Download Converting Repeating Decimals to Fractions

Converting Repeating
Decimals to Fractions
Lesson 2.1.3
1
Lesson
2.1.3
Converting Repeating Decimals to Fractions
California Standard:
What it means for you:
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.
You’ll see how to change
repeating decimals into fractions
that have the same value.
Key Words:
• fraction
• decimal
• repeating
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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
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Lesson
2.1.3
Converting Repeating Decimals to Fractions
Repeating Decimals Can Be “Subtracted Away”
Look at the decimal 0.33333..., or 0.3.
If you multiply it by 10, you get 3.33333..., 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.”
3.33333… – 0.33333… = 3
3.3 – 0.3 = 3
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Lesson
2.1.3
Example
Converting Repeating Decimals to Fractions
1
Find 3.3 – 0.3
Solution
The digits after the decimal point in both these numbers
are the same, since 0.3 = 0.3333… and 3.3 = 3.3333…
So when you subtract the numbers, the result has no
digits after the decimal point.
3.3333…
– 0.3333…
3.0000…
So 3.3 – 0.3 = 3.
or
3.3
– 0.3
3.0
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Solution follows…
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.
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Solution follows…
Lesson
2.1.3
Example
Converting Repeating Decimals to Fractions
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
1
x = , which can be simplified to x = .
9
3
7
Solution follows…
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
In Exercises 1–3, use x = 0.4.
1. Find 10x.
10x = 10 × 0.4 = 4.4
2. Use your answer to Exercise 1 to find 9x.
9x = 10x – x
= 4.4 – 0.4 = 4
3. Write x as a fraction in its simplest form. 9x = 4, divide both
sides by 9 to give x =
In Exercises 4–6, use y = 1.2.
4. Find 10y.
4
9
10y = 10 × 1.2 = 12.2
5. Use your answer to Exercise 4 to find 9y.
9y = 10y – y
= 12.2 – 1.2 = 11
6. Write y as a fraction in its simplest form. 9x = 11, divide both
11
sides by 9 to give x =
9
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Solution follows…
Lesson
2.1.3
Converting Repeating Decimals to Fractions
Guided Practice
Convert the numbers in Exercises 7–9 to fractions.
Let x = 2.5
7. 2.5
8. 4.1
9. –2.5
10x = 10 × 2.5 = 25.5
9x = 10x – x = 25.5 – 2.5 = 23
23
9x = 23, divide both sides by 9 to give x =
9
Let x = 4.1
10x = 10 × 4.1 = 41.1
9x = 10x – x = 41.1 – 4.1 = 37
37
9x = 37, divide both sides by 9 to give x =
9
Let x = –2.5
10x = 10 × –2.5 = –25.5
9x = 10x – x = –25.5 – –2.5 = –23
23
9x = –23, divide both sides by 9 to give x = –
9
9
Solution follows…
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.
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Lesson
2.1.3
Example
Converting Repeating Decimals to Fractions
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 = 23.23 – 0.23 = 23.
23
So 99x = 23, which means that x = .
99
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Solution follows…
Lesson
2.1.3
Example
Converting Repeating Decimals to Fractions
4
Convert 1.728 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 = 1728.728
Now subtract: 1000y – y = 1728.728 – 1.728 = 1727.
1727
So 999y = 1727, which means that y =
.
999
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Solution follows…
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.
10. 0.09
99(0.09) = 100(0.09) – 0.09
11. 0.18
= 9.09 – 0.09 = 9
1
so 0.09 =
11
99(0.18) = 100(0.18) – 0.18
12. 0.909
= 18.18 – 0.18 = 18
2
so 0.18 =
11
999(0.909) = 1000(0.909) – 0.909
= 909.909 – 0.909 = 909
101
so 0.909 =
111
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Solution follows…
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.
13. 0.123
999(0.123) = 1000(0.123) – 0.123
14. 2.12
= 123.123 – 0.123 = 123
41
so 0.123 =
333
99(2.12) = 100(2.12) – 2.12
15. 0.1234
= 212.12 – 2.12 = 210
70
so 2.12 =
33
9999(0.1234) = 10,000(0.1234) – 0.1234
= 1234.1234 – 0.1234 = 1234
1234
so 0.1234 =
9999
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Solution follows…
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.
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Lesson
Converting Repeating Decimals to Fractions
2.1.3
Example
5
Convert 3.43 to a fraction.
Solution
Call the number x.
There is one repeating digit in x, so multiply by 10.
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.
30.9
So 9x = 30.9, which means that x =
.
9
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Solution
Solution
continues…
follows…
Lesson
2.1.3
Example
Converting Repeating Decimals to Fractions
5
Convert 3.43 to a fraction.
Solution (continued)
30.9
So 9x = 30.9, which means that x =
.
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.
30.9 × 10 309
103
x=
=
, or more simply, x =
.
9 × 10
90
30
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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.
9(1.12) = 10(1.12) – 1.12
16. 1.12
17. 2.334
= 11.22 – 1.12 = 10.1
101
so 1.12 =
90
99(2.334) = 100(2.334) – 2.334
18. 0.54321
= 233.434 – 2.334 = 231.1
2311
so 2.334 =
990
999(0.54321) = 1000(0.54321) – 0.54321
= 543.21321 – 0.54321 = 542.67
18,089
so 0.54321 =
33,300
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Solution follows…
Lesson
Converting Repeating Decimals to Fractions
2.1.3
Independent Practice
Convert the numbers in Exercises 1–9 to fractions.
Give your answers in their simplest form.
1. 0.8
4. 0.26
8
9
2. 0.7
26
99
7. 0.142857
5. 4.87
1
7
7
9
3. 1.1
161
33
8. 3.142857
22
7
10
9
6. 0.246
82
333
9. 10.01
901
90
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Solution follows…
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.
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