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Topic

# Multiplying Decimals

Topic Progress:

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.

Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.

So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side in Table. Look for a pattern.

A B
(0.3)(0.7) (0.2)(0.46)
Convert to fractions.  
Multiply.  
Convert back to decimals. 0.21 0.092

There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.

How many decimal places would you expect for the product of (0.01)(0.004)? If you said “five”, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors—in this case two plus three—to get the number of decimal places in the product—in this case five.

Once we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors.

The rules for multiplying positive and negative numbers apply to decimals, too, of course.

### Multiplying Positive and Negative Numbers

When multiplying two numbers,

• if their signs are the same, the product is positive.
• if their signs are different, the product is negative.

When you multiply signed decimals, first determine the sign of the product and then multiply as if the numbers were both positive. Finally, write the product with the appropriate sign.

### Multiplying Decimal Numbers

1. Determine the sign of the product.
2. Write the numbers in vertical format, lining up the numbers on the right.
3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.
5. Write the product with the appropriate sign.

### Multiply: (3.9)(4.075).

Solution
 (3.9)(4.075) Determine the sign of the product. The signs are the same. The product will be positive. Write the numbers in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points. Place the decimal point. Add the number of decimal places in the factors (1+3).(1+3).Place the decimal point 4 places from the right. The product is positive. (3.9)(4.075)=15.8925

27.4815

### Multiply: 10.79(8.12).

87.6148

Watch this example of how Sal Khan solves a multiplication question with decimals:

## Multiply by Powers of 10

In many fields, especially in the sciences, it is common to multiply decimals by powers of 10. Let’s see what happens when we multiply 1.9436 by some powers of 10.

Look at the results without the final zeros. Do you notice a pattern?

• 1.9436(10) = 19.436
• 1.9436(100) = 194.36
• 1.9436(10001943.6

The number of places that the decimal point moved is the same as the number of zeros in the power of ten. Table summarizes the results.

Multiply by Number of zeros Number of places decimal point moves
10 1 1 place to the right
100 2 2 places to the right
1,000 3 3 places to the right
10,000 4 4 places to the right

We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of 10 and then move the decimal point that same of places to the right.

So, for example, to multiply 45.86 by 100, move the decimal point 2 places to the right.

Sometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let’s multiply 2.4 by 100. We need to move the decimal point 2 places to the right. Since there is only one digit to the right of the decimal point, we must write a 0 in the hundredths place.

### NOTE: Multiply a Decimal by a Power of 10

1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
2. Write zeros at the end of the number as placeholders if needed.

### Multiply 5.63 by factors of ⓐ10 ⓑ100 ⓒ1000.

Solution

By looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.

 ⓐ 56.3(10) There is 1 zero in 10, so move the decimal point 1 place to the right. 56.3
 ⓑ 5.63(100) There are 2 zeros in 100, so move the decimal point 2 places to the right. 563
 ⓒ 5.63(1000) There are 3 zeros in 1000, so move the decimal point 3 places to the right. A zero must be added at the end. 5,630

25.8
258
2,580

142
1,420
14,200

### Changing from a Decimal to a Fraction

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that $5.03 means 5 dollars and 3 cents. Since there are 100 cents in one dollar, 3 cents means  of a dollar, so 0.03 We convert decimals to fractions by identifying the place value of the farthest right digit. In the decimal 0.03, the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03. 0.03 =  For our$5.03 lunch, we can write the decimal 5.03 as a mixed number.
5.03 5 

Notice that when the number to the left of the decimal is zero, we get a proper fraction. When the number to the left of the decimal is not zero, we get a mixed number.

### Exercise: Write each of the following decimal numbers as a fraction or a mixed number.

4.09  3.7  −0.286

#### Solution

 ⓐ 4.09 There is a 4 to the left of the decimal point. Write "4" as the whole number part of the mixed number. Determine the place value of the final digit. Write the fraction. Write 9 in the numerator as it is the number to the right of the decimal point. Write 100 in the denominator as the place value of the final digit, 9, is hundredth. The fraction is in simplest form.

Did you notice that the number of zeros in the denominator is the same as the number of decimal places?

 ⓑ 3.7 There is a 3 to the left of the decimal point. Write "3" as the whole number part of the mixed number. Determine the place value of the final digit. Write the fraction. Write 7 in the numerator as it is the number to the right of the decimal point. Write 10 in the denominator as the place value of the final digit, 7, is tenths. The fraction is in simplest form.
 ⓒ -.286 There is a 0 to the left of the decimal point. Write a negative sign before the fraction. Determine the place value of the final digit and write it in the denominator. Write the fraction. Write 286 in the numerator as it is the number to the right of the decimal point.Write 1,000 in the denominator as the place value of the final digit, 6, is thousandths. We remove a common factor of 2 to simplify the fraction.

### Changing from a Fraction to a Decimal

To do so we need only recall that a fraction bar can also be a division symbol. Thus,  not only means "3 objects out of 4," but can also mean "3 divided by 4."

. Divide 3 by 4.

Thus,  =0.75.

. Divide 1 by 5.

Thus,  0.

. Divide 5 by 6.

0.833⋯ We are to round to two decimal places.

Thus,  0.83 to two decimal places.

5 . Note that 5  5+

Convert  to a decimal.

 = .125.

Thus, 5  5+ 5+.125 5.125.