it take the train to go 540 km?
A car gets 16 km per litre of gasoline. How many litres will the car need to go 128 km?
About 8 m is needed for one parking space. How many parking spaces can be made along a street that is 232 m long?
If you spend an average of 8 minutes on one math problem, how many problems can you finish in one hour? Will you have any time left? How much?
The Skating Club members decided to sell home-made candy to raise money. The boxes they bought will hold 45 pieces of candy. If everyone makes a double batch of fudge they will have 2590 pieces of How many boxes can they fill? How many pieces of fudge are left-over for them to eat?
A class of 334 students is going to Victoria by bus. Each bus holds 43 passengers. How many buses do they need? Will there be any empty seats? (Be careful with this one!)

Answers to Exercise 2

6 hours
8 L
29 parking spaces
7 problems, yes, 4 min
57 boxes, 25 pieces left over
8 buses, 10 empty seats

Unit Pricing

A unit price is the price for one of something. To find unit price, divide the total cost by the number of things bought.

Example E

5 shirts cost $60

To find the cost per shirt, $$60 \div 5 =$

$\begin{array}{r}12\\5\enclose{longdiv}{60}\end{array}$

The unit price is $12.

Example F

6 L of oil for $18

To find the cost per L, $$18 \div 6 =$

$\begin{array}{r}3\\6\enclose{longdiv}{18}\end{array}$

The unit price is $3.

Exercise 3

Solve the cost per unit price. Check your work using the answer key at the end of the exercise.

2 CDs for $26
3 cans of dog food for $6
4 air fresheners for $8
2 cat treats for $4
2 pizzas for $22
2 cans of peanuts for $8
2 ice cream for $12
4 boxes of chocolate bars for $48
2 WD-40 for $6
3 paint rollers for $9
4 tie downs for $20
3 boxes of diapers for $51
3 work shirts for $45
8 pairs of socks for $64

Answers to Exercise 3

Best Buy

The best buy is the lowest unit price.

Example G

You can buy 4 L of canola oil for $8 or 10 L of canola oil for $30. Which is the best buy?

$$8\div4=$ can be rewritten and solved: $\begin{array}{r}2\\4\enclose{longdiv}{8}\end{array}$

$$30 \div 10 =$ can be rewritten and solved: $\begin{array}{r}3\\10\enclose{longdiv}{30}\end{array}$

4 L of canola oil for $8 is a better buy since the unit price is $2 per L, while 10 L for $30 has a unit price of $3 per L.

Exercise 4

Solve the unit price and then underline the best buy. Check your work using the answer key at the end of the exercise.

2 L of engine oil for $8 or 5 L of engine oil for $15
4 tires for $240 or 2 tires for $110
6 jars of salad dressing for $24 or 3 jars of salad dressing for $15
7 kg of dog food for $21or 16 kg of dog food for $32
3 DVDs for $54 or 7 DVDs for $119
3 L of laundry soap for $6 or 17 L of laundry soap for $68

Answers to Exercise 4

$4, $3, 5 L for $15 is the best buy.
$60, $55, 2 tires for $110 is the best buy.
$4, $5, 6 salad dressing for $24 is the best buy.
$3, $2, 16 kg for $32 is the best buy.
$18, $17, 7 DVDs for $119 is the best buy.
$2, $4, 3 L for $6 is the best buy.

Topic F: Self Test

Mark /12 Aim 10/12

Solve these problems. (12 marks) 2 marks each – 1 for correct method, 1 for correct solution.
1. Enrique drove the 1 920 km from Dease Lake to Creston in 24 hours. What was his average speed in kilometres per hour?
2. The Evergreen Company employs 26 people. Its total payroll for last month was $84 162. What was the average monthly pay cheque per person?
3. The proud gardener grew a total crop of 135 cucumbers on 15 plants. What was the average crop per plant?
4. In a recent truckload sale, electric stoves were sold for $432. The gross income from the stove sale was $42336. How many stoves were sold?
5. The 39 farmers in Jones Valley had a total income last year of $2905500. What was their average income?
6. A store has an inventory (stock on hand) of chairs with a total value of $1738. Each chair is to be sold at $79. How many of these chairs are there?

Answers to Topic E Self-Test

Give an estimated quotient. Show your work.
1. 80 km/h
2. $3237 per month
3. 9 cucumbers per plant
4. 98 stoves
5. $74500
6. 22 chairs

Topic G: Mixed Problems

Carefully read again the steps in Topic F: Problem Solving in Adult Literacy Fundamentals Mathematics: Book 2, or ask your instructor for a copy of those pages. Read the problems to help you get a feel for the wording and problem situations you can expect for addition, subtraction, multiplication, and division problems. Ask your instructor for a list of key words that will point to the operation you should choose.

Exercise 1

Solve these problems using the five problem solving steps. Show your estimation and actual work. Write a sentence answer remembering to use the units. Check your work using the answer key at the end of the exercise.

Saika gave her students cinnamon hearts on Valentine’s Day. She bought a box of 1120 cinnamon hearts and gave each student 35 candies, using up the whole box. How many students does she have?
Each stamp cost 57 cents. Dolores bought 100 stamps. How much money did she spend on stamps (before taxes)?
Etienne planted 30 tomato plants in rows of 5 plants. How many rows did he plant?
There are 8 servings per large economy-size can of fruit. The restaurant ordered 5 cases with 24 cans per case. How many servings of fruit can the restaurant get from this order? (This is a two step problem – first find the total number of cans. Then find the number of servings.)
A train traveled 2250 km at a speed of 75 km per hour. How many hours did the trip take?
The regular mountain bike costs $499 and the fancy model is $675. How much more do you pay for the fancy mountain bike?
The members of the Shiny Wheels Bicycle Club ride at an average speed of 16 km/h. On their weekend trip they rode 2 hours Friday night, 7 hours on Saturday, spent two hours soaking their aching bones at the Hot Springs, and then rode a final five hours on Sunday. How many kilometres did they ride on this weekend trip? (2 steps)
Last week Mrs. Sanderson drove 29 km on Monday, 42 km on Tuesday, 5 km on Wednesday, and 21 km on Friday. How far did she drive last week?

Answers to Exercise 1

32 students
$57.00
6 rows
960 servings
30 hours
$176 more
224 km
97 km

Unit 2 Review: Division

You will now practice all the skills you learned in Unit 2. Check your work using the answer key at the end of the review.

Complete this chart

Multiplication Division Division “Say”

a) $5\times3=15$
$3\times5=15$ $15\div3=5$
$15\div5=3$ $\begin{array}{r}5\\3\enclose{longdiv}{15} \end{array}$
$\begin{array}{r}3\\5\enclose{longdiv}{15} \end{array}$ “15 divided by 3 is 5.”
“15 divided by 5 is 3.”

b) $3\times6=18$

c) $3\times7=21$

d) $5\times9=45$
Give the answer.
1. $56\div7=$
2. $40\div8=$
3. $54\div9=$
4. $6\enclose{longdiv}{42}$
5. $9\enclose{longdiv}{72}$
6. $8\enclose{longdiv}{32}$
Find the quotients.
1. $7\enclose{longdiv}{68}$
2. $4\enclose{longdiv}{29}$
3. $5\enclose{longdiv}{24}$
4. $6\enclose{longdiv}{53}$
For each number, identify if it is divisible by 2, 3, 5, and, 9.

