1. Changing Recurring Decimals
into
Fractions
0.833… =
… =
… =
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2. A recurring decimal can always be written as a fraction.
A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern. =
0.333…
1.1666…
0.8333… … … … 1 3 7 6 5 31 12
913
9990 2
A recurring decimal can always be written as a fraction.
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3. These recurring decimals are worth remembering.
A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern. =
0.333…
0.666…
0.1666…
0.8333… 1 3 2 6 5
These recurring decimals are worth remembering.
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4. A terminating decimal comes to an end after a number of decimal places.=
0.25
0.4
0.52
0.3875
2.2975 1 4 2 5 13 25 31 80
919
400
1511
1024
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5. THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM
It turns out that:
The number we are dividing (i.e. the numerator) plays no role into whether the decimal will be terminating or recurring.
The divisor (i.e. the denominator) is important:
If the denominator can be broken into prime factors of 2 and/or 5 only then the decimal will be terminating.
If the denominator contains any other prime factors then the decimal will be recurring.
THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM
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6. Converting Recurring Decimals
into
Fractions
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7. Convert 0.444… into a fraction
Let x =
0.444…
Since the recurring decimal has a one-digit pattern we multiply this expression by 10
10x =
4.444…
x =
0.444…
9x = 4.
... 4 9
x = 4 9
0.444… =
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8. Convert 0.363636… into a fraction
Let x = …
Since the recurring decimal has a two-digit pattern we multiply this expression by 100
100x = …
x =
0.3636…
99x =
36.
... 36 99 4 11
x = = 4 11
… =
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9. Convert 0.411411411… into a fraction
Let x = …
Since the recurring decimal has a three-digit pattern we multiply this expression by 1000
1000x = …
x = …
999x =
411.
...
411
999
137
333
x = =
137
333
… =
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10. Convert 0.3777… into a fraction
Let x =
0.3777…
Since the recurring decimal has a one-digit pattern we multiply this expression by 10
10x =
3.777…
x =
0.377…
9x = 3. 4
...
3.4 9 34 90 17 45
x = = = 17 45
0.3777… =
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11. Convert 1.01454545… into a fraction
Let x = …
Since the recurring decimal has a two-digit pattern we multiply this expression by 100
100x = …
x = …
99x =
100. 4 4
...
100.44 99
10044
9900
2511
2475
x = = =
279
275
… =
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12. Convert 2.9135135135… into a fraction
Let x = …
Since the recurring decimal has a three-digit pattern we multiply this expression by 1000
1000x = …
x = …
999x =
2910. 6
...
2910.6
999
29106
9990
539
185
x = = =
[HCF:54]
539
185
… =
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13. Convert 0.153846153846153846… into a fraction
Let x = …
Since the recurring decimal has a six-digit pattern we multiply this expression by
x = …
x = …
999999x =
...
153846
999999
17094
111111
5698
37037
518
3367 2 13
x = = = = = ÷9 ÷3
÷11
÷259 2 13
… =
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14. It is worth noting a pattern in some recurring decimals:4 9 31 99
107
999
0.444… =
… =
… = 7 9 8 99 23
999
0.777… =
… =
… = 5 9 37 99
163
999
2.555… = 2
… =
… = 1 3
This might save a bit of work when converting:
“write as a decimal” 5 11 x9 5 11 45 99 … = = x9
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15. © T Madas

16. by recognising patterns:17 33
Write as a recurring decimal
Method A 5 1 5 1 17 33
by division: 3 3 1 7
= … 5 17 5 17
Method B
by recognising patterns: x3 17 33 51 99 … = = x3
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17. © T Madas

18. by immediate recognition:
Calculate the mean of 0.6 and 0.16 giving your final answer as a recurring decimal
by immediate recognition:
Mean
add them
divide by 2 2 3
0.6
0.666… = = 1 6
0.16
0.1666… = =
Method 1
Method 2
x 2 2 3 1 6 4 6 1 6 5 6
0.6666… + +
x 2 5 12
0.1666… = = = + 2 2 2
0.8333… 1 4 1 6 6 4 1 6 6 1 2 5 2 2 8 3 3 3 8 8 8 1 1 1 5 12 5 12
= 0.416
= 0.416
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