1. Converting Repeating Decimals to Fractions
Lesson 2.1.3

2. 2.1.3 Converting Repeating Decimals to Fractions California Standard:
Lesson
2.1.3
Converting Repeating Decimals to Fractions
California Standard:
Number Sense 1.5
Know 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:
You’ll see how to change repeating decimals into fractions that have the same value.
Key Words:
fraction
decimal
repeating

3. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
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.
0.27 3 11

4. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Repeating Decimals Can Be “Subtracted Away”
Look at the decimal , or 0.3.
If you multiply it by 10, you get , or 3.3.
In both these numbers, the digits after the decimal point are the same.
So if you subtract one from the other, the decimal part of the number “disappears.”
… – … = 3
3.3 – 0.3 = 3

5. 2.1.3 Converting Repeating Decimals to Fractions Find 3.3 – 0.3
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 1
Find 3.3 – 0.3
Solution
The digits after the decimal point in both these numbers are the same, since 0.3 = … and 3.3 = …
So when you subtract the numbers, the result has no digits after the decimal point.
3.3333…
– …
3.0000…
3.3
– 0.3
3.0 or
So 3.3 – 0.3 = 3.
Solution follows…

6. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
This idea of getting repeating decimals to “disappear” by subtracting is used when you convert a repeating decimal to a fraction.
Solution follows…

7. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 2
If x = 0.3, find: (i) 10x, and (ii) 9x. Use your results to write x as a fraction in its simplest form.
Solution
(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: 3 9 1
x = , which can be simplified to x = .
Solution follows…

8. 2.1.3 Converting Repeating Decimals to Fractions Guided Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
In Exercises 1–3, use x = Find 10x 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 = Find 10y 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 4 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 11 9
9x = 11, divide both sides by 9 to give x =
Solution follows…

9. 2.1.3 Converting Repeating Decimals to Fractions Guided Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
Convert the numbers in Exercises 7–9 to fractions
8. 4.1
9. –2.5 23 9
Let x = x = 10 × 2.5 = x = 10x – x = 25.5 – 2.5 = 23 9x = 23, divide both sides by 9 to give x = 37 9
Let x = x = 10 × 4.1 = x = 10x – x = 41.1 – 4.1 = 37 9x = 37, divide both sides by 9 to give x = 23 9
Let x = –2.5 10x = 10 × –2.5 = –25.5 9x = 10x – x = –25.5 – –2.5 = –23 9x = –23, divide both sides by 9 to give x = –
Solution follows…

10. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
You May Need to Multiply by 100 or 1000 or 10,000...
If two digits are repeated forever, then multiply by 100 before subtracting.
If three digits are repeated forever, then multiply by 1000, and so on.

11. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 3
Convert 0.23 to a fraction.
Solution
Call the number x. There are two repeating digits in x, so you need to multiply by 100 before subtracting.
100x = 23.23
Now subtract: 100x – x = – 0.23 = 23. 23 99
So 99x = 23, which means that x = .
Solution follows…

12. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 4
Convert to a fraction.
Solution
Call the number y. There are three repeating digits in y, so you need to multiply by 1000 before subtracting.
1000y =
Now subtract: 1000y – y = – = 1727.
1727
999
So 999y = 1727, which means that y =
Solution follows…

13. 2.1.3 Converting Repeating Decimals to Fractions Guided Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
For Exercises 10–12, write each repeating decimal as a fraction in its simplest form.
99(0.09) = 100(0.09) – = 9.09 – 0.09 = 9 so 0.09 = 1 11
99(0.18) = 100(0.18) – = – 0.18 = 18 so 0.18 = 2 11
999(0.909) = 1000(0.909) – = – = 909 so =
101
111
Solution follows…

14. 2.1.3 Converting Repeating Decimals to Fractions Guided Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
For Exercises 13–15, write each repeating decimal as a fraction in its simplest form.
999(0.123) = 1000(0.123) – = – = 123 so = 41
333
99(2.12) = 100(2.12) – = – 2.12 = 210 so 2.12 = 70 33
9999(0.1234) = 10,000(0.1234) – = – = 1234 so =
1234
9999
Solution follows…

15. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
The Numerator and Denominator Must Be Integers
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.

16. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 5
Convert 3.43 to a fraction.
Solution
Call the number x. There is one repeating digit in x, so multiply by 10.
Using rather than 34.3 makes the subtraction easier.
10x = 34.33
Subtract as usual: 10x – x = – 3.43 = 30.9.
So 9x = 30.9, which means that x =
30.9 9
Solution continues…
Solution follows…

17. 2.1.3 Converting Repeating Decimals to Fractions
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Example 5
Convert 3.43 to a fraction.
Solution (continued)
So 9x = 30.9, which means that x =
30.9 9
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.
x = = , or more simply, x =
30.9 × 10
9 × 10
309 90
103 30

18. 2.1.3 Converting Repeating Decimals to Fractions Guided Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
For Exercises 16–18, write each repeating decimal as a fraction in its simplest form.
101 90
9(1.12) = 10(1.12) – = – 1.12 = 10.1 so 1.12 =
2311
990
99(2.334) = 100(2.334) – = – = so =
18,089
33,300
999( ) = 1000( ) – = – = so =
Solution follows…

19. 2.1.3 Converting Repeating Decimals to Fractions Independent Practice
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Independent Practice
Convert the numbers in Exercises 1–9 to fractions. Give your answers in their simplest form. 8 9 7 9 10 9 26 99
161 33 82
333 1 7 22 7
901 90
Solution follows…

20. 2.1.3 Converting Repeating Decimals to Fractions Round Up
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Round Up
This is a really handy 3-step method —
(i) multiply by 10, 100, 1000, or whatever,
(ii) subtract the original number, and
(iii) divide to form your fraction.