# Number 2 3 5 9

a) 135

b) 384

c) 4614

d) 495

e) 648

f) 745
Find the quotients.
1. $8\enclose{longdiv}{296}$
2. $6\enclose{longdiv}{252}$
3. $4\enclose{longdiv}{732}$
4. $5\enclose{longdiv}{175}$
Find the quotients.
1. $3\enclose{longdiv}{86}$
2. $4\enclose{longdiv}{97}$
3. $2\enclose{longdiv}{71}$
4. $5\enclose{longdiv}{59}$
Find the quotients.
1. $7\enclose{longdiv}{615}$
2. $2\enclose{longdiv}{647}$
3. $3\enclose{longdiv}{781}$
4. $9\enclose{longdiv}{839}$
Find the quotients.
1. $8956\div42=$
2. $32\enclose{longdiv}{832}$
3. $69140\div56=$
4. $312\enclose{longdiv}{9984}$
5. $41082\div334=$
6. $781\enclose{longdiv}{39752}$
7. $275\enclose{longdiv}{55661}$
8. $307\enclose{longdiv}{91838}$
Find the quotients.
1. $1000\enclose{longdiv}{38645}$
2. $18592\div100=$
3. $4923\div10=$
4. $100\enclose{longdiv}{17342}$
Quickly find the quotients. Remember to cancel the same number of zeros in both the divisor and dividend in each question.
1. $200\enclose{longdiv}{50000}$
2. $6000\enclose{longdiv}{360000}$
3. $40000\enclose{longdiv}{1600000}$
4. $70000\enclose{longdiv}{6300000}$
Give an estimated quotient. Show your rounding.
1. $37\enclose{longdiv}{15725}$
2. $54\enclose{longdiv}{8478}$
3. $768\enclose{longdiv}{63721}$
4. $6267\enclose{longdiv}{536497}$
Word Problems
1. At the Kaizen Factory, 14325 cars were put together in 5 days. Each day the same number of cars were built. How many cars were built each day?
2. The Blaster Rubber Company needs to make 6912 hockey pucks. Mr. Frost, the foreman, says that their machines can make the pucks in 12 hours. How many pucks would be made in one hour?
3. The distance between Fort St. John and Kimberley is 1092 km. What was your average speed if the trip took 12 hours?
4. The new stadium has 15981 seats divided evenly into 76 sections. Estimate how many seats are in each section?
Solve the cost per unit price.
1. 4 rolls of hockey tape for $8
2. 4 cans of butane fuel for $12
Solve the unit price and then underline the best buy.
1. 2 L of antifreeze for $6 or 5 L of antifreeze for $10
2. 8 kilograms of bird seed for $16 or 4 kilograms of bird seed for $12
Word Problems.
1. The bakery uses 43 kilograms of butter in each batch of shortbread cookies. How many batches of shortbread can be made from 3569 kilograms of butter?
2. Each crate that the men unloaded weighed 175 kilograms. If they unloaded 232 crates, how many kilograms did they unload?
3. The parts factory produced 4173 less parts this month than last month. The factory produced 49736 parts this month. How many parts did the factory produce last month?
4. Three Eastjet jets were flown 24826 kilometres, 9423 kilometres and 56015 kilometres. What is the total kilometres the three jets were flown?

	Multiplication	Division	Division	“Say”
a)	$5\times3=15$ $3\times5=15$	$15\div3=5$ $15\div5=3$	$\begin{array}{r}5\\3\enclose{longdiv}{15} \end{array}$ $\begin{array}{r}3\\5\enclose{longdiv}{15} \end{array}$	“15 divided by 3 is 5.” “15 divided by 5 is 3.”
b)	$3\times6=18$
c)	$3\times7=21$
d)	$5\times9=45$

Answers to Unit 2 Review

Complete this chart

#	Multiplication	Division	Division	“Say”
a)	$5\times3=15$ $3\times5=15$	$15\div3=5$ $15\div5=3$	$\begin{array}{r}5\\3\enclose{longdiv}{15} \end{array}$ $\begin{array}{r}3\\5\enclose{longdiv}{15} \end{array}$	15 divided by 3 is 5. 15 divided by 5 is 3.
b)	$3\times6=18$ $6\times3=18$	$18\div6=3$ $18\div3=6$	$\begin{array}{r}3\\6\enclose{longdiv}{18} \end{array}$ $\begin{array}{r}6\\3\enclose{longdiv}{18} \end{array}$	18 divided by 6 is 3. 18 divided by 3 is 6.
c)	$3\times7=21$ $7\times3=21$	$21\div7=3$ $21\div3=7$	$\begin{array}{r}3\\7\enclose{longdiv}{21} \end{array}$ $\begin{array}{r}7\\3\enclose{longdiv}{21} \end{array}$	21 divided by 7 is 3. 21 divided by 3 is 7.
d)	$5\times9=45$ $9\times5=45$	$45\div9=5$ $45\div5=9$	$\begin{array}{r}5\\9\enclose{longdiv}{45} \end{array}$ $\begin{array}{r}9\\5\enclose{longdiv}{45} \end{array}$	45 divided by 9 is 5. 45 divided by 5 is 9.

Multiplication

Division

“Say”

$5\times3=15$
$3\times5=15$

$15\div3=5$
$15\div5=3$

$\begin{array}{r}5\\3\enclose{longdiv}{15} \end{array}$
$\begin{array}{r}3\\5\enclose{longdiv}{15} \end{array}$

15 divided by 3 is 5.
15 divided by 5 is 3.

$3\times6=18$
$6\times3=18$

$18\div6=3$
$18\div3=6$

$\begin{array}{r}3\\6\enclose{longdiv}{18} \end{array}$
$\begin{array}{r}6\\3\enclose{longdiv}{18} \end{array}$

18 divided by 6 is 3.
18 divided by 3 is 6.

$3\times7=21$
$7\times3=21$

$21\div7=3$
$21\div3=7$

$\begin{array}{r}3\\7\enclose{longdiv}{21} \end{array}$
$\begin{array}{r}7\\3\enclose{longdiv}{21} \end{array}$

21 divided by 7 is 3.
21 divided by 3 is 7.

$5\times9=45$
$9\times5=45$

$45\div9=5$
$45\div5=9$

$\begin{array}{r}5\\9\enclose{longdiv}{45} \end{array}$
$\begin{array}{r}9\\5\enclose{longdiv}{45} \end{array}$

45 divided by 9 is 5.
45 divided by 5 is 9.

Give the answer.
1. 8
2. 5
3. 6
4. 7
5. 8
6. 4
Find the quotients.
1. 9 R5
2. 7 R1
3. 4 R4
4. 8 R5
For each number, identify if it is divisible by 2, 3, 5, and, 9.

Number 2 3 5 9

a) 135 No Yes Yes Yes

b) 384 Yes Yes No No

c) 4614 Yes Yes No No

d) 495 No Yes Yes Yes

e) 648 Yes Yes No Yes

f) 745 No No Yes No
Find the quotients.
1. 37
2. 42
3. 183
4. 35
Find the quotients.
1. 28 R2
2. 24 R1
3. 35 R1
4. 11 R4
Find the quotients.
1. 87 R6
2. 323 R1
3. 260 R1
4. 93 R2
Find the quotients.
1. 213 R10
2. 26
3. 1234 R36
4. 32
5. 123
6. 50 R702
7. 202 R111
8. 299 R45
Find the quotients.
1. 38 R645
2. 185 R92
3. 492 R3
4. 173 R42
Quickly find the quotients. Remember to cancel the same number of zeros in both the divisor and dividend in each question.
1. 250
2. 60
3. 40
4. 90
Give an estimated quotient. Show your rounding.
1. $1600 \div 40=400$
2. $8500\div50=170$
3. $64000\div800=80$
4. $540000\div6000=90$
Word Problems
1. 2865 cars per day
2. 576 pucks per hour
3. 91 km per hour
4. $16000\div80=200$ 200 seats per section
Solve the cost per unit price.
1. $2
2. $3
Solve the unit price and then underline the best buy.
1. $3, $2, 5 L of antifreeze for $10 is the best buy
2. $2, $3, 8 kilograms of bird seed for $16 is the best buy
Word Problems.
1. 83 batches
2. 40600 kilograms
3. 53909 parts
4. 90264 kilometres

	Number	2	3	5	9
a)	135	No	Yes	Yes	Yes
b)	384	Yes	Yes	No	No
c)	4614	Yes	Yes	No	No
d)	495	No	Yes	Yes	Yes
e)	648	Yes	Yes	No	Yes
f)	745	No	No	Yes	No

III

Unit 3: Change and The Metric System

Topic A: Counting to Make Change

Practice your counting by filling in the counting chart. Have your instructor check your chart when you are done.

0	1	2	3	4	5	6	7	8	9
10

Use your counting chart and start at 0. Count two and write down that number.

If you had a pile of toonies or two dollar coins and wanted to know how much money you have, you would count by 2’s.

Use your counting chart and start at 0. Count five and write down that number.

If you had a pile of nickels or five dollar bills and wanted to know how much money you have, you would count by 5’s.

Use your counting chart and start at 0. Count ten and write down that number.

If you had a pile of dimes or ten dollar bills and wanted to know how much money you have, you would count by 10’s.

Use your counting chart and start at 0. Count twenty and write down that number.

If you had a pile of twenty dollar bills and wanted to know how much money you have, you would count by 20’s.

Use your counting chart and start at 0. Count twenty-five and write down that number.

If you had a pile of quarters and wanted to know how much money you have, you would count by 25’s.

Exercise 1

Write the missing numerals. Check your work using the answer key at the end of the exercise.

Count by 5’s.

0 10 20 30 40

50 60 70 80 90
Count by 5’s.

0 5 15 25 35 45

55 65 75 85 95
Count by 5’s.

0
Count by 10’s.

0 10 30 50 70 90
Count by 10’s.

0 20 40 60 80 100
Count by 10’s.

0
Count by 20’s

0 40 80
Count by 20’s

0
Count by 25’s.

0 25 75
Count by 25’s.

0 50 100
Count by 25’s.

0

Answers to Exercise 1

Count by 5’s.

