Unit 2: Division
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
[latex]624 \div 24[/latex]
Rewrite as [latex]24\enclose{longdiv}{624}[/latex]
Step 1: Divide
- Think: [latex]2\enclose{longdiv}{6}[/latex] is [latex]3[/latex]. So [latex]24\enclose{longdiv}{62}[/latex] is about [latex]3[/latex].
Step 2: Multiply and subtract.
- [latex]\begin{array}{r}3 \ \ \ \\ 24\enclose{longdiv}{624}\\72 \ \ \ \\ \hline \end{array}[/latex]
- Since 72 > 62, 3 is too large.
Step 3: Try a smaller number, multiply and subtract.
- [latex]\begin{array}{r}2\ \ \ \\ 24\enclose{longdiv}{624}\\48 \ \ \ \\ \hline 14 \ \ \ \end{array}[/latex]
- Since 4 < 24, 2 is correct.
Step 4: Finish the problem.
Example B
[latex]630 \div 15[/latex]
Rewrite as [latex]15\enclose{longdiv}{630}[/latex]
Step 1: Divide
- 15 rounds to 20. Think: [latex]2\enclose{longdiv}{6}[/latex] is [latex]3[/latex]. So [latex]15\enclose{longdiv}{63}[/latex] is about [latex]3[/latex].
Step 2: Multiply and subtract.
- [latex]\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{630}\\45 \ \ \ \\ \hline 18 \ \ \ \end{array}[/latex]
- Since 18 > 15, 3 is too small.
Step 3: Try a larger number, multiply and subtract.
- [latex]\begin{array}{r}4 \ \ \ \\ 24\enclose{longdiv}{630}\\60 \ \ \\ \hline 3 \ \ \end{array}[/latex]
- 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.
- [latex]\begin{array}{r}2 \ \ \ \\ 25\enclose{longdiv}{475}\end{array}[/latex]
- [latex]\begin{array}{r}3 \ \ \ \\ 15\enclose{longdiv}{682}\end{array}[/latex]
- [latex]\begin{array}{r}5 \ \ \ \\ 18\enclose{longdiv}{813}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 25\enclose{longdiv}{810}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 33\enclose{longdiv}{891}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 18\enclose{longdiv}{819}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 27\enclose{longdiv}{727}\end{array}[/latex]
- [latex]\begin{array}{r}2 \ \ \ \\ 35\enclose{longdiv}{652}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 25\enclose{longdiv}{650}\end{array}[/latex]
- [latex]\begin{array}{r}4\ \ \ \\34\enclose{longdiv}{176}\end{array}[/latex]
- [latex]\begin{array}{r}4\ \ \ \\ 12\enclose{longdiv}{420}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 43\enclose{longdiv}{801}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 31\enclose{longdiv}{899}\end{array}[/latex]
- [latex]\begin{array}{r}4\ \ \ \\ 18\enclose{longdiv}{648}\end{array}[/latex]
- [latex]\begin{array}{r}4\ \ \ \\ 27\enclose{longdiv}{946}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 23\enclose{longdiv}{943}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 24\enclose{longdiv}{578}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 29\enclose{longdiv}{406}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 48\enclose{longdiv}{892}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{534}\end{array}[/latex]
- [latex]\begin{array}{r}3\ \ \ \\ 37\enclose{longdiv}{939}\end{array}[/latex]
- [latex]\begin{array}{r}2\ \ \ \\ 28\enclose{longdiv}{854}\end{array}[/latex]
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
[latex]78\enclose{longdiv}{2706}[/latex]
Since 78 rounds to 80, think [latex]8\enclose{longdiv}{27}[/latex]. 8 goes into 27 ≈ 3. 3 would be a good trial quotient.
[latex]\begin{array}{r}3\ \ \ \\ 78\enclose{longdiv}{2706}\\234\ \ \ \\ \hline 36\ \ \ \end{array}[/latex]
Since 36 < 78, 3 is a good trial quotient.
Example D
[latex]27\enclose{longdiv}{2205}[/latex]
Since 27 rounds to 30, think [latex]3\enclose{longdiv}{22}[/latex] . 3 goes into 22 ≈ 7. 7 would be a good trial quotient.
