3.3: Multiplying Decimals

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Multiplying decimal numbers involves two steps: (1) multiplying the numbers as whole numbers, ignoring the decimal point, and (2) placing the decimal point in the correct position in the product or answer.

For example, consider (0.7)(0.08), which asks us to find the product of “seven tenths” and “eight hundredths.” We could change these decimal numbers to fractions, then multiply.

\[ \begin{aligned} (0.7)(0.8) & = \frac{7}{10} \cdot \frac{8}{100} \\ & = \frac{56}{100} \\ & = 0.056 \end{aligned}\nonumber \]

The product is 56/1000, or “fifty six thousandths,” which as a decimal is written 0.056.

Important Observations

There are two very important observations to be made about the example (0.7)(0.08).

1. In fractional form

\[\frac{7}{10} \cdot \frac{8}{100} = \frac{56}{1000},\nonumber \]

note that the numerator of the product is found by taking the product of the whole numbers 7 and 8. That is, you ignore the decimal points in 0.7 and 0.08 and multiply 7 and 8 as if they were whole numbers.

2. The first factor, 0.7, has one digit to the right of the decimal point. Its fractional equivalent, 7/10, has one zero in its denominator. The second factor, 0.08, has two digits to the right of the decimal point. Its fractional equivalent, 8/100, has two zeros in its denominator. Therefore, the product 56/1000 is forced to have three zeros in its denominator and its decimal equivalent, 0.056, must therefore have three digits to the right of the decimal point.

Let’s look at another example.

Example 1

Simplify: (2.34)(1.2).

Solution

Change the decimal numbers “two and thirty four hundredths” and “one and two tenths” to fractions, then multiply.

\[ \begin{aligned} (2.34)(1.2) = 2 \frac{34}{100} \cdot 1 \frac{2}{10} ~ & \textcolor{red}{ \text{ Change decimals to fractions.}} \\ = \frac{234}{100 \cdot \frac{12}{10} ~ & \textcolor{red}{ \text{ Change mixed to improper fractions.}} \\ = \frac{2808}{1000} ~ & \textcolor{red}{ \text{ Multiply numerators and denominators.}} \\ = 2 \frac{808}{1000} ~ & \textcolor{red}{ \text{ Change to mixed fraction.}} \\ = 2.808 ~ & \textcolor{red}{ \text{ Change back to decimal form.}} \end{aligned}\nonumber \]

Exercise

Multiply: (1.86)(9.5)

Answer: 17.67

Important Observations

We make the same two observations as in the previous example.

If we treat the decimal numbers as whole numbers without decimal points, then (234)(12) = 2808, which is the numerator of the fraction 2808/1000 in the solution shown in Example 1. These are also the same digits shown in the answer 2.808.
There are two digits to the right of the decimal point in the first factor 2.34 and one digit to the right of the decimal point in the second factor 1.2. This is a total of three digits to the right of the decimal points in the factors, which is precisely the same number of digits that appear to the right of the decimal point in the answer 2.808.

The observations made at the end of the previous two examples lead us to the following method.

Multiplying Decimal Numbers

To multiply two decimal numbers, perform the following steps:

Ignore the decimal points in the factors and multiply the two factors as if they were whole numbers.
Count the number of digits to the right of the decimal point in each factor. Sum these two numbers.
Place the decimal point in the product so that the number of digits to the right of the decimal point equals the sum found in step 2.

Example 2

Use the steps outlined in Multiplying Decimal Numbers to find the product in Example 1.

Solution

We follow the steps outlined in Multiplying Decimal Numbers.

1. The first step is to multiply the factors 2.34 and 1.2 as whole numbers, ignoring the decimal points.

\[ \begin{array}{r} 234 \\ \times 12 \\ \hline 468 \\ 234 \\ \hline 2808 \end{array}\nonumber \]

2. The second step is to find the sum of the number of digits to the right of the decimal points in each factor. Note that 2.34 has two digits to the right of the decimal point, while 1.2 has one digit to the right of the decimal point. Thus, we have a total of three digits to the right of the decimal points in the factors.

3. The third and final step is to place the decimal point in the product or answer so that there are a total of three digits to the right of the decimal point. Thus,

(2.34)(1.2) = 2.808.

Note that this is precisely the same solution found in Example 1.

What follows is a convenient way to arrange your work in vertical format.

\[ \begin{array}{r} 2.34 \\ \times 1.2 \\ \hline 468 \\ 2 ~ 34 \\ \hline 2.808 \end{array}\nonumber \]

Exercise

Multiply: (5.98)(3.7)

Answer: 22.126

Example 3

Simplify: (8.235)(2.3).

Solution

We use the convenient vertical format introduced at the end of Example 2.

\[ \begin{array}{r} 8.235 \\ \times 2.3 \\ \hline 2~4705 \\ 16 ~ 470 \\ \hline 18.9405 \end{array}\nonumber \]

The factor 8.235 has three digits to the right of the decimal point; the factor 2.3 has one digit to the right of the decimal point. Therefore, there must be a total of four digits to the right of the decimal point in the product or answer.

