Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.

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4a Decimal Arithmetic

Adding Decimals

Recall the process used for adding decimal numbers.

Adding Decimals. To add decimal numbers, proceed as follows:

1. Place the numbers to be added in vertical format, aligning the decimal points.

2. Add the numbers as if they were whole numbers.

3. Place the decimal point in the answer in the same column as the decimal points above it.

You Try It!

EXAMPLE 1. Jane has $4.35 in her purse. Jim has $5.62 in his wallet. If Alice has $8.63 in her purse

and Joanna has $2.29. If they combine their money, what is the total?

they sum their money, what is the total?

Solution. Arrange the numbers in vertical format, aligning decimal points, then add.

$4.35 + $5.62

$9.97

Together they have $9.97, nine dollars and ninety seven cents.

Answer: $10.91

! If there are multiple decimal values, we can follow the same procedure.

Sometimes it is convenient to add in some additional zeros to make the numbers line up visually.

You Try It!

EXAMPLE 2. Find the sum: 12.2 + 8.352 + 22.44.

Add: 12.9 + 4.286 + 33.97

Solution. Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to improve the columnar alignment.

12.200

8.352 + 22.440

42.992

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Hence, 12.2 + 8.352 + 22.44 = 42.992. Answer: 51.156

!

Subtracting Decimals

As with addition, we add trailing zeros to the fractional part of the decimal numbers to help columnar alignment.

Subtracting Decimals. To subtract decimal numbers, proceed as follows:

1. Place the numbers to be subtracted in vertical format, aligning the deci- mal points.

2. Subtract the numbers as if they were whole numbers.

3. Place the decimal point in the answer in the same column as the decimal points above it.

You Try It!

EXAMPLE 3. Find the difference: 13.3 − 8.572. Subtract: 15.2 − 8.756 Solution. Arrange the numbers in vertical format, aligning their decimal

points in a column. Note that we add trailing zeros to the fractional part of 13.3 to improve columnar alignment.

13.300 − 8.572 4.728

Hence, 13.3 − 8.572 = 4.728. Answer: 6.444

!

Adding and Subtracting Signed Decimal Numbers

The rules for adding integers with like signs also apply to adding decimals with like signs.

Adding Two Decimals with Like Signs. To add two decimals with like signs, proceed as follows:

1. Add the magnitudes of the decimal numbers.

2. Prefix the common sign.

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The rules for adding integers with unlike signs also apply to adding decimals with unlike signs.

Adding Two Decimals with Unlike Signs. To add two decimals with unlike signs, proceed as follows:

1. Subtract the smaller magnitude from the larger magnitude.

2. Prefix the sign of the decimal number with the larger magnitude.

You Try It!

EXAMPLE 4. Simplify: −3.2 + (−18.95).

Simplify: −5.7 + (−83.85)

Solution. To add like signs, first add the magnitudes.

3.20 + 18.95 22.15

Prefix the common sign. Hence, −3.2 + (−18.95) = −22.15 Answer: −89.55

! Remember that subtraction means add the opposite.

You Try It!

EXAMPLE 5. Simplify: −8.567 − (−12.3).

Simplify: −2.384 − (−15.2)

Solution. Subtraction must first be changed to addition by adding the oppo- site.

−8.567 − (−12.3) = −8.567 + 12.3

We now have unlike signs. Subtract the smaller magnitude from the larger magnitude.

12.300 − 8.567 3.733

Prefix the sign of the decimal number with the larger magnitude. Hence:

−8.567 − (−12.3) = −8.567 + 12.3

= 3.733 Answer: 12.816

!

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Multiplying Decimals

When multiplying decimal numbers, use the following method.

Multiplying Decimal Numbers. To multiply two decimal numbers, perform the following steps:

1. Ignore the decimal points in the factors and multiply the two factors as if they were whole numbers.

2. Count the number of digits to the right of the decimal point in each factor. Sum these two numbers.

3. 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.

You Try It!

EXAMPLE 6. Simplify: (8.235)(2.3). Multiply: (9.582)(8.6)

Solution. Using the convenient vertical format, write

8.235 × 2.3 2 4705 16 470 18.9405

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. Answer: 82.4052

!

You Try It!

EXAMPLE 7. Simplify: (5.68)(−0.012). Multiply: (9.23)(−0.018)

Solution. Ignore the signs to do the multiplication, then consider the signs in

the final answer.

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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 neces- sitates prepending a zero in front of our product.

5.68 × 0.012 1136 568 0.06816

Unlike signs give a negative prod- uct. Hence:

(5.68)(−0.012) = −0.06816

Answer: −0.16614

Powers of Ten Consider:

10

¹

= 10

10

²

= 10 · 10 = 100 10

³

= 10 · 10 · 10 = 1, 000 10

⁴

= 10 · 10 · 10 · 10 = 10, 000

Note the answer for 10

⁴

, 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, 10

ⁿ

will be a 1 followed by n zeros.

Multiplying Decimal Numbers by Powers of Ten

Let’s multiply 1.234567 by 10

³

, or equivalently, by 1,000. Ignore the decimal point and multiply the numbers as whole numbers.

1.234567

×1000 1234.567000

The sum total of digits to the right of the decimal points in the factors 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

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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 10

ⁿ

will move the decimal point n places to the right.

You Try It!

EXAMPLE 8. Simplify: 1.234567 · 10

⁴

Simplify: 1.234567 · 10

²

Solution. Multiplying by 10

⁴

(or equivalently, by 10,000) moves the decimal

4 places to the right. Thus, 1.234567 · 10, 000 = 12345.67. Answer: 123.4567

!

Dividing Decimals

We begin by reviewing the division algorithm for dividing a decimal number by a whole number.

Dividing a Decimal by a Whole Number. To divide a decimal number by a whole number, proceed as follows:

1. Set up the long division as you would the division of two whole numbers.

2. Perform the division as if the numbers were both whole numbers, adding zeros to the right of the decimal point in the dividend as necessary to complete the division.

3. Place the decimal point in the quotient immediately above the decimal point in the dividend.

You Try It!

EXAMPLE 9. Divide: 155.2 ÷ 25. Divide: 42.55 ÷ 23

Solution. Use long division and arrange as if dividing whole numbers.

6.2 25)155.2 150

5 2

5 0

2

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We still have a nonzero remainder. Adding one zero does no good.

6.20 25)155.20

150 5 2 5 0 20

However, if we add tw zeros, the division completes with a zero remainder.

6.208 25)155.200

150 5 2 5 0

200 200 0 Thus, 155.2 divided by 25 is 6.208.

Answer: 1.85

! If you are dividing a decimal number by another decimal number, then there are additional steps that must be included in the division algorithm.

Dividing by a Decimal Divisor. If the divisor contains a decimal, proceed as follows:

1. Move the decimal to the end of the divisor.

2. Move the decimal in the dividend an equal number of places.

You Try It!

EXAMPLE 10. Divide: −0.03 ÷ 0.024.

Divide: −0.0113 ÷ 0.05

Solution. First, divide the magnitudes. Move the decimal in the divisor to the end of the divisor. Move the decimal in the dividend an equal number of places (three places) to the right. Note that this requires writing a trailing zero in the dividend.

0.024 )0.030

Our problem then becomes:

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24)30

We can now follow the algorithm for dividing a decimal number by a whole number. Note that we have to add two trailing zeros in the dividend to complete the division with a zero remainder.

1.25 24)30.00

24 6 0 4 8 1 20 1 20 0

Finally, because the quotient of unlike signs is negative, −0.03 divided by 0.024 is −1.25. That is,