1Divide a decimal by a whole number.

2Divide a decimal by a decimal.

3Round a decimal quotient.

4Estimate quotients of decimals.

5Divide decimals by powers of 10.

6Divide signed decimals.

7Use the order of operations rule.

8Evaluate formulas.

9Solve application problems by dividing decimals.

Dividing Decimals

In Chapter 1, we used a process called long division to divide whole numbers.

Long division form

In this section, we consider division problems in which the divisor, the dividend, or both are decimals.

1 Divide a decimal by a whole number.

To develop a rule for decimal division, let’s consider the problem 47 10. If we rewrite the division as 4710 , we can use the long division method from Chapter 3 for changing an improper fraction to a mixed number to find the answer:

divisor

40 7

To perform this same division using decimals, we write 47 as 47.0 and divide as we would divide whole numbers.

Self Check 1

Divide: 20.8 4. Check the result.

Now Try Problem 15

Dividing a Decimal by a Whole Number

To divide a decimal by a whole number:

1.Write the problem in long division form and place a decimal point in the quotient (answer) directly above the decimal point in the dividend.

2.Divide as if working with whole numbers.

3.If necessary, additional zeros can be written to the right of the last digit of the dividend to continue the division.

Divide: 42.6 6. Check the result.

Strategy Since the divisor, 6, is a whole number, we will write the problem in long division form and place a decimal point directly above the decimal point in 42.6. Then we will divide as if the problem was 426 6.

Self Check 2

Divide: 101.44 32

Now Try Problem 19

Self Check 3

Divide: 42.8 8

Now Try Problem 23

Division with a Decimal Divisor

To divide with a decimal divisor:

1.Write the problem in long division form.

2.Move the decimal point of the divisor so that it becomes a whole number.

3.Move the decimal point of the dividend the same number of places to the right.

4.Write the decimal point in the quotient (answer) directly above the decimal point in the dividend. Divide as if working with whole numbers.

5.If necessary, additional zeros can be written to the right of the last digit of the dividend to continue the division.

Strategy We will move the decimal point of the divisor, 0.36, two places to the right and we will move the decimal point of the dividend, 0.2592, the same number of places to the right.

WHY We can then use the rule for dividing a decimal by a whole number.

Solution We begin by writing the problem in long division form.

.

0 36 0 25 . 92 Move the decimal point two places to the right in the divisor and

Since the divisor is now a whole number, we can use the rule for dividing a decimal by a whole number to find the quotient.

0.72

36 25.92 Now divide as with whole numbers.

25 2

72

72

0

Check:

0.7236

432

2160

Self Check 4

Divide:

0.6045

0.65

Now Try Problem 27

Self Check 5

Divide: 12.82 0.9. Round the quotient to the nearest hundredth.

Now Try Problem 33

3 Round a decimal quotient.

In Example 4, the division process stopped after we obtained a 0 from the second subtraction. Sometimes when we divide, the subtractions never give a zero remainder, and the division process continues forever. In such cases, we can round the result.

Divide: 9.350.7 . Round the quotient to the nearest hundredth.

Strategy We will use the methods of this section to divide to the thousandths column.

WHY To round to the hundredths column, we need to continue the division process for one more decimal place, which is the thousandths column.

We begin by writing the problem in long division form.

.

to the right. Do the same for the dividend. Place the decimal point in the quotient (answer) directly above the decimal point in the dividend.

2 5

2 1

4035

5049 1

Since the test digit 7 is 5 or greater, we will round 13.357 up to approximate the quotient to the nearest hundredth.

9.35

0.7 13.36 Read as “is approximately equal to.”

Check:

13.36The approximation of the quotient

0.7 The original divisor

Self Check 6

Estimate the quotient: 6,229.249 68.9

Now Try Problems 35 and 39

Self Check 7

Find each quotient:

a. 721.3 100 1.07

b.1,000

c.19.4407 0.0001

Now Try Problems 43 and 49

Self Check 8

Divide:

a.100.624 15.2

23.9

b.0.1

Note that the quotients, 0.813 and 81.3, and the dividend, 8.13, are the same except for the location of the decimal points. The first quotient, 0.813, can be easily obtained by moving the decimal point of the dividend one place to the left.The second quotient, 81.3, is easily obtained by moving the decimal point of the dividend one place to the right. These observations illustrate the following rules for dividing a decimal by a power of 10.

Dividing a Decimal by 10, 100, 1,000, and So On

To find the quotient of a decimal and 10, 100, 1,000, and so on, move the decimal point to the left the same number of places as there are zeros in the power of 10.

Dividing a Decimal by 0.1, 0.01, 0.001, and So On

To find the quotient of a decimal and 0.1, 0.01, 0.001, and so on, move the decimal point to the right the same number of decimal places as there are in the power of 10.

Strategy We will identify the divisor in each division. If it is a power of 10 greater than 1, we will count the number of zeros that it has. If it is a power of 10 less than 1, we will count the number of decimal places that it has.

WHY Then we will know how many places to the right or left to move the decimal point in the dividend to find the quotient.

Solution

a. 16.74 10 1.674 Since the divisor 10 has one zero, move the decimal point one

placeto the left.

b. 8.6 10,000 .00086 Since the divisor 10,000 has four zeros, move the decimal

point in 290.623 two places to the right.

6 Divide signed decimals.

The rules for dividing integers also hold for dividing signed decimals. The quotient of two decimals with like signs is positive, and the quotient of two decimals with unlike signs is negative.

Strategy In part a, we will use the rule for dividing signed decimals that have different (unlike) signs. In part b, we will use the rule for dividing signed decimals that have the same (like) signs.

163

Move the decimal point in the divisor and the dividend one place to the right.

Write the decimal point in the quotient (answer) directly above the decimal point in the dividend.

Divide as if working with whole numbers.

38.6770.1

386.77

Since the divisor 0.1 has one decimal place, move the decimal point in 38.677 one place to the right. Since the dividend and divisor have like signs, the quotient is positive.

Self Check 9

2.7756 3( 0.63)

Evaluate:

0.4 1.2

Now Try Problem 59

Self Check 10

Evaluate the formula l Aw for A 5.511 and w 1.002.

Now Try Problem 63

Self Check 11

FRUIT CAKES A 9-inch-long fruitcake loaf is cut into 0.25-inch- thick slices. How many slices are there in one fruitcake?

Now Try Problem 95

8 Evaluate formulas.

Strategy In the given formula, we will replace the letter A with 15.36 and h with 6.4.

WHY Then we can use the order of operations rule to find the value of the expression on the right side of the symbol.

9 Solve application problems by dividing decimals.

Recall that application problems that involve forming equal-sized groups can be solved by division.

are there in one loaf?

Analyze

Form Cutting a loaf of French bread into equally thick slices indicates division.We translate the words of the problem to numbers and symbols.

Solve When we write 25 0.625 in long division form, we see that the divisor is a decimal.

0.625 25.000 To write the divisor as a whole number, move the decimal

Self Check 12

U.S. NATIONAL PARKS Use the following data to determine the average number of visitors per year to the national parks for the years 2004 through 2008. (Source: National Park Service)

2008 2.749

2007 2.756

2006 2.726

2005 2.735

2004 2.769

Now Try Problem 103