Multiplying and Dividing Decimals
Multiplying Decimals:
Multiplying decimals is much like multiplying whole numbers, except a decimal must be
inserted into the final answer. We must decide where to place the decimal in the answer. To
decide where to place the decimal in the answer, we must count the number of total places
behind the decimals in the multiplicands (the numbers being multiplied). The answer will equal
the total number of places behind the answer in the product (answer).
Examples:
1.) Multiply: 12.4 × 1.5
12.4
× 1.5
Multiply as if the numbers were whole numbers
124
× 15
620
1240
1860
12.4
x 1.5
620
1240
1860
There are two total numbers behind the decimal points (two in the first number + one in the
second number). So, there will be three total decimal places in the final answer.
12.4
x 1.5
620
1240
18.60 Count from right to left.
or
18.6
2.) Find the product of: 9.73 × 6.4
9.73 Multiply as if the numbers are whole numbers.
× 6.4
3892
58380
62272
9.73
× 6.4
3892
58380
62.272 (There are three total places behind the decimal).
Using Order of Operations with Decimals
Simplify: 429 – 32.6 × 3.2
= 429 – 104.32 The multiplication must be done before the subtraction.
Now subtract.
8 9 10
4 2 9. 0 0 Fill in zeros, line up the decimal points, and regroup.
1 0 4. 3 2
3 2 4. 6 8 Final answer
Dividing Decimals:
I. Changing a fraction to a decimal.
To change a fraction to a decimal, we must use long division. The denominator is divided into
the numerator. You must add a decimal and zeros to the dividend. The decimal point is placed in
the quotient (answer) directly above the decimal point in the dividend. If the remainder is not
zero, keep adding zeros until the division comes out with a remainder of zero OR round as
indicated in the instructions.
Examples:
11
1.) Change to a decimal.
20
Divisor Dividend Quotient
20 11
Put the decimal in the answer above the decimal in the divisor.
20 11.0 Ask: How many times does 20 go into 110? The answer is 5. Place the 8 behind
the decimal, above the zero.
0.5
20 11.0
Now multiply back out. 5 × 20 = 100. Then subtract.
0.5
20 11.0
100
10
Add another zero.
0.55
20 11.00 The remainder is zero, so the problem is completed.
100
100
100
0
Final Answer: .55
25
2.) Change to a decimal.
4
4 25
6
4 25 Since the division does not come out evenly, add a decimal point and zero after the 25
and continue dividing.
6.
4 25.0
24 Since the remainder is not zero, add another zero in the dividend.
10
6.2
4 25.0
24
10
8
2
6.25
4 25.00
24
10
8 The remainder is now zero, so the problem is completed.
20
20
0
The final answer is: 6.25
1
3.) Change 7 to a decimal.
4
1 29
7 = Now divide as in example 3 above.
4 4
4 29
7
4 29
28 The remainder is not zero, so add another zero to the dividend.
1
7.2
4 29.0
28
10
8
7.2
4 29.0
28
10 The remainder is not zero, so add another zero to the dividend.
8
2
7.25
4 29.00
28
10
8 The remainder is now zero so the problem is completed.
20
20
0
Answer: 7.25
Sometimes the remainder will never come out to be zero. In these cases, you will be told to
round to a particular place value. When this occurs, you should carry the division out one more
place than what you will be rounding to. For example, if you are to round to the hundredths
place, divide until you have a digit in the thousandths place.
Examples:
5
1.) Convert to a decimal. Round to the nearest thousandths place.
12
(Note: Since we must round to the hundredths place, we will carry the answer out 1 place beyond
the hundredths place.)
12 5
12 5.0
.4
12 5.0
48 The remainder is not zero, so add another zero to the dividend.
2
.41
12 5.00
4 8 The remainder is not zero, so add another zero to the dividend.
20
12
.416
12 5.000
4 8
20
1 2
80
72
8
.4166
12 5.0000
4 8
20
12
80
72
80
The remainder is still not zero. The remainder continues to be 8. This is a repeating decimal.