[latex]\begin{array}{r}7\ \ \ \\ 27\enclose{longdiv}{2205}\\189\ \ \ \\ \hline 31\ \ \ \end{array}[/latex]
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.
- [latex]43\enclose{longdiv}{1772}[/latex] [latex]\begin{array}{r}4\\ 4\enclose{longdiv}{17}\\16\\ \hline 1\end{array}[/latex]. 1 < 4
- [latex]64\enclose{longdiv}{3276}[/latex]
- [latex]28\enclose{longdiv}{6008}[/latex]
- [latex]33\enclose{longdiv}{2731}[/latex]
- [latex]59\enclose{longdiv}{4164}[/latex]
- [latex]75\enclose{longdiv}{2420}[/latex]
- [latex]54\enclose{longdiv}{3316}[/latex]
- [latex]38\enclose{longdiv}{2759}[/latex]
- [latex]46\enclose{longdiv}{387}[/latex]
- [latex]35\enclose{longdiv}{3316}[/latex]
- [latex]83\enclose{longdiv}{7237}[/latex]
- [latex]77\enclose{longdiv}{6763}[/latex]
- [latex]93\enclose{longdiv}{3724}[/latex]
- [latex]52\enclose{longdiv}{4690}[/latex]
- [latex]86\enclose{longdiv}{2089}[/latex]
- [latex]26\enclose{longdiv}{1417}[/latex]
- [latex]72\enclose{longdiv}{1462}[/latex]
- [latex]27\enclose{longdiv}{6939}[/latex]
- [latex]32\enclose{longdiv}{7840}[/latex]
- [latex]24\enclose{longdiv}{7605}[/latex]
- [latex]16\enclose{longdiv}{8640}[/latex]
- [latex]45\enclose{longdiv}{3060}[/latex]
- [latex]38\enclose{longdiv}{2158}[/latex]
- [latex]42\enclose{longdiv}{1491}[/latex]
Answers to Exercise 2
- 4
- 5
- 2
- 8
- 7
- 3
- 6
- 7
- 8
- 4
- 8
- 8
- 4
- 9
- 2
- 5
- 2
- 2
- 2
- 3
- 5
- 6
- 5
- 3
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
[latex]964 \div 75 =[/latex]
Step 1: Divide
- Does 75 go into 9? NO
- Does 75 go into 96? YES
- Estimate
- Round 75 to 80 – think “8”
- Round 96 to 100 – think “10”
- How many 8’s in 10? (8 1 = 8, 10 8 = 1)
- The estimate for the first digit in the trial quotient is 1.
- Write 1 in the quotient above the 6 tens.
[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\end{array}[/latex]
Step 2: Multiply
[latex]1 \times 75 = 75[/latex]
- Write 75 under 96.
[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \ \ \\ \hline \end{array}[/latex]
Step 3: Subtract
[latex]96 - 75 = 21[/latex]
- Check 21 < 75? YES!
[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75\ \ \\ \hline 21 \ \ \end{array}[/latex]
Step 4: Bring down the next digit in the dividend. 214 is now the number to be divided.
[latex]\begin{array}{r}1\ \ \ \\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}[/latex]
REPEAT
Step 1: Divide
- Does 75 go into 214? YES
- Estimate 75 as 80 – think “8”
- Estimate 214 as 200 – think “20”
- 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.
[latex]\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214\end{array}[/latex]
Step 2: Multiply
[latex]2 \times 75 = 150[/latex]
- Write 150 under the 214.
[latex]\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline\end{array}[/latex]
Step 3: Subtract and check that the remainder is less than the divisor.
[latex]\begin{array}{r}12\\ 75\enclose{longdiv}{964}\\75 \downarrow \\ \hline 214 \\ 150 \\ \hline 64\end{array}[/latex]
Step 4: Bring down – no more digits in dividend.
To check your answer
[latex]\begin{array}{rrr}&75\\\times&12\\\hline&150 \\ &750 \\ \hline &900 \\ + &64 \\ \hline &964\end{array}[/latex]
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.