Exercise

Multiply: (9.582)(8.6)

Answer: 82.4052

Multiplying Signed Decimal Numbers

The rules governing multiplication of signed decimal numbers are identical to the rules governing multiplication of integers.

Like Signs. The product of two decimal numbers with like signs is positive. That is:

(+)(+) = + and (−)(−) = +

Unlike Signs. The product of two decimal numbers with unlike signs is negative. That is:

(+)(−) = − and (−)(+) = −

Example 4

Simplify: (−2.22)(−1.23).

Solution

Ignore the signs to do the multiplication, then consider the signs in the final answer behind.

As each factor has two digits to the right of the decimal point, there should be a total of 4 decimals to the right of the decimal point in the product.

\[ \begin{array}{r} 2.22 \\ \times 1.23 \\ \hline \\ 666 \\ 444 \\ 1~11 \\ \hline 1.6206 \end{array}\nonumber \]

Like signs give a positive product. Hence:

(−2.22)(−1.23) = 1.6206

Exercise

Multiply: (−3.86)(−5.77)

Answer: 22.2722

Example 5

Simplify: (5.68)(−0.012).

Solution

Ignore the signs to do the multiplication, then consider the signs in the final answer below.

The first factor has two digits to the right of the decimal point, the second factor has three. Therefore, there must be a total of five digits to the right of the decimal point in the product or answer. This necessitates prepending an extra zero in front of our product.

\[ \begin{array}{r} 5.68 \\ \times 0.012 \\ \hline 1136 \\ 568 \\ \hline 0.06816 \end{array}\nonumber \]

Unlike signs give a negative product. Hence:

(5.68)(−0.012) = −0.06816

Exercise

Multiply: (9.23)(−0.018)

Answer: −0.16614

Order of Operations

The same Rules Guiding Order of Operations also apply to decimal numbers.

Rules Guiding Order of Operations

When evaluating expressions, proceed in the following order.

Evaluate expressions contained in grouping symbols first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.
Evaluate all exponents that appear in the expression.
Perform all multiplications and divisions in the order that they appear in the expression, moving left to right.
Perform all additions and subtractions in the order that they appear in the expression, moving left to right.

Powers of Ten

Consider:

\[ \begin{array}{l}10^1 = 10 \\ 10^2 = 10 \cdot 10 = 100 \\ 10^3 = 10 \cdot 10 \cdot 10 = 1,000 \\ 10^4 = 10 \cdot 10 \cdot 10 \cdot 10 = 10,000 \end{array}\nonumber \]

Note the answer for 10⁴, a one followed by four zeros! Do you see the pattern?

Powers of Ten

In the expression 10 ⁿ , the exponent matches the number of zeros in the answer. Hence, 10ⁿ will be a 1 followed by n zeros.

Example 7

Simplify: 10⁹.

Solution

10⁹ should be a 1 followed by 9 zeros. That is,

\[10^9 = 1, 000, 000, 000,\nonumber \]

or “one billion.”

Exercise

Simplify: 10⁶

Answer: 1,000,000

Multiplying Decimal Numbers by Powers of Ten

Let’s multiply 1.234567 by 10³, or equivalently, by 1,000. Ignore the decimal point and multiply the numbers as whole numbers.

\[ \begin{array}{r} 1.234567 \\ \times 1000 \\ \hline 1234.567000 \end{array}\nonumber \]

The sum total of digits to the right of the decimal point in each factor is 6. Therefore, we place the decimal point in the product so that there are six digits to the right of the decimal point.

However, the trailing zeros may be removed without changing the value of the product. That is, 1.234567 times 1000 is 1234.567. Note that the decimal point in the product is three places further to the right than in the original factor. This observation leads to the following result.

Multiplying a Decimal Number by a Power of Ten

Multiplying a decimal number by 10ⁿ will move the decimal point n places to the right.

Example 8

Simplify: 1.234567 · 10⁴

Solution

Multiplying by 104 (or equivalently, by 10,000) moves the decimal 4 places to the right. Thus, 1.234567 · 10, 000 = 12345.67.

Exercise

Simplify: 1.234567 · 10²

Answer: 123.4567

Exercises

Multiply the decimals.

1. (6.7)(0.03)

5. (4.1)(4.6)

7. (75.3)(0.4)

25. (3.02)(6.7)

29. (−9.41)(0.07)

39. (−39.3)(−0.8)

51. (1.02)(−0.2)

Multiply the decimal by the given power of 10.

57. 24.264 · 10

59. 53.867 · 10⁴

61. 5.096 · 10³

65. 61.303 · 100

Simplify the given expression.

69. (0.36)(7.4) − (−2.8)²

71. 9.4 − (−7.7)(1.2)²

75. 6.3 − 4.2(9.3)²

79. (−8.1)(9.4) − 1.8²

Answers

1. 0.201

5. 18.86

7. 30.12

25. 20.234

29. −0.6587

39. 31.44

51. −0.204

57. 242.64

59. 538670

61. 5096

65. 6130.3

69. −5.176

71. 20.488

75. −356.958

79. −79.38

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