0 5 10 15 20 25 30 35 40 45

50 55 60 65 70 75 80 85 90 95
Count by 5’s.

0 5 10 15 20 25 30 35 40 45

50 55 60 65 70 75 80 85 90 95
Count by 5’s.

0 5 10 15 20 25 30 35 40 45

50 55 60 65 70 75 80 85 90 95
Count by 10’s.

0 10 20 30 40 50 60 70 80 90 100
Count by 10’s.

0 10 20 30 40 50 60 70 80 90 100
Count by 10’s.

0 10 20 30 40 50 60 70 80 90 100
Count by 20’s

0 20 40 60 80 100
Count by 20’s

0 20 40 60 80 100
Count by 25’s.

0 25 50 75 100
Count by 25’s.

0 25 50 75 100
Count by 25’s.

0 25 50 75 100

Topic B: Making Change

Making change is a vital everyday skill that helps you give exact cash quickly and accurately. In this chapter, you will learn how to round amounts to the nearest 0 or 5 and then use the fewest bills or coins to complete the transaction. Mastering these steps will boost your confidence and speed whenever you handle cash.

When you make change, your first goal is to get a number that ends in 0 or 5. For example, if you bought something for $4, the first thing to do would be to get to $5 (check out example A below).

Example A

$4 to $5

To get from $4 to $5, you could use 1 loonie.

Example B

$23 to $25

To get from $23 to $25, you could use 1 toonie.

Example C

$55 to $60

To get from $55 to $60, you could use 1 five-dollar bill.

Exercise 1

Circle the number of coins or bills you would need to get from the first number to the second number. Make sure to use the least number of coins or bills. Check your work using the answer key at the end of the exercise.

$33 to $35
$48 to $50
$16 to $20
$68 to $70
$10 to $15
$35 to $40
$55 to $60
$85 to $90
$60 to $70
$90 to $100
$30 to $40
$40 to $50
$30 to $50
$70 to $80
$80 to $100
$45 to $50
$21 to $25
$55 to $60
$45 to $50
$40 to $50
$70 to $80

Answers to Exercise 1

1 toonie
1 toonie
2 toonies
1 toonie
1 $5 bill
1 $5 bill
1 $5 bill
1 $5 bill
1 $10 bill
1 $10 bill
1 $10 bill
1 $10 bill
1 $20 bill
1 $10 bill
1 $20 bill
1 $5 bill
2 toonies
1 $5 bill
1 $5 bill
1 $10 bill
1 $10 bil

Exercise 2

State the number and kind of coins and bills you would need to get from the first number to the second number. Make sure you use the least number of coins and bills as possible. Check your work using the answer key at the end of the exercise.

Example: $56 to $60. You require 2 toonies to get to $60.

$28 to $30
$35 to $40
$90 to $100
$32 to $50
$54 to $55
$30 to $50
$65 to $70
$45 to $50

Answers to Exercise 2

1 toonie
1 $5 bill
1 $10 bill
1 $10 bill, 1 $5 bill, 1 toonie, and 1 loonie
1 loonie
1 $20 bill
1 $5 bill
1 $5 bill

Example D

$28 to $50
Need	To get to
1 toonie	$30
1 $20 bill	$50

Example E

$36 to $50
Need	To get to
2 toonies	$40
1 $10 bill	$50

Example F

$63 to $80
Need	To get to
1 toonie	$65
1 $5 bill	$70
1 $10 bill	$80

Exercise 3

Example:

$67 to $80
Need	To get to
1 loonie	$68
1 toonie	$70
1 $10 bill	$80

$26 to $40
Need To get to
$47 to $60
Need To get to
$69 to $80
Need To get to
$18 to $20
Need To get to
$36 to $50
Need To get to
$51 to $60
Need To get to
$78 to $100
Need To get to
$82 to $100
Need To get to
$93 to $100
Need To get to
$3 to $10
Need To get to

Answers to Exercise 3

$26 to $40
Need To get to

2 toonies $30

1 $10 bill $40
$47 to $60
Need To get to

1 loonie $48

1 toonie $50

1 $10 $60
$69 to $80
Need To get to

1 loonie $70

1 $10 bill $80
$18 to $20
Need To get to

1 toonie $20
$36 to $50
Need To get to

2 toonies $40

1 $10 bill $50
$51 to $60
Need To get to

2 toonies $55

1 $5 bill $60
$78 to $100
Need To get to

1 toonie $80

1 $20 bill $100
$82 to $100
Need To get to

1 loonie $83

1 toonie $85

1 $5 bill $90

1 $10 bill $100
$93 to $100
Need To get to

1 toonie $95

1 $5 bill $100
$3 to $10
Need To get to

1 toonie $5

1 $5 bill $10

Exercise 4

State the number and kind of coins and bills you would need to get change from $100. Make sure you use the least number of coins and bills as possible. Check your work using the answer key at the end of the exercise.

Example: $67 to $100

Need	To get to
1 loonie	$68
1 toonie	$70
1 $10 bill	$80
1 $20 bill	$100

$26
Need To get to
$13
Need To get to
$57
Need To get to
$49
Need To get to
$38
Need To get to
$74
Need To get to

Answers to Exercise 4

$26
Need To get to

2 toonies $30

1 $10 bill $40

3 $20 bills $100
$13
Need To get to

1 toonie $15

1 $5 bill $20

4 $20 bills $100
$57
Need To get to

1 loonie $58

1 toonie $60

2 $20 bills $100
$49
Need To get to

1 loonie $50

1 $10 bill $60

2 $20 bills $100
$38
Need To get to

1 toonie $40

3 $20 $100
$74
Need To get to

3 toonies $80

1 $20 bill $100

Exercise 5

Example A: $2.70 to $5.00

Need	To get to
1 nickel	$2.75
1 quarter	$3.00
1 toonie	$5.00

Example B: $6.30 to $10.00

Need	To get to
2 dimes	$6.50
2 quarters	$7.00
1 loonie	$8.00
1 toonie	$10.00

Example C: $12.50 to $20.00

Need	To get to
2 quarters	$13.00
1 toonie	$15.00
1 $5 bill	$20.00

$2.20 to $10.00
Need To get to
$6.50 to $20.00
Need To get to
$8.70 to $20.00
Need To get to
$3.35 to $10.00
Need To get to
$17.85 to $20.00
Need To get to
$50.25 to $60.00
Need To get to
$20.55 to $40.00
Need To get to
$37.75 to $50.00
Need To get to
$19.90 to $50.00
Need To get to
$4.40 to $5.00
Need To get to
$44.55 to $60.00
Need To get to
$29.15 to $40.00
Need To get to
$65.80 to $80.00
Need To get to
$42.00 to $60.00
Need To get to
$97.70 to $100.00
Need To get to
$32.05 to $35.00
Need To get to
$58.30 to $100.00
Need To get to
$61.15 to $80.00
Need To get to
$72.85 to $100.00
Need To get to
$83.95 to $100.00
Need To get to
$5.25 to $20.00
Need To get to
$19.60 to $40.00
Need To get to

Answers to Exercise 5

$2.20 to $10.00
Need To get to

1 nickel $2.25

3 quarters $3

1 toonie $5

1 $5 bill $10
$6.50 to $20.00
Need To get to

2 quarters $7

1 loonie $8

1 toonie $10

1 $10 bill $20
$8.70 to $20.00
Need To get to

1 nickel $8.75

1 quarter $9

1 loonie $10

1 $10 bill $20
$3.35 to $10.00
Need To get to

1 nickel $3.40

1 dime $3.50

2 quarters $4

1 loonie $5

1 $5 bill $10
$17.85 to $20.00
Need To get to

1 nickel $17.90

1 dime $18

1 toonie $20
$50.25 to $60.00
Need To get to

3 quarters $51

2 toonies $55

1 $5 bill $60
$20.55 to $40.00
Need To get to

2 dimes $20.75

1 quarter $21

2 toonies $25

1 $5 bill $30

1 $10 bill $40
$37.75 to $50.00
Need To get to

1 quarter $38

1 toonie $40

1 $10 bill $50
$19.90 to $50.00
Need To get to

1 dime $20

1 $10 bill $30

1 $20 bill $50
$4.40 to $5.00
Need To get to

1 dime $4.50

2 quarters $5
$44.55 to $60.00
Need To get to

2 dimes $44.75

1 quarter $45

1 $5 bill $50

1 $10 bill $60
$29.15 to $40.00
Need To get to

1 dime $29.25

3 quarters $30

1 $10 bill $40
$65.80 to $80.00
Need To get to

2 dimes $66

2 toonies $70

1 $10 bill $80
$42.00 to $60.00
Need To get to

1 loonie $43

1 toonie $45

1 $5 bill $50

1 $10 bill $60
$97.70 to $100.00
Need To get to

1 nickel $97.75

1 quarter $98

1 toonie $100
$32.05 to $35.00
Need To get to

2 dimes $32.25

3 quarters $33

1 toonie $35
$58.30 to $100.00
Need To get to

2 dimes $58.50

2 quarters $59

1 loonie $60

2 $20 bills $100
$61.15 to $80.00
Need To get to

1 dime $61.25

3 quarters $62

1 loonie $63

1 toonie $65

1 $5 bill $70

1 $10 bill $80
$72.85 to $100.00
Need To get to

1 nickel $72.90

1 dime $73

1 toonie $75

1 $5 bill $80

1 $20 bill $100
$83.95 to $100.00
Need To get to

1 nickel $84

1 loonie $85

1 $5 bill $90

1 $10 bill $100
$5.25 to $20.00
Need To get to

3 quarters $6

2 toonies $10

1 $10 bill $20
$19.60 to $40.00
Need To get to

1 nickel $19.65

1 dime $19.75

1 quarter $20

1 $20 bill $40

Exercise 6

$1.75 to $5.00
$54.05 to $60.00
$96.45 to $100.00
$28.20 to $30.00
$73.95 to $80.00
$32.85 to $50.00
$9.40 to $20.00
$85.25 to $100.00

