Unit 2: Adding and Subtracting Decimals
Topic B: Subtracting Decimals
Line Up the Decimals
To subtract decimals you must subtract each digit from the digit of the same place value.
- Subtract thousandths from thousandths.
- Subtract hundredths from hundredths.
- Subtract tenths from tenths.
The best way to do this is to line up your decimals.
How to Subtract Decimals
The same techniques that you used in adding decimals are helpful when you subtract decimals.
- Rewrite the problem. Write the first number. Put the amount you are subtracting underneath so the decimal points are in a straight column.
[latex]\begin{array}{rrl}0.468&-&0.3=\\ \uparrow &&\uparrow \\ \text{starting number}&&\text{subtracting this much}\end{array}[/latex]
[latex]\begin{array}{ll}&0.468\\-&0.3\\ \hline\end{array}[/latex] - Put zeros at the end of the decimals so that all the decimals in the question have the same number of decimal places.
[latex]0.468-0.3=\begin{array}[t]{ll}&0.468\\-&0.3\mathbf{00}\\ \hline\end{array}[/latex] - Subtract the numbers, keeping the decimal point in the answer directly beneath the other decimal points.
[latex]\begin{array}{ll}&0.468\\-&0.300\\ \hline &\phantom{0}\mathbf{.}\end{array}[/latex]
Example A
[latex]2.536-0.59=[/latex]
- Rewrite the problem, lining up the decimals:
[latex]\begin{array}{ll}&2.536\\- &0.59\\ \hline \end{array}[/latex] - Add zeros so that there are the same number of decimal places for each.
[latex]\begin{array}{ll}&2.536\\- &0.59\mathbf{0}\\ \hline \end{array}[/latex] - Subtract the numbers. You will need to borrow.
[latex]\begin{array}{ll}&\tiny{\,1\;4\,1}\\&2.536\\- &0.590\\ \hline &1.946 \end{array}[/latex]
Vocabulary Review.
Write the definition.
[latex]8-5 = 3[/latex] ← Difference
Difference:
Exercise 1
Subtract to find the differences.
- [latex]\begin{array}[t]{ll}&2.75\\-&0.68\\ \hline&2.07\end{array}[/latex]
- [latex]\begin{array}[t]{ll}&9.64\\-&7.15\\ \hline&2.49\end{array}[/latex]
- [latex]\begin{array}[t]{ll}&3.85\\-&1.75 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{ll}&1.17\\-&0.92 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{ll}&27.3\\-&18.9 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{ll}&0.732\\-&0.651 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{ll}&0.362\\-&0.177 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{ll}&6.85\\-&1.28 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{rr}&18.5\\-&7.9 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{rr}&98.6\\-&45.8 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{rr}&5.276\\-&3.298 \\ \hline\\ \end{array}[/latex]
- [latex]\begin{array}[t]{rr}&5.251\\-&2.738 \\ \hline\\ \end{array}[/latex]
Answers to Exercise 1
- 2.07
- 2.49
- 2.10
- 0.25
- 8.4
- 0.081
- 0.185
- 5.57
- 10.6
- 52.8
- 1.978
- 2.513
Subtracting a Decimal from a Whole Number
Follow these steps to subtract a decimal from a whole number:
- Put a decimal point after the whole number.
16. − 0.4 = - Rewrite the problem. Write the first number. Put the amount you are subtracting underneath so the decimal points are in a straight column.
[latex]\begin{array}[t]{rr}&16.\hspace{0.5em}\\-&0.4\\ \hline\end{array}[/latex] - Put zeros after the decimal point as needed.
[latex]\begin{array}[t]{rr}&16.\mathbf{0}\\-&0.4\\ \hline\end{array}[/latex] - Do the subtraction as usual. See that you will need to borrow right away.
Example B
[latex]32-0.12 =[/latex]
[latex]\begin{array}{rr}&32.00\\-&0.12\\ \hline\end{array}[/latex]
- Rename the 2 in the ones place as 1 and 10 tenths.
[latex]\begin{array}{rr}&\tiny{1\, 10\;\;}\\&32.00\\-&0.12\\ \hline\end{array}[/latex] - Now rename the 10 tenths as 9 tenths and 10 hundredths. You are ready to subtract.
