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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply decimals. By the end of the module students should understand the method used for multiplying decimals, be able to multiply decimals, be able to simplify a multiplication of a decimal by a power of 10 and understand how to use the word "of" in multiplication.

Section overview

  • The Logic Behind the Method
  • The Method of Multiplying Decimals
  • Calculators
  • Multiplying Decimals By Powers of 10
  • Multiplication in Terms of “Of”

The logic behind the method

Consider the product of 3.2 and 1.46. Changing each decimal to a fraction, we have

( 3 . 2 ) ( 1 . 46 ) = 3 2 10 1 46 100 = 32 10 146 100 = 32 146 10 100 = 4672 1000 = 4 672 1000 = four and six hundred seventy-two thousandths = 4 . 672

Thus, ( 3 . 2 ) ( 1 . 46 ) = 4 . 672 size 12{ \( 3 "." 2 \) \( 1 "." "46" \) =" 4" "." "672"} {} .

Notice that the factor

3.2 has 1 decimal place, 1.46 has 2 decimal places, and the product 4.672 has 3 decimal places. 1 + 2 = 3

Using this observation, we can suggest that the sum of the number of decimal places in the factors equals the number of decimal places in the product.

Vertical multiplication. 1.46 times 3.2. The first round of multiplication yields a first partial product of 292. The second round yields a second partial product of 438, aligned in the tens column. Take note that 2 decimal places in the first factor and 1 decimal place in the second factor sums to a total of three decimal places in the product. The final product is 4.672.

The method of multiplying decimals

Method of multiplying decimals

To multiply decimals,
  1. Multiply the numbers as if they were whole numbers.
  2. Find the sum of the number of decimal places in the factors.
  3. The number of decimal places in the product is the sum found in step 2.

Sample set a

Find the following products.

6 . 5 4 . 3 size 12{6 "." "5 " cdot " 4" "." 3} {}

Vertical multiplication. 6.5 times 4.3. The first round of multiplication yields a first partial product of 195. The second round yields a second partial product of 260, aligned in the tens column. Take note that 1 decimal place in the first factor and 1 decimal place in the second factor sums to a total of two decimal places in the product. The final product is 27.95.

Thus, 6 . 5 4 . 3 = 27 . 95 size 12{6 "." 5 cdot 4 "." 3="27" "." "95"} {} .

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23 . 4 1 . 96 size 12{"23" "." 4 cdot 1 "." "96"} {}

Vertical multiplication. 23.4 times 1.96. The first round of multiplication yields a first partial product of 1404. The second round yields a second partial product of 2106, aligned in the tens column. The third round yields a third partial product of 234, aligned in the hundred column. Take note that 1 decimal place in the first factor and 2 decimal places in the second factor sums to a total of three decimal places in the product. The final product is 45.864.

Thus, 23 . 4 1 . 96 = 45 . 864 size 12{"23" "." 4 cdot 1 "." "96"="45" "." "864"} {} .

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Find the product of 0.251 and 0.00113 and round to three decimal places.

Vertical multiplication. 0.251 times 0.00113. The first round of multiplication yields a first partial product of 753. The second round yields a second partial product of 251, aligned in the tens column. The third round yields a third partial product of 251, aligned in the hundred column. Take note that 3 decimal places in the first factor and 5 decimal places in the second factor sums to a total of eight decimal places in the product. The final product is 0.00028363.

Now, rounding to three decimal places, we get

0.251 times 0.00113 = 0.000, if the product is rounded to three decimal places.

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Practice set a

Find the following products.

5 . 3 8 . 6 size 12{5 "." 3 cdot " 8" "." 6} {}

45.58

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2 . 12 4 . 9 size 12{2 "." "12" cdot " 4" "." 9} {}

10.388

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1 . 054 0 . 16 size 12{1 "." "054 " cdot " 0" "." "16"} {}

0.16864

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0 . 00031 0 . 002 size 12{0 "." "00031 " cdot " 0" "." "002"} {}

0.00000062

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Find the product of 2.33 and 4.01 and round to one decimal place.

9.3

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10 5 . 394 size 12{"10 " cdot " 5" "." "394"} {}

53.94

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100 5 . 394 size 12{"100 " cdot " 5" "." "394"} {}

539.4

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1000 5 . 394 size 12{"1000" cdot " 5" "." "394"} {}

5,394

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10,000 5 . 394 size 12{"10,000 " cdot " 5" "." "394"} {}

59,340

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Calculators

Calculators can be used to find products of decimal numbers. However, a calculator that has only an eight-digit display may not be able to handle numbers or products that result in more than eight digits. But there are plenty of inexpensive ($50 - $75) calculators with more than eight-digit displays.

Sample set b

Find the following products, if possible, using a calculator.

2 . 58 8 . 61 size 12{2 "." "58 " cdot " 8" "." "61"} {}

Display Reads
Type 2.58 2.58
Press × 2.58
Type 8.61 8.61
Press = 22.2138

The product is 22.2138.

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0 . 006 0 . 0042 size 12{0 "." "006 " cdot " 0" "." "0042"} {}

Display Reads
Type .006 .006
Press × .006
Type .0042 0.0042
Press = 0.0000252

We know that there will be seven decimal places in the product (since 3 + 4 = 7 size 12{"3 "+" 4 "=" 7"} {} ). Since the display shows 7 decimal places, we can assume the product is correct. Thus, the product is 0.0000252.

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0 . 0026 0 . 11976 size 12{0 "." "0026 " cdot " 0" "." "11976"} {}

Since we expect 4 + 5 = 9 size 12{"4 "+" 5 "=" 9"} {} decimal places in the product, we know that an eight-digit display calculator will not be able to provide us with the exact value. To obtain the exact value, we must use "hand technology." Suppose, however, that we agree to round off this product to three decimal places. We then need only four decimal places on the display.

Display Reads
Type .0026 .0026
Press × .0026
Type .11976 0.11976
Press = 0.0003114

Rounding 0.0003114 to three decimal places we get 0.000. Thus, 0 . 0026 0 . 11976 = 0 . 000 size 12{0 "." "0026 " cdot " 0" "." "11976 "=" 0" "." "000"} {} to three decimal places.

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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