The document provides steps for converting recurring decimals to fractions:
1. Let x equal the recurring decimal and multiply it by an appropriate power of 10 so that the recurring part is at the front.
2. Subtract the multiplied decimal from the original to isolate the recurring part.
3. Divide the recurring part by the result of the subtraction to obtain the fraction.
Several examples are worked through demonstrating this process.
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Recurring decimals
1. How to start…
Write the following in full to 9 decimal places…
a) 0.6
b) 0.53
c) 0.72
d) 0.479
e) 0.328
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●
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2. How to start…
Write the following in full to 9 decimal places…
a) 0.6
b) 0.53
c) 0.72
d) 0.479
e) 0.328
●
●
● ●
● ●
● ●
→ 0.666666667
→ 0.533333333
→ 0.727272727
→ 0.479479479
→ 0.328282828
3. How do we convert
recurring decimals to
fractions?
10. Fraction to decimal
Sometimes the decimal will be a recurring decimal
3 1. 0 0 0
0.3 3 3
e.g.
1
3
= 1 ÷ 3 = Because the same
division is repeating
over and over we can
say it is recurring. The
digits which recur are
marked with a dot
above them.
11. Fraction to decimal
Sometimes the decimal will be a recurring decimal
3 1. 0 0 0
0.3 3 3
e.g.
1
3
= 1 ÷ 3 = Because the same
division is repeating
over and over we can
say it is recurring. The
digits which recur are
marked with a dot
above them.
1
3
= 0. 3
37. Let’s change 0.7 into a fraction
The algebra of
recurring decimals
38. Let x= 0.7 so
. .
Let’s change 0.7 into a fraction
The algebra of
recurring decimals
39. Let x= 0.7 so 10x = 7.7
. .
Let’s change 0.7 into a fraction
Multiply!
The algebra of
recurring decimals
40. Let x= 0.7 so 10x = 7.7
. .
10x= 7.7777777777777777...
x= 0.7777777777777777...
Let’s change 0.7 into a fraction
Multiply!
Subtract!
The algebra of
recurring decimals
41. Let x= 0.7 so 10x = 7.7
. .
10x= 7.7777777777777777...
x= 0.7777777777777777...
9x=7
Let’s change 0.7 into a fraction
Multiply!
Subtract!
The algebra of
recurring decimals
42. Let x= 0.7 so 10x = 7.7
. .
10x= 7.7777777777777777...
x= 0.7777777777777777...
9x=
x= 7
9
Let’s change 0.7 into a fraction
Multiply!
Subtract!
Divide!
The algebra of
recurring decimals
7
43. Change 0.47 into a fraction
. .
The algebra of
recurring decimals
44. Change 0.47 into a fraction
. .
Let x= 0.47
. .
The algebra of
recurring decimals
45. Change 0.47 into a fraction
. .
Let x= 0.47
. .
Multiply!
The algebra of
recurring decimals
With what
this time?
46. Change 0.47 into a fraction
. .
Let x= 0.47
. .
Multiply!
The algebra of
recurring decimals
With 100
47. Change 0.47 into a fraction
. .
Let x= 0.47
. .
so 100x = 47.47
. .
Multiply!
The algebra of
recurring decimals
48. Change 0.47 into a fraction
. .
Let x= 0.47
100x= 47.4747474747474747...
x= 0.4747474747474747...
. .
so 100x = 47.47
. .
Multiply!
Subtract!
The algebra of
recurring decimals
49. Change 0.47 into a fraction
. .
Let x= 0.47
100x= 47.4747474747474747...
x= 0.4747474747474747...
. .
so 100x = 47.47
. .
Multiply!
Subtract!
The algebra of
recurring decimals
50. Change 0.47 into a fraction
. .
Let x= 0.47
100x= 47.4747474747474747...
x= 0.4747474747474747...
99x= 47
. .
so 100x = 47.47
. .
Multiply!
Subtract!
Divide!
The algebra of
recurring decimals
51. Change 0.47 into a fraction
. .
Let x= 0.47
100x= 47.4747474747474747...
x= 0.4747474747474747...
99x=
x= 47
99
. .
so 100x = 47.47
. .
Multiply!
Subtract!
Divide!
The algebra of
recurring decimals
47
52. Steps:
1. Write the recurring decimal out to nine decimal places
2. Multiply the decimal by powers of 10 until the part of the
decimal that repeats is in full at the front
3. Write the amount that you multiplied by ‘𝑥’ = the result from
above
4. Write ‘𝑥 = ′ the original decimal
5. Subtract the two
6. Divide to get 𝑥 =
The algebra of
recurring decimals
66. Change 0.213 into a fraction
. .
Let x= 0.213
. .
so 1000x = 213.213
. .
Recurring
Decimals
67. Change 0.213 into a fraction
. .
Let x= 0.213
. .
so 1000x = 213.213
. .
Multiply!
Recurring
Decimals
68. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
. .
so 1000x = 213.213
. .
Multiply!
Recurring
Decimals
69. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
. .
so 1000x = 213.213
. .
Multiply!
Subtract!
Recurring
Decimals
70. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
. .
so 1000x = 213.213
. .
Multiply!
Subtract!
Recurring
Decimals
71. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
999x=
. .
so 1000x = 213.213
. .
Multiply!
Subtract!
213
Recurring
Decimals
72. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
999x=
. .
so 1000x = 213.213
. .
Multiply!
Subtract!
Divide!
213
Recurring
Decimals
73. Change 0.213 into a fraction
. .
Let x= 0.213
1000x= 213.213213213213213...
x= 0.213213213213213...
999x=
x= 213
999
. .
so 1000x = 213.213
. .
Multiply!
Subtract!
Divide!
213
Recurring
Decimals
74. Change these recurring decimals into fractions.
Recurring
Decimals
1. 0. 45
2. 0. 235
3. 0.28
4. 0.213 (This is different from the previous example.)
(= 0.235235235….)