[latex]\begin{array}{rr}&\scriptsize{9\quad}\\&\scriptsize{1 \cancel{10}10}\\&3\cancel{2.00}\\-&0.12\\ \hline&31.88\end{array}[/latex]
Example C
[latex]\$14 - \$3.49[/latex]
[latex]\begin{array}{rr}&$14.00\\-&$3.49\\ \hline\end{array}[/latex]
[latex]\begin{array}{rr}&\scriptsize{9\quad}\\& \scriptsize{3\cancel{10}10}\\&$14.\cancel{00}\\-&$3.49\\ \hline&$10.51\end{array}[/latex]
Exercise 2
Rewrite each question in columns and find the difference.
- 6 − 3.42 =
[latex]\begin{array}{rr}&\scriptsize{5\cancel{10}10\;\;}\\&6.\cancel{00}\\-&3.42\\ \hline &2.58\end{array}[/latex] - 14 − 9.23 =
[latex]\begin{array}{rr}&14.00\\-&9.23\\ \hline \\ \end{array}[/latex]
- 11 − 3.821 =
- 2 − 1.98 =
- 7 − 3.976 =
Answers to Exercise 2
- 2.58
- 4.77
- 7.179
- 0.02
- 3.024
If you had problems with this, go over your subtraction method with your instructor before you continue.
Exercise 3
Rewrite each question in columns and find the difference.
- 163.682 − 41.5 =
[latex]\begin{array}{rr}&163.682\\-&41.500\\ \hline &122.182\end{array}[/latex]
- $60 − $44.28 =
- $260.06 − $3 =
- 89.0309 − 6.3 =
- $100 − $13.75 =
Answers to Exercise 3
- 122.182
- $15.72
- $257.06
- 82.7309
- $86.25
Word Problems Using Subtraction of Decimals
Some key words that point to subtraction include:
- difference
- balance
- minus
- amount left
- subtracted from
- decreased by
- reduced by
- taken away
- less
- compare
A math question may ask you to compare or find the difference between two amounts. Look for such words as “how much more” (or larger, taller, greater) or “how much less” (or smaller, shorter). What are the savings?
Subtract to find the answer.
Exercise 4
Use your skills in subtracting decimal fractions to do the following problems. Underline key words in the problems that will help you to recognize subtraction problems. Remember to draw a picture first!
- Brad is 1.8 m tall. He just did the best high jump of his life, clearing 1.89 m. How much less is his own height than the height he jumped? (add the information to the drawing)
- Estimation:
- Actual Solution:
- Susan’s best track and field event is long jump. She leapt 6.16 m. Her mom used to long jump in high school and jumped 5.52 m. How much farther did Susan jump than her mom?
- Estimation (to tenths):
- Actual Solution:
- Joe had a bank balance of $438. He wrote a cheque for $111.59 to pay for a phone bill. What is the balance in his bank account now?
- Estimation:
- Actual Solution:
- A plumber needs to replace 11.5 m of pipe in a home. She has 6.5 m in her truck. How much more pipe does she need?
- Estimation:
- Actual Solution:
- Lee is going to install base boards in the bachelor suite he has built in his basement. The room is 5.8 metres square. The baseboard material is expensive, so he will be sure to deduct 1 m for each of the two doorways. How much baseboard material does he need to buy? (this question involves addition and subtraction)
- Estimation (to tenths):
- Actual Solution:
Answers to Exercise 4
- Estimation: 1.9 m − 1.8 ≈ 0.1 m
Actual Solution: 1.89 − 1.8 = 0.09 m
Answer: Brad‘s height is 0.09 m less than the height he jumped. - Estimation: 6.2 − 5.5 ≈ 0.7 m
Actual Solution: 6.16 − 5.52 = 0.64 m
Answer: Susan jumped 0.64 m farther than her mom. - Estimation: $440 − $100 ≈ $340
Actual Solution: $438 − $111.59 = $326.41
Answer: Joe‘s bank balance is now $326.41. - Estimation: 12 m − 7 m ≈ 5 m
Actual Solution: 11.5 m – 6.5 m = 5 m
Answer: The plumber needs 5 m more of pipe. - Estimation: 6 m × 4 = 24 m, 24 m − 2 m (doors) ≈ 22 m
Actual Solution: 5.8 m × 4 = 23.2 m, 23.2 m − 2 m (doors) = 21.2 m
Answer: Lee will need to buy 21.2 m worth of base board material.