Answers to Exercise 6

$1.75 to $5.00
- 1 quarter
- 1 loonie
- 1 toonie
$54.05 to $60.00
- 2 dimes
- 3 quarters
- 1 $5 bill
$96.45 to $100.00
- 1 nickel
- 2 quarters
- 1 loonie
- 1 toonie
$28.20 to $30.00
- 1 nickel
- 3 quarters
- 1 loonie
$73.95 to $80.00
- 1 nickel
- 1 loonie
- 1 $5 bill
$32.85 to $50.00
- 1 nickel
- 1 dime
- 1 toonie
- 1 $5 bill
- 1 $10 bill
$9.40 to $20.00
- 1 dime
- 2 quarters
- 1 $10 bill
$85.25 to $100.00
- 3 quarters
- 2 toonies
- 1 $10 bill

Exercise 7

State the number and kind of coins and bills you would need to get change after making each purchase with a $100 bill. Make sure you use the least number of coins and bills as possible. Check your work using the answer key at the end of the exercise.

1 kg of chicken for $24.60
DVD Player for $25.75
Metal shelf for $75.60
Heavy duty drill costs $89.90
Skates cost $67.50
Floor lamp costs $73.95
coffee maker costs $22.40
a package of toilet paper costs $6.75
a box of laundry detergent costs $13.45
a can of baby formula costs $30.25

Answers to Exercise 7

1 nickel, 1 dime, 1 quarter, 1 $5 bill, 1 $10 bill, 3 $20 bills
1 quarter, 2 toonies, 1 $10 bill, 3 $20 bills
1 nickel, 1 dime, 1 quarter, 2 toonies, 1 $20 bill
1 dime, 1 $10 bill
2 quarters, 1 toonie, 1 $10 bill, 1 $20 bill
1 nickel, 1 loonie, 1 $5 bill, 1 $20 bill
1 dime, 2 quarters, 1 toonie, 1 $5 bill, 1 $10 bill, 3 $20 bills
1 quarter, 1 loonie, 1 toonie, 1 $10 bill, 4 $20 bills
1 nickel, 2 quarters, 1 loonie, 1 $5 bill, 4 $20 bills
3 quarters, 2 toonies, 1 $5 bill, 3 $20 bills

Exercise 8

Mrs. Bhabra bought a food processor that cost $56.00. What change will she get from $100?
Pablo bought Lego for $30.25. What change will she get from $100?
A case of 6 bottles of motor oil costs $43.70. What change will you get from?
Shampoo costs $3.95 and conditioner costs $4.95. Together with taxes, they cost $9.35. What change will you get from $100?

Answers to Exercise 8

2 toonies, 2 $20 bills
3 quarters, 2 toonies, 1 $5 bill, 3 $20 bills
1 nickel, 1 quarter, 1 loonie, 1 $5 bill, 1 $10 bill, 2 $20 bills
1 nickel, 1 dime, 2 quarters, 1 $10 bill, 4 $20 bills

Topic B: Self-Test

Mark /21 Aim 17/21

Circle the number of coins and bills needed to get from the first number to the second number. Use the least number of coins. (4 marks)
1. $76 to $80
2. $22 to $25
3. $40 to $50
4. $55 to $60
State the number and kind of coins or bills are needed to get from the first number to the second number. (4 marks)
1. $48 to $50
2. $76 to $80
3. $95 to $100
4. $40 to $50
State the number and kind of coins and bills you would need to get from the first number to the second number. Make sure you use the least number of coins and bills as possible. (4 marks)
1. $37 to $50
2. $16 to $50
3. $42 to $60
4. $81 to $100
State the number and kind of coins and bills you would need to get change from $100. Make sure you use the least number of coins and bills as possible. (9 marks)
1. $23
2. $41
3. $68
4. $72
5. groceries cost $89.65
6. telephone bill was $51.10
7. gas for the car cost $61.60
8. Kristina bought a new vacuum cleaner for $86.35. What change will she get from $100?
9. Uchida bought a new frying pan for $39.20. How much change will she get back from $100?

Answers to Topic B Self-Test

1. 2 toonies
2. 1 loonie, 1 toonie
3. 1 $10 bill
4. 1 $5 bill
1. 1 toonie
2. 2 toonies
3. 1 $5 bill
4. 1 $10 bill
1. 1 loonie, 1 toonie, 1 $10 bill
2. 2 toonies, 1 $10 bill, 1 $20 bill
3. 1 loonie, 1 toonie, 1 $5 bill, 1 $10 bill
4. 2 toonies, 1 $5 bill, 1 $10 bill
1. 1 toonie, 1 $5 bill, 1 $10 bill, 3 $20 bill
2. 2 toonies, 1 $5 bill, 1 $10 bill, 2 $20 bills
3. 1 toonie, 1 $10 bill, 1 $20 bills
4. 1 loonie, 1 toonie, 1 $5 bill, 1 $20 bill
5. 1 dime, 1 quarter, 1 $10 bill
6. 1 nickel, 1 dime, 3 quarters, 1 loonie, 1 toonie, 1 $5 bill, 2 $20 bills
7. 1 nickel, 1 dime, 1 quarter, 1 loonie, 1 toonie, 1 $5 bill, 1 $10 bill, 1 $20 bill
8. 1 nickel, 1 dime, 2 quarters, 1 loonie, 1 toonie, 1 $10 bill
9. 1 nickel, 3 quarters, 3 $20 bills

Topic C: The Metric System

When you measure something, you are comparing. Measurement is comparing something with a standard unit of measure.

In the past, units of measure were based on things found in a community. Often, lengths of parts of the body were used to measure:

Inch: the width of a thumb
Span: the distance from tip of thumb to tip of little finger when the hand is spread out
Foot: the distance from big toe to heel of foot
Yard (becomes “rod): the distance from tip of nose to tip of thumb of an outstretched hand and arm
Cubit: the distance from the tip of the middle finger to the elbow
Fathom: the distance from the tip of one hand to the tip of the other with both arms stretched out
Mile: 1000 military double steps in the Roman army (mile passuum means “1000 paces”)

The problem with these units was the distances would be different based on the size of the person doing the measuring. The problem grew even more when trade was started between cities.

Over a long period of time, people in different countries came up with different standard units of measure.

The British came up with imperial units such as inch, foot, yard and mile.

The French came up with metric units such as metre, centimetre, litre and gram. In Canada, we use both metric and imperial units.

Measuring Length

The basic unit of measure for length is the metre (abbreviation m). If you stretch your arm straight out beside you, a metre is about the distance from the tip of your nose to the tip of your middle finger.

Exercise 1

Consider the following distances. Answer yes or no if you would use a metre to measure. Check your work using the answer key at the end of the exercise.

Example:

The length of a table – Yes
The width of your watchband – No

length of your classroom
thickness of a piece of paper
length of your pen or pencil
height of the door
distance from Prince George to Fort Nelson
the width of your baby finger
the size of a postage stamp
the thickness of a quarter
the length of the hallway outside your classroom
the distance from your home to college
the height of your coffee mug
the width of your book
the distance from your home to Moncton, New Brunswick
the length of a city bus

Answers to Exercise 1

Exercise 2

Estimate the length of each item. Remember a metre is about the distance from the tip of your nose to the tip of your middle finger when your arm is stretched out beside you. Be sure to include the unit of measure in your answer. Have your instructor check your work.

the height of the doorway
the height of your table
the width of the doorway
the length of your arm
the distance from the floor to hip
the distance from the floor to waist
the height of the ceiling
the width of the window
the length of your classroom
the width of your classroom

Exercise 3

Circle the most reasonable measure. Remember a metre is about the distance from the tip of your nose to the tip of your middle finger when your arm is stretched out beside you. Check your work using the answer key at the end of the exercise.