- [latex]10\enclose{longdiv}{720}[/latex]
- [latex]12\enclose{longdiv}{564}[/latex]
- [latex]21\enclose{longdiv}{882}[/latex]
- [latex]22\enclose{longdiv}{946}[/latex]
- [latex]32\enclose{longdiv}{1632}[/latex]
- [latex]23\enclose{longdiv}{943}[/latex]
- [latex]62\enclose{longdiv}{2528}[/latex]
- [latex]71\enclose{longdiv}{2414}[/latex]
- [latex]24\enclose{longdiv}{7578}[/latex]
- [latex]82\enclose{longdiv}{2958}[/latex]
- [latex]18\enclose{longdiv}{6250}[/latex]
- [latex]25\enclose{longdiv}{1550}[/latex]
- [latex]19\enclose{longdiv}{9595}[/latex]
- [latex]47\enclose{longdiv}{3854}[/latex]
- [latex]58\enclose{longdiv}{6500}[/latex]
- [latex]24\enclose{longdiv}{9648}[/latex]
- [latex]49\enclose{longdiv}{1312}[/latex]
- [latex]67\enclose{longdiv}{7683}[/latex]
Answers to Exercise 3
- 72
- 47
- 42
- 43
- 51
- 41
- 40 R48
- 34
- 24 R2
- 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.
- Divide: Trial estimate is 4.
- Multiply:
- [latex]4 \times 23 = 92[/latex]
- 92 is larger than 78, so 4 is too large an estimate. Erase it. Try 3. [latex]3 \times 23 = 69[/latex]
- 3 is the correct estimate and you can complete the division.
|
1. |
[latex]23\enclose{longdiv}{784}[/latex] |
| 2. | [latex]\begin{array}{r}4\ \ \ \\ 23\enclose{longdiv}{784}\\92\ \ \ \\ \hline \end{array}[/latex] |
| 3. | [latex]\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}[/latex] |
If the estimate is too small, the result of the subtraction will be larger than the divisor.
- Divide: Trial estimate is 5.
- Multiply: [latex]5 \times 72 = 360[/latex]
- Subtract:
- [latex]448 - 72 = 360[/latex]
- Check 88 < 72? NO, 88 is greater than 72.
- So 5 is too small. Erase it and use a larger number. 6 will be a better estimate.
- Divide: [latex]448 \div 82 \approx 6[/latex]
- Multiply: [latex]6 \times 72 = 432[/latex]
- Subtract:
- [latex]448 - 432 = 16[/latex]
- Check 16<72? YES!
- Bring down the next digit and complete the division.
| 1. | [latex]72\enclose{longdiv}{4487}[/latex] |
| 2. | [latex]\begin{array}{r}5\ \ \ \\ 72\enclose{longdiv}{4487}\\360\ \ \ \\ \hline88\hspace{0.8em} \end{array}[/latex] |
| 3. | [latex]\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}[/latex] |
Exercise 4
Divide and check your work by multiplying. Check your work using the answer key at the end of the exercise.
- [latex]18\enclose{longdiv}{648}[/latex]
- [latex]26\enclose{longdiv}{6766}[/latex]
- [latex]52\enclose{longdiv}{1968}[/latex]
- [latex]84\enclose{longdiv}{8640}[/latex]
- [latex]72\enclose{longdiv}{2883}[/latex]
- [latex]94\enclose{longdiv}{8126}[/latex]
- [latex]20\enclose{longdiv}{4060}[/latex]
- [latex]47\enclose{longdiv}{1728}[/latex]
- [latex]33\enclose{longdiv}{1886}[/latex]
- [latex]25\enclose{longdiv}{5750}[/latex]
- [latex]79\enclose{longdiv}{2765}[/latex]
- [latex]42\enclose{longdiv}{8442}[/latex]
- [latex]57\enclose{longdiv}{9144}[/latex]
- [latex]96\enclose{longdiv}{20160}[/latex]
- [latex]75\enclose{longdiv}{23550}[/latex]
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.
[latex]10\enclose{longdiv}{324} = 32 R4[/latex]
- 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.