We need our answer rounded to the nearest thousandths place, so we are ready to round.
0.4166 rounded to the nearest thousandths place is 0.417
Final answer: 0.417.
II. Dividing Decimals when the Divisor Contains a Decimal Point.
To Divide when the Divisor Contains a Decimal Point:
1.) Move the decimal point in the divisor to the right end of the number.
2.) Move the decimal point in the dividend the same number of places to the right as the decimal
point was moved in the divisor.
3.) Insert a decimal point in the answer (quotient) directly above the decimal point in the
dividend.
4.) Divide similar to how whole numbers are divided. Add zeros if necessary in the dividend.
Examples:
1.) Divide 0.544 0.68
.68 .544 The decimal point must be moved to the end of the divisor.
.68 .544 Move the decimal two places to the right in the divisor and the dividend.
0.
68 54.4 Place a decimal point in the answer (quotient) directly above the decimal point in
the dividend.
0.8
68 5.44
5 44 Divide 68 into 54.4.
0
The remainder is zero, so the problem is completed.
Answer: 0.8 or .8
2.) Find the quotient: 0.0558 0.006
0.006 0.0558 The decimal point must be moved to the end of the divisor.
0.006 0.0558 Move the decimal three places to the right in the divisor and the dividend.
Note: If there were not three decimal places in the dividend, we would add zeros as place
holders.
6. 55.8 The decimal has now been moved.
or
6 55.8
Now, divide just like whole numbers.
9
6 55.8
54
1
Place the decimal in the quotient (answer) directly above the decimal in
the dividend.
9.
6 55.8
54
18
9.3
6 55.8
54
18
18
0
The final answer is 9.3.
3.) Divide: 12 0.03
0.03 12 The decimal point must be moved to the end of the divisor.
0.03 1200 Move the decimal two places to the right in the divisor and the dividend.
Since there are not two decimal places in the dividend, we must add zeros
as place holders.
3 1200 The decimal has now been moved. Now divide. Just like whole numbers.
4
3 12 00
12
00
40
3 12 00
12
00
0
00
40 0
3 12 00
12
00
0
00
0
0
The remainder is zero, so the problem is completed.
The final answer is 400
III. Notes for Solving Word Problems-Decimals
Step 1: Read the problem through completely. Look for key words.
Step 2: Write the number sentence from the problem.
Step 3: Solve the problem.
1. Elliot Jones has 103.4 pounds of candy to place in boxes. Each box holds 4.7 pounds of
candy. How many boxes will Elliot need to hold the candy?
Here we need to divide. Elliot is dividing the candy into boxes.
Numeric Problem: 103.4 ÷ 4.7 Answer: 22
2. A bookstore ordered 175 copies of a textbook at a cost of $58.25 each. What is the total
cost of the textbooks?
Numeric Problem: 175 × $58.25 Answer: $10,193.75
3. The total ticket receipts for a play were $2362.50. If the tickets were $4.50 each, how
many tickets were sold?
Numeric Problem: $2362.50 ÷ $4.50 Answer: 525
Elliot Jones has 245 pounds of candy to place in boxes. Each box holds 17.5 pounds of candy.
How many boxes will Elliot need to hold the candy?
Here we need to divide. Elliot is dividing up the candy into boxes.
17.5 245
175. 2450. Move the decimal one place to the right in the divisor and the dividend.
1
175. 2450.
- 175
7 00
14
175. 2450.
- 175
700
- 700
0 The remainder is now zero, so the problem is completed.
The answer: Elliot will need 14 boxes for the candy.
Practice:
1.) 125.8 13.6 2.) 106.8/17.56 (Round to the hundredths place)
3.) The Spanish club collected $13.45 from each of its nine members. How much did the Spanish
club collect?
4.) Henry earned $596.45 for working 32 hours. What is Henry’s hourly wage? Round to the
nearest cent.
Answers:
1.) 9.25 2.) 6.08 3.) $121.05 4.) $18.64