A person’s height
1. 2m
2. 20m
3. 200m
The height of a child
1. 10 m
2. 100 m
3. 1 m
The length of a house
1. 150 m
2. 15 m
3. 1 m
The length of a bed
1. 200 m
2. 2 m
3. 20 m
The height of a building
1. 1 m
2. 100 m
3. 10 m
The height of a refrigerator
1. 20 m
2. 200 m
3. 2 m

Answers to Exercise 3

Measuring Small Lengths and Long Distances

It is hard to measure small things using a metre. To measure small things, you can use centimetre (cm) or millimetre (mm).

A centimetre (cm) is about the width of your baby finger. Remember it is just a guide. A millimetre (mm) is about the thickness of your fingernail.

It is hard to measure long distances using a metre. To measure long distances, you can use kilometre (km). A kilometre is 1000 metres.

Exercise 4

For each item, circle the unit of measure you would use. Check your work using the answer key at the end of the exercise

Item

Unit of Measure

the width of a room

m, cm, mm

the thickness of a coin

m, cm, mm

the length of your pencil

m, cm, mm

the length of the hall

m, cm, mm

the length of this page

m, cm, mm

the length of a screw

m, cm, mm

the height of your cup

m, cm, mm

the length of a bus

m, cm, mm

the thickness of a window pane

m, cm, mm

the width of a chair

m, cm, mm

Answers to Exercise 4

Exercise 5

For each item, circle the letter of the most reasonable unit of measure. Check your work using the answer key at the end of the exercise.

The width of a doorway
1. 50 mm
2. 1 m
3. 50 m
The length of your pencil
1. 20 m
2. 20 mm
3. 20 cm
The height of a tall building
1. 1 m
2. 100 m
3. 10 m
The height of a refrigerator
1. 20 m
2. 200 m
3. 2 m
The diameter of a quarter
1. 24 mm
2. 24 cm
3. 24 m
The height of the kitchen counter
1. 9 m
2. 9 cm
3. 90 cm
The distance around your wrist
1. 15 mm
2. 15 cm
3. 15 m
The width of a small TV screen
1. 28 mm
2. 128 cm
3. 28 m
The length of a car
1. 5 m
2. 5 cm
3. 5 mm
The height of a bookcase
1. 2 cm
2. 2 mm
3. 2 m

Answers to Exercise 5

Exercise 6

Fill in the blank with the most reasonable unit of measure. Check your work using the answer key at the end of the exercise

Most hand held calculators are about 15 long.
The CN Tower in Toronto is 555 tall.
Many young men have an 80 waist.
Computer monitor screens are 28 wide.
The handle of a hammer is 20 long.
A table is about 65 long.
The seat of a chair is about 30 above the floor.
The window is about 3 long.
A roll of tape is about 13 wide.
A rope is about 7 thick.

Answers to Exercise 6

Measuring Capacity (Volume)

Volume is a measure of how much space something takes up. The basic unit of measure for volume is the Litre (L).

Can you think of two things that we buy in litres? We buy gasoline and milk in litres.

We use millilitres (mL) to measure small volumes. For example, a small cup of coffee is about 180 mL. Can you think of two things we buy in mL?

Exercise 7

Write the unit of measure you would use to measure each item (L or mL). Check your work using the answer key at the end of the exercise.

Bottle of pop
Gasoline
Car window wash
Can of beans
Large bottle of juice
Liquid dish soap
Cough syrup
Mixing bowls
Ketchup
Shampoo
Vinegar
Bathtub
Ice cube tray
Paint

Answers to Exercise 7

mL
L
L
mL
L
mL or L
mL
L
mL or L
mL
L
L
mL
L

Exercise 8

Circle the letter of the most reasonable unit of measure. Check your work using the answer key at the end of the exercise.

A can of soup
1. 3 L
2. 30 mL
3. 300 mL
A large container of ice cream
1. 5 L
2. 500 L
3. 50L
A hot water heater
1. 200 mL
2. 50 L
3. 200 L
A cup of tea
1. 18 mL
2. 180 mL
3. 218 L
A garbage can
1. 120 L
2. 120 mL
3. 12 L
A saucepan
1. 2 L
2. 20 L
3. 2 mL
A dose of cough syrup
1. 40 L
2. 4 mL
3. 40 mL
The gas tank of a car
1. 500 mL
2. 5 L
3. 50 L
A jar of mustard
1. 150 mL
2. 15 L
3. 15 mL
A large mixing bowl
1. 6 mL
2. 60 L
3. 6 L

Answers to Exercise 8

Measuring Mass

Mass is a measure of matter in something. The basic unit of measure for mass is the gram (g). We buy sliced meats, bulk food, spices and cereal in grams.

We use kilogram (kg) to measure the mass of large things such as cars, people, flour and sugar. Can you think of two things that we buy in kilograms?

We use milligram (mg) to measure the mass of small things such as medicine or vitamins. A milligram is very small. Can you think of two things that we buy in milligrams?

Exercise 9

Circle the unit of measure you would use to measure each item. Check your work using the answer key at the end of the exercise

Item

Unit of Measure

pasta

mg, g, kg

vitamin C

mg, g, kg

peanut butter

mg, g, kg

flour

mg, g, kg

nails

mg, g, kg

dry dog food

mg, g, kg

rice

mg, g, kg

apples

mg, g, kg

spices

mg, g, kg

cement

mg, g, kg

medicine tablets

mg, g, kg

the family pet

mg, g, kg

a coin

mg, g, kg

Answers to Exercise 9

g or kg
mg
g or kg
kg
kg
kg
kg
g
kg
g
kg
mg
kg
g

Exercise 10

Circle the letter of the most reasonable unit of measure. Check your work using the answer key at the end of the exercise.

A nickel
1. 5 kg
2. 5 g
3. 50 g
A small television
1. 8 g
2. 8 kg
3. 80 g
A flashlight battery
1. 8 g
2. 8 kg
3. 80 g
A baby
1. 30 kg
2. 3 kg
3. 300 kg
A dinner fork
1. 50 g
2. 5 g
3. 5 kg
A slice of bread
1. 2 g
2. 20 g
3. 2 kg
A sugar cube
1. 2 mg
2. 20 g
3. 2 g
A refrigerator
1. 120 g
2. 120 kg
3. 12 kg
A bag of potatoes
1. 5 g
2. 5 kg
3. 50 mg
A car
1. 100 kg
2. 1000 kg
3. 10 kg
A chocolate bar
1. 300 mg
2. 300 mg
3. 300 g
A back pack
1. 12 kg
2. 12 g
3. 12 mg

Answers to Exercise 10

Metric Prefixes

In the metric system a prefix is used to tell if something is large or small. A prefix is a part of a word that is added to the start of word to change the meaning.

The base units of measure in the metric system are the metre (m), litre (L) and gram (g).

If the prefix kilo is added to one of the base units of measures, such as kilometre or kilogram, you know that these are large amounts.

A kilometre is 1000 metres.

A kilogram is 1000 grams.

If the prefix centi or milli is added to one of the base units of measure, such as centimetre or milligram, you know that these are small amounts.

It takes 100 centimetres to make a metre. It takes 1000 milligrams to make a gram.

Measures

Large

Base

Small

length

kilometre (km)

metre (m)

centimetre (cm) millimetre (mm)

volume

litre (L)

millilitre (mL)

mass

kilogram (kg)

gram (g)

milligram (mg)

Exercise 11

Write large, small or base on the line. Check your work using the answer key at the end of the exercise.

kilometre .
millimetre .
metre .
gram .
litre .
millimetre .
milligram .
kilogram .
centimetre .

Answers to Exercise 11

large
small
base
base
base
small
small
large
small

Exercise 12

Write the base unit of measure and then the prefix if one is needed. Check your work using the answer key at the end of the exercise.

