S k i l l
i n
A R I T H M E T I C

FRACTIONS INTO DECIMALS

Lesson 23 Section 2

In the previous section, we saw the most frequent
and therefore the most important decimal and percent equivalents. Nevertheless, we now ask the following:

3.	What is a general method for changing a fraction to a decimal?
4 11

	Divide the numerator by the denominator.

Example 1. Write

4
11

as a decimal.

Solution.

4
11

= 4 ÷ 11. As they said in the Little Red School House,

"Let 4 fall into the house"

11 does not go into 4. Write 0 in the quotient, place a decimal point,

and add a 0 onto the dividend. (Lesson 11)

"11 goes into 40 three (3) times (33) with 7 left over.

"11 goes into 70 six (6) times (66) with 4 left over."

Since we are dividing 11 into 40 again, we see that this division will never be exact. We will have 36 repeated as a pattern:

4
11

0.363636. . .

By writing three dots (called ellipsis), we mean, "A decimal for

4
11

will never be complete or exact. However we can approximate it with as many decimal digits as we please according to the indicated pattern; and

the more decimal digits we write, the closer we will be to

4
11

You sometimes hear that when we write

0.363636. . .,

it means that the decimal goes on forever. But if the decimal really went on forever (whatever that means), then it would not be a number. Why not? Because numbers have names. It is not that we will never finish naming a so-called infinite decimal. We cannot even begin

Thus we cannot express

4
11

exactly as a decimal. Therefore if we want

to use that number as a decimal, we must approximate it. Let us approximate it with three decimal digits (Lesson 11):

4
11

0.364

Example 2. Write 5

4
11

as a decimal.

Answer. According to the previous problem,

4
11

5.364

We say then that any decimal for

4
11

is inexact. But the decimal for ¼,

for example, which is .25, is exact.

The decimal .363636 in and of itself is exact. But
as a value for	4 11	, it is inexact.

Fractions, then, when expressed as decimals, will be either exact or inexact. Inexact decimals nevertheless exhibit a pattern of digits. The

pattern for

4
11

is .3636363636.

Which fractions will have exact decimals? Only those in which the factors of the denominator are made up of 2's and/or 5's. They are fractions with these denominators: 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, and so on.

Example 3.

a) Show the decimal pattern that

1
9

generates.

9 goes into 1 zero (0).

9 goes into 10 one (1) time with 1 left over.

Again, 9 goes into 10 one (1) time with 1 left over.

And so on. This division will never be exact -- we will keep getting 1's in the quotient.

1 9	= 0.111111. . .

b) Use that value for

1
9

to find the value of

8
9

Solution.

8
9

= 8 ×

1
9

= 8 × 0.111111. . . = 0.8888888. . .

c) Round off

8
9

to three decimal digits.

8
9

0.889.

See Problem 15 at the end the Lesson.

Example 4. Calculator problem. Write as a decimal:

73
96

Answer. Divide 73 by 96. Press

Displayed is

0.7604166

Therefore, to three decimal digits,

73
96

.760

Example 5. In a class of 52 students, 29 were women.

a) What fraction were women?

Answer. Since 29 out of 52 were women, then

29
52

were women.

(Lesson 19, Question 10.)

b) Use a calculator to express that fraction as a decimal.

Answer. Press

See

0.5576923

This is approximately .558.

c) What percent were women?

Answer. To change a number to a percent, multiply it by 100.

.558 = 55.8%

(Lesson 3.)

In summary, look at what we have done:

29 out of 52

29
52

29 ÷ 52

.558

55.8%

"Out of," with a calculator, always signifies division. Division of a smaller number by a larger.

We will return to this in Lesson 30.

Please "turn" the page and do some Problems.

Continue on to the next Lesson.

Section 1 of this Lesson

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