3.2 ADDING AND SUBTRACTING DECIMAL NUMBERS
Diane stopped at her local market on her way home from work.
She purchased the following items:
1 bunch of bananas $1.59
6 apples
$2.49
1 head of lettuce
$0.99
1 carton ice cream
$3.50
1 pound cheese
$4.80
1 gallon milk
$3.29
1 package pasta
$1.39
sales tax total:
$1.26
If she paid for her purchases
with a $20 bill, how much
change did she receive?
0.69
$______________
Assess your readiness to complete this activity. Rate how well you understand:
Not
ready
Almost
ready
Bring
it on!
• the importance of proper decimal point and place value alignment when
adding or subtracting decimal numbers
• the use of trailing zeros when adding or subtracting decimal numbers
• the validation of addition by successive subtractions
• the validation of subtraction by addition
•
Adding any set of decimal numbers
– neatness of presentation
– proper use of trailing zeros
– validation of the answer
•
Subtracting any two decimal numbers
– neatness of presentation
– proper use of trailing zeros
– validation of the answer
139
Chapter 3 — Decimal Numbers
In order to add or subtract decimal numbers, you must align the numbers according to their base ten place
values. You can then proceed column by column as you do with whole numbers, carrying in addition and
borrowing in subtraction when necessary. The methodology includes a step to ensure that you will add or
subtract the correct decimal place digits (Step 2).
Example 1: Add: 3 + 5.76 + 19.871
Example 2: Subtract: 83.4 – 24.57
Steps in the Methodology
Step 1
Set up the
problem.
Step 2
Fill in
trailing
zeros.
Step 3
Add or
subtract in
columns.
Example 1
Example 2
Align the numbers by lining up their
decimal points and their place digits.
3.
5.76
+19.871
83.4
−24.57
Use trailing zeros to make all terms have
the same number of decimal places.
3.000
5.760
+19. 871
83.40
−24.57
Add or subtract the digits with the same
place value, starting with the right-most
column.
1
For addition, carry to the next higher place
value column when necessary.
For subtraction, borrow from the
next higher place value column when
necessary.
1
1
3 .000
5.760
+ 19.871
2 8 631
7 12
1
7 12
1
3
8 3 . 410
−2 4 . 5 7
5 8 83
Just as for whole numbers, the meaning
of carrying and borrowing comes from an
understanding of decimal place values.
Step 4
Bring down
the decimal
point.
Align the decimal point in the answer with
the decimal points of the terms.
1
1
1
3 .000
5.760
+ 19.871
3
8 3 . 410
−2 4 . 5 7
5 8.83
28.631
Step 5
Present your answer.
28.631
Present the
answer.
Step 6
Validate
your
answer.
Validate your answer by using the
opposite operation.
15
278. 6131
−5 . 7 60
2 2 . 8 71
22.871
−19.871
3.000
3.000 = 3
140
58.83
1 1
1
5 8.8 3
+24.57
83.4 0
83.40 = 83.4
Activity 3.2 — Adding and Subtracting Decimal Numbers
Subtract: 76.92 – 39
6
1
Step 1
76.92
−39.
Step 4
7 6.9 2
−3 9 .
3 7.9 2
Step 2
76. 92
−39.00
Step 5
37.92
Step 3
6
Step 6
Validate:
1
7 6.9 2
−3 9 .
37 92
Make Your Own Model
1
3 7.9 2
+39.00
76.9 2 Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
141
Chapter 3 — Decimal Numbers
1. Why is it important to align numbers by place value when adding or subtracting decimals?
Place value columns must be aligned during addition so that like values can be added or subtracted easier and with
reduced errors. Like in adding and subtracting whole numbers, you can only add numbers arranged in columns
of the same place value – ones with ones, tens with tens, and so on. Decimals are a continuation of the whole
number place value system, therefore, you have to continue to add or subtract with the same place value column.
In carrying and borrowing, you are re-grouping by either borrowing “ten” from one column to another, or adding
“ten” to the column to the left.
2. When do you use trailing zeros in decimal addition or subtraction?
Trailing zeros should be used to fill missing place value digits so that all the place value columns are aligned for
addition and subtraction.
3. How do you validate a subtraction problem?
Validate subtraction of decimals just as you validate subtraction of whole numbers: add the difference to the
subtrahend to get the minuend. In other words add the answer to the bottom number to get the top number.
4. How can you validate an addition problem?
There are two ways to validate addition. One way is to independently add the numbers in reverse order to check the
total. Another, more reliable way is to subtract each addend from the total until only one addend remains, which
should check with the last performed subtraction.
5. When adding decimal numbers, why can you carry from the tenths place to the ones place?
Because ten tenths is equal to one whole.
6. What does it mean to borrow from the ones column?
It means that you are borrowing ten-tenths.
7. When adding or subtracting decimal numbers greater than one (1), what differences, if any, are there
between how you deal with the whole number parts and the decimal parts?
The most important thing to remember is that you must align the numbers by lining up the decimal points. In a
whole number, the decimal is implied. It is always at the end of the number. When adding or subtracting, line up
decimal points and make all numbers have the same number of decimal places by appending trailing zeros. The
addition or subtraction is then done just like with whole numbers and the decimal point is carried straight down in
the answer.
142
Activity 3.2 — Adding and Subtracting Decimal Numbers
8. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
Perform the indicated operations and validate your answers.
Problem
1)
3.92 + 5.6
2)
3.5 + 14.9 + 0.637 + 8
3)
5.4 – 3.26
4)
Subtract 6.24 from 9.
Worked Solution
Validation
143
Chapter 3 — Decimal Numbers
Worked Solution
Problem
5)
Validation
3.006 – 2.9
6) Ben went to the store and
purchased the following
items:
1 can soup
$1.89
1 box crackers
$1.79
1 box cereal
$4.29
1 gallon juice
$3.99
1 bag carrots
$2.50
a. How much will Ben pay
for his groceries?
b. How much change will
Ben get back if he gives
the cashier a fifty dollar
bill?
7. You have decided to attempt refinishing an old chest of drawers. At the hardware store you purchase a
small paint brush for $ 3.99, a few sheets of sand paper for $ 1.79, tack cloth for $ 2.49, a scraping tool for
$ 2, and a small can of primer for $ 4.50. Tax amounts to $ .89. How much is your change if you pay with
a $100 bill?
$ 3.99
1.79
2.49
2.00
4.50
.89
Total cost
144
$ 15.66
$ 100.00
-15.66
$ 84.34 is the change for the $100 bill
Activity 3.2 — Adding and Subtracting Decimal Numbers
Add or subtract as indicated. Validate your answers.
1. 29.48 + 0.037
29.517
4. 5.47 + 21 + 2.3
28.77
2. 43 + 0.562
43.562
5. 16 – 0.055
15.945
3. 9.003 – 0.05
8.953
6. 2.78 – 0.623
2.157
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) Subtract:
5.29 – 0.672
Identify Errors
or Validate
Correct Process
Validation
Validate:
1
1
4.618
+ 0.672
5.290
≈ 5.29
Correct
2) 19.28 – 4.5 =
Didn’t line up
decimal points.
145
Chapter 3 — Decimal Numbers
Worked Solution
What is Wrong Here?
Identify Errors
or Validate
3) Add:
3.76 + 12 + 2.92
The decimal point
in a whole number
is to the right of the
number.
4) 62 – 4.35 =
Need to add enough
zeros in order to
properly borrow and
subtract.
146
Correct Process
Validation