Item

Base

Prefix (if needed)

Height of a tree

A bottle of vanilla

A cold tablet

Distance between Vancouver and Toronto

Thickness of a piece of paper

Length of your foot

Length of a piece of lumber

A bottle of hand lotion

A granola bar

Diameter of a DVD

Mass of a book

Water in a hot tub

Distance around the Earth

Gap in a spark plug

Answers to Exercise 12

Item

Base

Prefix (if needed)

Height of a tree

A bottle of vanilla

milli

A cold tablet

milli

Distance between Vancouver and Toronto

kilo

Thickness of a piece of paper

milli

Length of your foot

centi

Length of a piece of lumber

A bottle of hand lotion

milli

A granola bar

Diameter of a DVD

centi

Mass of a book

Water in a hot tub

Distance around the Earth

kilo

Gap in a spark plug

milli

Exercise 13

Write the unit of measure you would use for each item below. Check your work using the answer key at the end of the exercise.

coffee in a cup – millilitres (mL)
bag of potatoes
gas for a car
length of the hall
vitamin C tablet
thickness of glass
width of a page
box of cereal
distance from Vancouver to Halifax
height of a child
can of soup
window wash for the car
dose of heart medicine
length of a machine bolt
cheese

Answers to Exercise 13

coffee in a cup – millilitres (mL)
bag of potatoes – kilogram (kg)
gas for a car – litre (L)
length of the hall – metre (m)
vitamin C tablet – milligram (mg)
thickness of glass – millimetre (mm)
width of a page – centimetre (cm)
box of cereal – gram (g)
distance from Vancouver to Halifax – kilometre (km)
height of a child – metre (m)
can of soup – millilitre (mL)
window wash for the car – litre (L)
dose of heart medicine – milligram (mg)
length of a machine bolt – millimetre (mm)
cheese – kilogram (kg) or gram (g)

Area of Rectangles and Squares

Area is the amount of surface within a figure. Area is described using square units.

For example,

If the figure is a room, the area is the floor surface.
If the figure is a tabletop, the area is the top surface of the table. If the figure is a roof, the area would be the shingled surface.
If the figure is property, the area is the ground within the property lines.
If the figure is this page, the area is the entire flat page that you are reading.

Example A

Find the area of this rectangle.

length 10cm, width 4cm

The measurements of this rectangle are given in centimetres. To measure the area, we will use squares which are 1 cm by 1 cm ( a square centimetre). How many square centimetres will fit on the surface of this rectangle? Count the 1 cm squares drawn within the Example A rectangle.

The area of this rectangle is square centimetres.

This is written as cm²

Square centimetres is usually written cm² which is said as “square centimetres” or “centimetres squared”. The small number is called an exponent and if the exponent is 2, it means multiplied by itself, so cm² means $\text{cm}\times \text{cm}$ .

Square kilometres are written km². Square metres are written m².

Example B

This is a diagram of a large cattle ranch. The measurements of the ranch are shown as kilometres. What is the area of this ranch?

length 16 km, width 10 km

The square units to measure will be square kilometres. The area of this ranch will be the number of units 1 km by 1 km (km²). The diagram shows how many square kilometres would fit in an area 16 km by 10 km. Count them. The area of the Lone Bar Ranch is km².

Did you find a quick method for counting the square units? Did you multiply the number of rows by the number of square kilometres in each row? The quick method of finding the area of a rectangle is to multiply the length by the width.

Use this formula to find the area of a rectangle:

$\text{Area of a rectangle} = \text{length (}l\text{)} \times \text{width (}w\text{)}$

Length times width can also be expressed as lw,

so $\text{A}_{\text{rectangle}}= lw$

The answer must be expressed in square units.

Example C

Give the area of a soccer field that is 100 m by 45 m.

A_rectangle = lw

$\text{Area of the soccer field} = 100 \text{ m} \times 45 \text{ m} = 4500 \text{ m}^2$

Exercise 14

Find the area of the rectangles described below. The measures of the length (l) and the width (w) have been given. Draw and label a picture for each. Be sure to write the unit of measure for each answer. Check your work using the answer key at the end of the exercise.

$l = 10 \text{ cm}$ , $w = 6 \text{ cm}$
$l = 100 \text{ km}$ , $w = 70 \text{ km}$
$l = 400 \text{ km}$ , $w = 100 \text{ km}$
$l = 975 \text{ cm}$ , $w = 35 \text{ cm}$
$l = 196 \text{ cm}$ , $w = 28 \text{ cm}$
$l = 82 \text{ km}$ , $w = 12 \text{ km}$
$l = 60 \text{ cm}$ , $w = 250 \text{ cm}$
$l = 90 \text{ cm}$ , $w = 2 \text{ cm}$

Answers to Exercise 14

$60 \text{ cm}^2$
$7000 \text{ km}^2$
$40000 \text{ km}^2$
$34125 \text{ cm}^2$
$5488 \text{ cm}^2$
$984 \text{ km}^2$
$1500 \text{ cm}^2$
$180 \text{ cm}^2$

Squares are rectangles with all four sides congruent (the same length). So to find the area of a square you still use the same formula of multiplying the length times the width. But since the length and the width of a square are the same, you are multiplying the measure of the side (s) by itself. The formula for finding the area of a square is often written using an exponent.

A_square = s²

For example, to find the area of a square piece of property, multiply the length of one side by itself. If the measure of one side of a property is 75 m,

Area of this piece of property: $75^2 = 75\text{ m} \times75\text{ m} = 5625\text{ m}^2$

Exercise 15

Find the area of the squares. The measure of the side has been given. Draw and label a picture for each. Be sure to write the unit of measure for each answer. Check your work using the answer key at the end of the exercise.

A_square, if $s = 5 \text{ cm}$
A_square, if $s = 125 \text{ km}$
A_square, if $s = 45 \text{ mm}$
A_square, if $s = 100 \text{ m}$
A_square, if $s = 14 \text{ km}$
A_square, if $s = 25 \text{ cm}$

Answers to Exercise 15

$25 \text{ cm}^2$
$15625 \text{ km}^2$
$2025 \text{ mm}^2$
$10000 \text{ m}^2$
$196 \text{ km}^2$
$625 \text{ cm}^2$

Problems Using Area

Exercise 16

Do these problems by following the five problem-solving steps. Be sure to draw a picture. Check your work using the answer key at the end of the exercise.

A garden in the shape of a rectangle is 34 m long and 20 m wide. What is the area of the garden?
Mr. Pavelski has a lot in the shape of a square. If the side of his square lot is 50 m, what is the area of the lot?
A rectangular board is 58 cm wide and 64 cm long. How much area will this board cover?
Kyoko bought some curtain material that is 198 cm long and 40 cm wide. How many square centimetres of material did she buy?
The distance between bases of a baseball diamond (a square) is 27 m. What is the area of the baseball diamond?
The janitor waxed a floor that was 24 m long and 18 m wide. How many square metres of floor did he wax?

Answers to Exercise 16

$680 \text{ m}^2$
$2500 \text{ m}^2$
$3712 \text{ cm}^2$
$7920 \text{ cm}^2$
$729 \text{ m}^2$
$432 \text{ m}^2$

Perimeter and Area of Rectangles and Squares

Rectangle

Perimeter means distance around. To find the perimeter of a rectangle, use the formula

$P = 2 \times \text{length} + 2 \times \text{width}$ .

Example D

$\begin{array}{l}P = 2 \times \text{length} + 2 \times \text{width} \\\hspace{1em} = 2\times 4 \text{ cm}+ 2 \times 2 \text{ cm}\\ \hspace{1em}= 8 \text{ cm}+4 \text{ cm}\\ \hspace{1em}=12 \text{ cm}\end{array}$

Remember: Multiply first and then add.

Area means the amount of surface within a shape. To find the area of a rectangle, use the formula

$A = \text{length} \times \text{width}$

Example E

$\begin{array}{l}A = \text{length} \times \text{width} \\\hspace{1em} = 4 \text{ cm}\times 2 \text{ cm}\\\hspace{1em} = 8 \text{ cm}^2 \end{array}$

Exercise 17

For each rectangle, find both the perimeter (distance around) and the area (amount of surface within). Check your work using the answer key at the end of the exercise.

Raoul’s lot is shaped like a rectangle. If it is 55 m long and 15 m wide, what is the perimeter and area of his lot?
The playground is shaped like a rectangle. Its length is 140 m and its width is 60 What is the perimeter and area of the playground?

Answers to Exercise 17

P = 16 m, A = 12 m²
P = 38 cm, A= 84 cm²
P = 52 km, A = 153 km²
P = 40 mm, A 96 mm²
P = 94 m, A = 480 m²
P – 166 m, A = 1170 m²
P = 140 m, A = 825 m²
P = 400 m, A = 8400 m²

Square

Perimeter means distance around. To find the perimeter of a square, use the formula

P = 4 × side or P_square = 4s

Example F

square. length and width 9 cm

$\begin{array}{l}P = 4\times \text{side} \\\hspace{1em} = 4 \times 9 \text{ cm}\\\hspace{1em} = 36 \text{ cm}\end{array}$

Area means the amount of surface within a shape. To find the area of a square, use the formula

A = side² or A_square = s²

Example G

square. length and width 4 cm

$\begin{array}{l}A = \text{side}^2 \\\hspace{1em} = (4 \text{ cm})^2\\\hspace{1em} = 16 \text{ cm}^2\end{array}$

Exercise 18

For each square, find both the perimeter (distance around) and the area (amount of surface within). Check your work using the answer key at the end of the exercise.