Design Your Own House Part 1: Drawing and Measuring
- On a sheet of graph paper, use a ruler to draw a one-story house (bird’s-eye-view). Use the scale of 1 cm2 = 1m2.
- You may make your house any shape you choose (square, rectangle, L-shaped, etc.) but one wall must measure 15.5 metres (it will be 15.5 cm on your graph paper).
- Clearly label the length of each wall and show your calculations! Be precise and careful with your sketching. Feel free to add colour or get creative.
- Be sure to organize and label all your work!
- What is the perimeter of the entire house? (perimeter = side + side + side + side)
- Give your house two bedrooms. Clearly label the length of each wall.
- Include a bathroom, a living room, and a kitchen. Label the length of each wall.
- What is the perimeter of each room?
- Bedroom 1:
- Bedroom 2:
- Bathroom:
- Kitchen:
- Living room:
- What is the perimeter of each room?
- On your sketch, give your house two exterior doors and four exterior windows, each 1m wide. Because it is two-dimensional bird’s eye view, you can’t see the height of your doors and windows, but imagine that each door would be 2.3 m tall and each window would be 1.1 m tall. Sketch one window and one door here, and label the measurements.
- Each window needs external trim and internal trim. How much trim would you need to buy to have enough to go around all of the windows?
- Each door would also need external and internal trim. How much trim would you need to go around both doors? Keep in mind that the bottom of the door will not need trim.
Marking Checklist of House Project Part 1
/50 marks
| Mark | Criteria |
|---|---|
| /2 | A ruler was used for drawing lines. |
| /2 | The house has five rooms. |
| /5 | All wall lengths (internal and external) are marked clearly on the sketch. |
| /2 | Locations for doors and windows have been marked on the sketch. |
| /2 | One door and one window have been drawn with measurements marked. |
| /2 | The house has a yard and measurements have been clearly marked on the sketch. |
| Mark | Criteria |
|---|---|
| /2 | The perimeter of the house has been calculated correctly. |
| /10 | The perimeter of each room has been calculated correctly. |
| /2 | Bedroom 1 |
| /2 | Bedroom 2 |
| /2 | Bathroom |
| /2 | Kitchen |
| /2 | Living room |
| /2 | Window and door locations have been marked on the graph paper sketch. |
| /2 | One window and one door has been sketched and measurements labelled. |
| /5 | The amount of trim for the windows has been calculated correctly. |
| /5 | The amount of trim for the doors has been calculated correctly. |
Topic B: Self-Test
Mark /6 Aim 5/6
- Subtract. (4 marks)
- [latex]\begin{array}[t]{rr}&72.04\\-&13.98\\ \hline\\ \end{array}[/latex]
- [latex]19.6-6.254 =[/latex]
- [latex]\begin{array}[t]{rr}&11.21\\-&3.875\\ \hline\\ \end{array}[/latex]
- [latex]\$140-\$102.73 =[/latex]
- Problems (2 marks)
- Gail spent $273.24 on her groceries. She had $300 with her. How much of her money is left?
- Estimation:
- Actual Solution:
- Gail spent $273.24 on her groceries. She had $300 with her. How much of her money is left?
Answers to Topic B Self-Test
- Subtract.
- 58.06
- 13.346
- 7.335
- $37.27
- Problems.
- Estimation: $300 − $275 ≈ $25
- Actual Solution: $300 − $273.24 = $26.76. Gail had $26.76 left after buying groceries.
Any of the ten numerals (0 to 9) are digits. This term comes from our ten fingers which are called digits. The numerals came to be called "digits" from the practice of counting on the fingers!
We understand numbers by the way the digits (numerals) are arranged in relationship to each other and to the decimal point. Each position has a certain value. Our number system is a decimal system. The place value is based on ten.
The result of a subtraction question, the answer. Subtraction gives the difference between two numbers.
Balance has many meanings. In money matters, the balance is the amount left. It might be the amount left in a bank account (bank balance) or it might be the amount you still must pay on a bill (balance owing).