- [latex]10\enclose{longdiv}{46}[/latex]
- [latex]10\enclose{longdiv}{75}[/latex]
- [latex]10\enclose{longdiv}{136}[/latex]
- [latex]10\enclose{longdiv}{832}[/latex]
- [latex]10\enclose{longdiv}{674}[/latex]
- [latex]10\enclose{longdiv}{952}[/latex]
- [latex]10\enclose{longdiv}{2457}[/latex]
- [latex]10\enclose{longdiv}{3685}[/latex]
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
- [latex]100\enclose{longdiv}{386}[/latex]
- [latex]100\enclose{longdiv}{995}[/latex]
- [latex]100\enclose{longdiv}{269}[/latex]
- [latex]100\enclose{longdiv}{175}[/latex]
- [latex]100\enclose{longdiv}{2948}[/latex]
- [latex]100\enclose{longdiv}{4671}[/latex]
- [latex]100\enclose{longdiv}{92045}[/latex]
- [latex]100\enclose{longdiv}{43821}[/latex]
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.
- [latex]1000\enclose{longdiv}{2398}[/latex]
- [latex]1000\enclose{longdiv}{6475}[/latex]
- [latex]1000\enclose{longdiv}{4835}[/latex]
- [latex]1000\enclose{longdiv}{63291}[/latex]
- [latex]1000\enclose{longdiv}{82405}[/latex]
- [latex]1000\enclose{longdiv}{293591}[/latex]
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 ([latex]4\times 4 = 16[/latex])
- Your estimate is 4. Write 4 in the quotient above the 0 in the dividend.
[latex]\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \end{array}[/latex]
Step 2: Multiply
- 4 × 381 = 1524
[latex]\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \ \ \ \\ \hline \end{array}[/latex]
Step 3: Subtract
- 1790 − 1524 = 266
- Check 266 < 381? YES!
[latex]\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \ \ \ \\ \hline 266 \ \ \ \end{array}[/latex]
Step 4: Bring down the 2. 2662 is now the number to be divided
[latex]\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \end{array}[/latex]
REPEAT
Step 1: Divide [latex]2662 \div 381 =[/latex]
- Estimate 381 as 400. Think of 4.
- Estimate 2662 as 2700. Think 27.
- 4 goes into 27 ≈ 6 times ([latex]4 \times 6 = 24[/latex]).
- Place this estimate in the quotient above the 2
[latex]\begin{array}{r}4 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \end{array}[/latex]
Step 2: Multiply
- [latex]6\times381 = 2286[/latex]
[latex]\begin{array}{r}46 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \\ 2286 \\ \hline \end{array}[/latex]
Step 3: Subtract
- 2662 − 2286 = 376
- Check 376 < 381? YES!
[latex]\begin{array}{r}46 \ \ \ \\ 381\enclose{longdiv}{17902} \\ 1524 \downarrow \\ \hline 2662 \\ 2286 \\ \hline 376 \end{array}[/latex]
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.
- [latex]115\enclose{longdiv}{8682}[/latex]
- [latex]205\enclose{longdiv}{2384}[/latex]
- [latex]325\enclose{longdiv}{66321}[/latex]
- [latex]241\enclose{longdiv}{13284}[/latex]
- [latex]860\enclose{longdiv}{262412}[/latex]
- [latex]659\enclose{longdiv}{270190}[/latex]
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)
- [latex]185 \div 10 =[/latex]
- [latex]4\enclose{longdiv}{92}[/latex]
- [latex]408 \div 50 =[/latex]
- [latex]72\enclose{longdiv}{6768}[/latex]
- [latex]67\enclose{longdiv}{5963}[/latex]
- [latex]53\enclose{longdiv}{4856}[/latex]
- [latex]91\enclose{longdiv}{8736}[/latex]
- [latex]265\enclose{longdiv}{133624}[/latex]
- [latex]606\enclose{longdiv}{26094}[/latex]
- [latex]1000\enclose{longdiv}{83652}[/latex]
Answers to Topic D Self-Test
- Divide and check your work for questions b and f using multiplication.
- 18 R5
- 8 R8
- 38
- 94
- 89
- 91 R33
- 96
- 504 R64
- 43 R36
- 83 R652