Luisa bought a 115 cm square tablecloth. What is the perimeter and area of the tablecloth?
Liang bought a square tarp whose side was 31 m. What is the perimeter and area of the tarp?

Answers to Exercise 18

P = 20 m, A = 25 m²
P = 36 cm, A= 81 cm²
P = 60 cm, A = 255 cm²
P = 152 cm, A = 1444 cm²
P = 168 cm, A = 1764 cm²
P – 288 mm, A = 3249 mm²
P = 460 cm, A = 13225 cm²
P = 124 m, A = 961 m²

Topic C: Self Test

Mark /38 Aim 30/38

For each item, circle the unit of measure you would use. (8 marks)

Item

Unit of Measure

Height of a doorknob above the floor

m, cm, mm, km

Thickness of a piece of wire

m, cm, mm, km

Length of a ski

m, cm, mm, km

Thickness of a piece of string

m, cm, mm, km

Height of a fence

m, cm, mm, km

Length of a finger

m, cm, mm, km

Length of a football field

m, cm, mm, km

How far you travelled on your holiday

m, cm, mm, km

Circle the letter of the most reasonable measure. (6 marks)
1. Storage box
  1. 66 mL
  2. 66 L
  3. 6 L
2. Baby Shampoo
  1. 593 mL
  2. 593 L
  3. 59 L
3. Antifreeze
  1. 40 L
  2. 4 L
  3. 40 mL
4. Wastebasket
  1. 42 mL
  2. 4 L
  3. 42 mL
5. Deodorant
  1. 354 L
  2. 35 mL
  3. 354 mL
6. Liquid laundry soap
  1. 975 mL
  2. 97 L
  3. 975 L
Circle the unit of measure you would use. (6 marks)

Item Unit of Measure

a) can of peanuts mg, g, kg

b) cat litter mg, g, kg

c) an antacid tablet mg, g, kg

d) bag of potato chips mg, g, kg
Fill in the chart with the right metric prefix. (6 marks)

Measure
Small Base Large
Find the perimeter and area for each shape. (12 marks)
5. A double-size bed cover measures 135 cm wide and 190 cm What is the perimeter and area of the bed cover?
6. An Olympic size swimming is 50 m long and 25 m wide. What is the perimeter and area of the swimming pool?

Answers to Topic D Self-Test

1. cm
2. mm
3. cm
4. mm
5. m or cm
6. cm
7. m
8. km
1. ii
2. i
3. ii
4. ii
5. iii
6. i
1. g
2. kg
3. mg
4. g
Measure
Small Base Large

volume milli Litre

length milli metre kilo

mass milli gram kilo
1. P = 20 cm, A = 21 cm²
2. P = 24 km, A = 36 km²
3. P = 392 m, A = 2208 m²
4. P = 348 mm, A = 7569 mm²
5. P = 650 cm, A = 25650 cm²
6. P = 150 m, A = 1250 m²

Unit 3 Review: Change and the Metric System

You will now practice all the skills you learned in Unit 3. Check your work using the answer key at the end of the review

Circle the number of coins or bills you would need to get from the first number to the second number. Make sure to use the least number of coins or bills.
1. $48 to $50
2. $59 to $60
3. $73 to $80
4. $33 to $40
State the number and kind of coins and bills you would need to get from the first number to the second number. Make sure you use the least number of coins and bills as possible.
1. $23 to $25
2. $31 to $35
3. $85 to $90
4. $70 to $90
State the number and kind of coins and bills you would need to get from the first number to the second number. Make sure you use the least number of coins and bills as possible.
1. $37 to $50
  Need To get to
2. $53 to $60
  Need To get to
3. $77 to $100
  Need To get to
4. $21 to $50
  Need To get to
State the number and kind of coins and bills you would need to get from the first number to the second number. Make sure you use the least number of coins and bills as possible.
1. $63.55 to $80
  Need To get to
2. $32.65 to $50
  Need To get to
3. $20.35 to $40
  Need To get to
4. $72.20 to $100
  Need To get to
State the number and kind of coins and bills you would need to get change from $100. Make sure you use the least number of coins and bills as possible.
1. printer for $78.40
2. cordless phone for $55.65
3. Mrs. Kono bought a new cordless kettle for $44.80. How much change will Mrs. Kono get from $100?
Circle the letter of the most reasonable measure.
1. Diameter of a hockey puck
  1. 76 mm
  2. 76 m
  3. 76 cm
2. Distance from the mall to home
  1. 10 km
  2. 10 m
  3. 10 cm
3. Thickness of a blanket
  1. 10 m
  2. 10 cm
  3. 10 mm
4. Height of a tree
  1. 28 mm
  2. 28 m
  3. 28 cm
Choose the most reasonable measure.
1. Carlos drinks
  1. 500 L of milk
  2. 500 mL of milk
  3. 5 mL of milk
2. A thermos holds
  1. 360 mL
  2. 360 L
  3. 36 L
3. A swimming pool holds 3758 of water
4. A tube of lotion is 50 .
Choose the most reasonable measure.
1. A dog weighs
  1. 17 g
  2. 17 kg
  3. 17 mg
2. A nickel has a mass of
  1. 5 g
  2. 5 mg
  3. 5 kg
3. A paper clip has mass of
  1. 1 kg
  2. 1 mg
  3. 1 g
4. Six math books have mass of
  1. 2 kg
  2. 2 mg
  3. 2 g
1. Elena took 400 of vitamin A
2. Suki bought 10 of potatoes
Write the base unit of measure and then the prefix if one is needed.

Item Base Prefix (if needed)

a) length of a garden hose

b) a bottle of olive oil

c) a child’s multivitamin

d) distance between Jupiter and Mars

e) thickness of a tissue
Write the unit of measure you would use for each item below.
1. Paint thinner
2. Cat litter
3. Deodorant
4. Length of the street
5. Aspirin
Find the area of each shape.
3. Red Square in Moscow measures 330 m long and 70 m wide. What is the area of Red Square?
4. Yoshiro is going to build a square patio whose side is 23 metres. What is the area of the patio?
Find the perimeter and area of each shape.
3. Nadal wants to fence his garden. It is 15 m wide and 26 m long. How much fencing does Nadal need? How much space does Nadal have to plant a garden?
4. Yolanda would like to buy fencing and sod for her new lawn that is 54 m long and 42 m wide. How much fencing does she need? How much sod does she need?

Answers to Unit 3 Review

Circle the number of coins or bills you would need.
1. 1 toonie
2. 1 loonie
3. 1 toonie, 1 $5 bill
4. 1 toonie, 1 $5 bill
State the number and kind of coins and bills you would need.
1. 1 toonie
2. 2 toonies
3. 1 $5 bill
4. 1 $20 bill
State the number and kind of coins and bills you would need.
1. $37 to $50
  Need To get to
  
  1 loonie $38
  
  1 toonie $40
  
  1 $10 bill $50
2. $53 to $60
  Need To get to
  
  1 toonie $55
  
  1 $5 bill $60
3. $77 to $100
  Need To get to
  
  1 loonie $78
  
  1 toonie $80
  
  1 $20 bill $100
4. $21 to $50
  Need To get to
  
  2 toonies $25
  
  1 $5 bill $30
  
  1 $20 bill $50
State the number and kind of coins and bills you would need.
1. $63.55 to $80
  Need To get to
  
  2 dimes $63.75
  
  1 quarter $64
  
  1 loonie $65
  
  1 $5 bill $70
  
  1 $10 bill $80
2. $32.65 to $50
  Need To get to
  
  1 dime $32.75
  
  1 quarter $33
  
  1 toonie $35
  
  1 $5 bill $40
  
  1 $10 bill $50
3. $20.35 to $40
  Need To get to
  
  1 nickel $20.40
  
  1 dime $20.50
  
  2 quarters $21
  
  2 toonies $25
  
  1 $5 bill $30
  
  1 $10 bill $40
4. $72.20 to $100
  Need To get to
  
  1 nickel $72.25
  
  3 quarters $73
  
  1 toonie $75
  
  1 $5 bill $80
  
  1 $20 bill $100
State the number and kind of coins and bills you would need.
1. 1 dime, 2 quarters, 1 loonie, 1 $20 bill
2. 1 dime, 1 quarter, 2 toonies, 2 $20 bills
3. 2 dimes, 1 $5 bill, 1 $10 bill, 2 $20 bills
Circle the letter of the most reasonable measure.
1. i
2. i
3. iii
4. ii
Choose the most reasonable measure.
1. ii
2. i
3. L
4. mL
Choose the most reasonable measure.
1. ii
2. i
3. iii
4. i
5. mg
6. kg

Write the base unit of measure and then the prefix if one is needed.

