1 Lesson 2.1.3 Converting Repeating Decimals to Fractions Converting Repeating Decimals to Fractions

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Lesson 2.1.3Lesson 2.1.3

Converting RepeatingDecimals to FractionsConverting RepeatingDecimals to Fractions

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Lesson

2.1.3Converting Repeating Decimals to FractionsConverting Repeating Decimals to Fractions

California Standard:Number Sense 1.5Know that every rational number is either a terminating or a repeating decimal and be able to convert terminating decimals into reduced fractions.

What it means for you:

Key Words:

You’ll see how to change repeating decimals into fractions that have the same value.

• fraction• decimal• repeating

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Lesson


You’ve seen how to convert a terminating decimal into a fraction. But repeating decimals are also rational numbers, so they can be represented as fractions too.

That’s what this Lesson is all about — taking a repeating decimal and finding a fraction with the same value.

0.273

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Repeating Decimals Can Be “Subtracted Away”

Lesson


Look at the decimal 0.33333..., or 0.3.

In both these numbers, the digits after the decimal point are the same.

If you multiply it by 10, you get 3.33333..., or 3.3.

So if you subtract one from the other, the decimal part of the number “disappears.”

3.33333… – 0.33333… = 3

3.3 – 0.3 = 3

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Example 1

Solution follows…

Lesson


Solution

Find 3.3 – 0.3

The digits after the decimal point in both these numbers are the same, since 0.3 = 0.3333… and 3.3 = 3.3333…

3.3333…

– 0.3333…

3.0000…

3.3

– 0.3

3.0or

So 3.3 – 0.3 = 3.

So when you subtract the numbers, the result has no digits after the decimal point.

6Solution follows…

Lesson


This idea of getting repeating decimals to “disappear” by subtracting is used when you convert a repeating decimal to a fraction.

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Example 2

Solution follows…

Lesson


Solution

If x = 0.3, find: (i) 10x, and (ii) 9x.Use your results to write x as a fraction in its simplest form.

(i) 10x = 10 × 0.3 = 3.3.

(ii) 9x = 10x – x = 3.3 – 0.3 = 3 (from Example 1).

You now know that 9x = 3.So you can divide both sides by 9 to find x as a fraction:

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9

1

3x = , which can be simplified to x = .

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Guided Practice

Solution follows…

Lesson


In Exercises 1–3, use x = 0.4.

1. Find 10x.

2. Use your answer to Exercise 1 to find 9x.

3. Write x as a fraction in its simplest form.

In Exercises 4–6, use y = 1.2.

4. Find 10y.

5. Use your answer to Exercise 4 to find 9y.

6. Write y as a fraction in its simplest form.

10x = 10 × 0.4 = 4.4

9x = 10x – x = 4.4 – 0.4 = 4

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9

9x = 4, divide both sides by 9 to give x =

10y = 10 × 1.2 = 12.2

9y = 10y – y = 12.2 – 1.2 = 11

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9x = 11, divide both sides by 9 to give x =

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Guided Practice

Solution follows…

Lesson


Convert the numbers in Exercises 7–9 to fractions.

7. 2.5

8. 4.1

9. –2.5

23

9

Let x = 2.510x = 10 × 2.5 = 25.59x = 10x – x = 25.5 – 2.5 = 239x = 23, divide both sides by 9 to give x =

37

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Let x = 4.110x = 10 × 4.1 = 41.19x = 10x – x = 41.1 – 4.1 = 379x = 37, divide both sides by 9 to give x =

23

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Let x = –2.510x = 10 × –2.5 = –25.59x = 10x – x = –25.5 – –2.5 = –239x = –23, divide both sides by 9 to give x = –

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Lesson


If two digits are repeated forever, then multiply by 100 before subtracting.

You May Need to Multiply by 100 or 1000 or 10,000...

If three digits are repeated forever, then multiply by 1000, and so on.

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Call the number x. There are two repeating digits in x, so you need to multiply by 100 before subtracting.

Example 3

Solution follows…

Lesson


Solution

Convert 0.23 to a fraction.

100x = 23.23

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99So 99x = 23, which means that x = .

Now subtract: 100x – x = 23.23 – 0.23 = 23.

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Call the number y. There are three repeating digits in y, so you need to multiply by 1000 before subtracting.

Example 4

Solution follows…

Lesson


Solution


1727

999So 999y = 1727, which means that y = .

1000y = 1728.728

Now subtract: 1000y – y = 1728.728 – 1.728 = 1727.

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For Exercises 10–12, write each repeating decimal as a fraction in its simplest form.

10. 0.09

11. 0.18

12. 0.909

Guided Practice

Solution follows…

Lesson


999(0.909) = 1000(0.909) – 0.909

= 909.909 – 0.909 = 909

so 0.909 = 101

111

99(0.18) = 100(0.18) – 0.18

= 18.18 – 0.18 = 18

so 0.18 = 2

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99(0.09) = 100(0.09) – 0.09

= 9.09 – 0.09 = 9

so 0.09 = 1

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13. 0.123

14. 2.12

15. 0.1234

Guided Practice

Solution follows…

Lesson


999(0.123) = 1000(0.123) – 0.123

= 123.123 – 0.123 = 123

so 0.123 = 41

333

99(2.12) = 100(2.12) – 2.12

= 212.12 – 2.12 = 210

so 2.12 = 70

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9999(0.1234) = 10,000(0.1234) – 0.1234

= 1234.1234 – 0.1234 = 1234

so 0.1234 = 1234

9999

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The Numerator and Denominator Must Be Integers

Lesson


You won’t always get a whole number as the result of the subtraction.

If this happens, you may need to multiply the numerator and denominator of the fraction to make sure they are both integers.

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Example 5

Solution follows…

Lesson


Solution


Call the number x. There is one repeating digit in x, so multiply by 10.

So 9x = 30.9, which means that x = .30.9

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10x = 34.33 Using 34.33 rather than 34.3 makes the subtraction easier.

Subtract as usual: 10x – x = 34.33 – 3.43 = 30.9.

Solution continues…

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But the numerator here isn’t an integer, so multiply the numerator and denominator by 10 to get an equivalent fraction of the same value.

Example 5

Lesson


Solution (continued)


So 9x = 30.9, which means that x = .30.9

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x = = , or more simply, x = .30.9 × 10

9 × 10

309

90

103

30

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Guided Practice

Solution follows…

Lesson



16. 1.12

17. 2.334

18. 0.54321

2311

990

99(2.334) = 100(2.334) – 2.334

= 233.434 – 2.334 = 231.1

so 2.334 =

101

90

9(1.12) = 10(1.12) – 1.12

= 11.22 – 1.12 = 10.1

so 1.12 =

18,089

33,300

999(0.54321) = 1000(0.54321) – 0.54321

= 543.21321 – 0.54321 = 542.67

so 0.54321 =

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Independent Practice

Solution follows…

Lesson


Convert the numbers in Exercises 1–9 to fractions. Give your answers in their simplest form.

1. 0.8 2. 0.7 3. 1.1

4. 0.26 5. 4.87 6. 0.246

7. 0.142857 8. 3.142857 9. 10.01

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9

8

9

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9

26

99

161

33

82

333

1

7

22

7901

90

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(iii) divide to form your fraction.

Round UpRound Up

Lesson


This is a really handy 3-step method —

(ii) subtract the original number, and (i) multiply by 10, 100, 1000, or whatever,

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1 Lesson 2.1.3 Converting Repeating Decimals to Fractions Converting Repeating Decimals to Fractions