Item

Base

Prefix (if needed)

length of a garden hose

a bottle of olive oil

milli

a child’s multivitamin

milli

distance between Jupiter and Mars

kilo

thickness of a tissue

milli

Write the unit of measure you would use for each item below.
1. L
2. kg
3. mL
4. km
5. mg
Find the area of each shape.
1. $A = 592 \text{ m}^2$
2. $A = 6724 \text{ cm}^2$
3. $A = 23100 \text{ m}^2$
4. $A = 529 \text{ m}^2$
Find the perimeter and area of each shape.
1. $P = 236 \text{ m}, A = 3312 \text{ m}^2$
2. $P = 156 \text{ mm}, A = 1521 \text{ mm}^2$
3. $P = 82 \text{ m}, A = 390 \text{ m}^2$
4. $P = 192 \text{ m}, A = 2268\text{ m}^2$

Unit	Date of Test A	Test A	Date of Test B	Test B
Example	September 4, 2020	25/33	September 7, 2020	25/33
1
2
3
Final Test

Number	Divisible by 2	Divisible by 3	Divisible by 5
474	Yes	Yes	No
615	No	Yes	Yes
412	Yes	No	No
865	No	No	Yes
300	Yes	Yes	Yes
831	No	Yes	No
525	No	Yes	Yes
350	Yes	No	Yes
710	Yes	No	Yes
429	No	Yes	No
906	Yes	Yes	No
634	Yes	No	No
430	Yes	No	Yes
275	No	No	Yes

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
453	No	Yes	No	No
320	Yes	No	Yes	No
216	Yes	Yes	No	Yes
726	Yes	Yes	No	No
712	Yes	No	No	No
425	No	No	Yes	No
630	Yes	Yes	Yes	Yes
375	No	Yes	Yes	No
990	Yes	Yes	Yes	Yes
210	Yes	Yes	Yes	No

Need	To get to
2 dimes	$20.75
1 quarter	$21
2 toonies	$25
1 $5 bill	$30
1 $10 bill	$40

Need	To get to
1 dime	$61.25
3 quarters	$62
1 loonie	$63
1 toonie	$65
1 $5 bill	$70
1 $10 bill	$80

Need	To get to
1 nickel	$72.90
1 dime	$73
1 toonie	$75
1 $5 bill	$80
1 $20 bill	$100

Need	To get to
2 dimes	$63.75
1 quarter	$64
1 loonie	$65
1 $5 bill	$70
1 $10 bill	$80

Need	To get to
1 dime	$32.75
1 quarter	$33
1 toonie	$35
1 $5 bill	$40
1 $10 bill	$50

Need	To get to
1 nickel	$20.40
1 dime	$20.50
2 quarters	$21
2 toonies	$25
1 $5 bill	$30
1 $10 bill	$40

Need	To get to
1 nickel	$72.25
3 quarters	$73
1 toonie	$75
1 $5 bill	$80
1 $20 bill	$100

Version	Date	Change	Details
1.00	October 3, 2014	Book published using Microsoft Word.
2.00		Book updated and republished in Pressbooks as a 2nd Edition.	Removed coverage of time in last unit Reduced the number of exercises per example. Created a “How to Deal with Math Anxiety” front matter section, which is now standardized across all ALF Math books. Deleted “Topic A: Emotions and Learning” since that content is now covered in the “How to Deal with Math Anxiety” front matter. Re-lettered all units and topics in unit 1. Moved content from Word into Pressbooks. Added textboxes to visually identify examples and exercises. Added some headings to give long chapters more structure.
2.01	June 12, 2024	Corrected answers.	In Topic B: Two- and Three-Digit Multipliers in Unit 1: Multiplication, the answers to many of the problems in Exercise 2 were corrected.
2.02	March 7, 2025	Corrected content and formatting.	Corrected content and formatting errors throughout the book, including wrong answers, misplaced formatting, and outdated references to pennies.
2.03	September 19, 2025	Corrected errors and formatting.	Corrected errors and added content in the following sections: Topic B: Two- and Three-Digit Multipliers in Unit 1 Topic D: Multiplication Problems in Unit 1 Topic A: Introduction and Division Facts in Unit 2 Topic D: Dividing by Two- and Three- Digit Divisors in Unit 2 Topic E: Estimating Quotients in Unit 2 Topic F: Division Problems in Unit 2 Unit 2 Review: Division Topic B: Making Change in Unit 3 Topic C: The Metric System in Unit 3 Unit 3 Review: Change and the Metric System Book 3 Review

Number	Divisible by 2	Divisible by 3	Divisible by 5
474	Yes	Yes	No
615	No	Yes	Yes
412	Yes	No	No
865	No	No	Yes
300	Yes	Yes	Yes
831	No	Yes	No
525	No	Yes	Yes
350	Yes	No	Yes
710	Yes	No	Yes
429	No	Yes	No
906	Yes	Yes	No
634	Yes	No	No
430	Yes	No	Yes
275	No	No	Yes

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
453	No	Yes	No	No
320	Yes	No	Yes	No
216	Yes	Yes	No	Yes
726	Yes	Yes	No	No
712	Yes	No	No	No
425	No	No	Yes	No
630	Yes	Yes	Yes	Yes
375	No	Yes	Yes	No
990	Yes	Yes	Yes	Yes
210	Yes	Yes	Yes	No

Adult Literacy Fundamental Mathematics: Book 3 – 2nd Edition

Adult Literacy Fundamental Mathematics: Book 3 – 2nd Edition

Contents

Accessibility Statement

Accessibility of This Resource

Known Accessibility Issues and Areas for Improvement

Let Us Know if You are Having Problems Accessing This Book

For Students: How to Access and Use this Textbook

Tips for Using This Textbook

About BCcampus Open Education

To the Learner

How to Use This Book

Grades Record

How to Deal with Math Anxiety

Emotions and Learning

Do You Suffer from Math Anxiety?

How to Deal with Math Anxiety

Use Breathing Exercises

Think Positive Math Messages

Know Your Textbook

Understand Test-Taking Anxiety

1. Not feeling prepared for the test

2. Not sure how to write the test in the best way

3. Feeling too much mental pressure

4. Poor health habits before writing a test

Unit 1: Multiplication

Topic A: Multiplying Larger Numbers

Renaming and Carrying

Topic A: Self-Test

Answers to Topic A Self-Test

Topic B: Two- and Three-Digit Multipliers

Multiplying by Two-Digit Multipliers

Multiplying by Three-Digit Multipliers

Multiplying by 10, 100, and 1000

Topic B: Self-Test

Answers to Topic B Self-Test

Topic C: Estimating Products

Topic C: Self-Test

Answers to Topic C Self-Test

Topic D: Multiplication Problems

Topic D: Self-Test

Answers to Topic D Self-Test

Unit 1 Review: Multiplication

Answers to Unit 1 Review

Unit 2: Division

Topic A: Introduction and Division Facts

Remainders

Topic A: Self-Test

Answers to Topic A Self-Test

Topic B: Divisibility

Divisibility by 2

Divisibility by 3

Divisibility by 5

Divisibility by 9

Topic B: Self-Test

Answers to Topic B Self-Test

Topic C: Dividing Larger Numbers by One-Digit Divisors

One-Digit Divisors without Remainders

One-Digit Divisors with Remainders

Check Division by Multiplying

Topic C: Self-Test

Answers to Topic C Self-Test

Topic D: Dividing by Two- and Three- Digit Divisors

Finding Trial Quotients

Two Digit Divisors

Dividing by 10, 100, 1000…

What is the pattern? When you divide by 10

When you divide by 100

When you divide by 1000

Three-Digit Divisors

Topic D: Self-Test

Answers to Topic D Self-Test

Topic E: Estimating Quotients

Divide Numbers That Both End With Zeros

Rounding Division Questions to Estimate

Topic E: Self-Test

Answers to Topic E Self-Test

Topic F: Division Problems

Unit Pricing

Best Buy

Number	Divisible by 2	Divisible by 3	Divisible by 5
474	Yes	Yes	No
615	No	Yes	Yes
412	Yes	No	No
865	No	No	Yes
300	Yes	Yes	Yes
831	No	Yes	No
525	No	Yes	Yes
350	Yes	No	Yes
710	Yes	No	Yes
429	No	Yes	No
906	Yes	Yes	No
634	Yes	No	No
430	Yes	No	Yes
275	No	No	Yes

Number	Divisible by 2	Divisible by 3	Divisible by 5	Divisible by 9
453	No	Yes	No	No
320	Yes	No	Yes	No
216	Yes	Yes	No	Yes
726	Yes	Yes	No	No
712	Yes	No	No	No
425	No	No	Yes	No
630	Yes	Yes	Yes	Yes
375	No	Yes	Yes	No
990	Yes	Yes	Yes	Yes
210	Yes	Yes	Yes	No