5
6
5
33
313
1998
© T Madas
A recurring decimal is a never ending decimal,
whose decimal part repeats with a pattern.
1 = 0.333…
3
7 = 1.1666…
6
5 =
0.8333…
6
A recurring decimal can always
be written as a fraction.
31 =
2.58333…
12
913 = 0.0913913913…
9990
2 = 0.285714285714285714…
7
© T Madas
A recurring decimal is a never ending decimal,
whose decimal part repeats with a pattern.
1 = 0.333…
3
2 = 0.666…
3
1 =
0.1666…
6
These recurring decimals are
worth remembering.
5 =
0.8333…
6
© T Madas
A terminating decimal comes to an end after a
number of decimal places.
1 = 0.25
4
2 = 0.4
5
13 =
0.52
25
31 =
0.3875
80
919 = 2.2975
400
1511 = 1.4755859375
1024
© T Madas
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
© T Madas
© T Madas
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 = 4.000...
x= 4
9
4
0.444… = 9
© T Madas
Convert 0.363636… into a fraction
Let x = 0.363636…
Since the recurring decimal has a two-digit
pattern we multiply this expression by 100
100x = 36.3636…
x=
0.3636…
99x = 36.0000...
36 = 4
x=
99 11
4
0.363636… = 11
© T Madas
Convert 0.411411411… into a fraction
Let x = 0.411411411…
Since the recurring decimal has a three-digit
pattern we multiply this expression by 1000
1000x = 411.411411…
x=
0.411411…
999x = 411. 000000...
411 = 137
x=
999 333
137
0.411411411… = 333
© T Madas
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 = 3.400...
3.4 = 34 = 17
x=
90 45
9
17
0.3777… = 45
© T Madas
Convert 1.01454545… into a fraction
Let x = 1.01454545…
Since the recurring decimal has a two-digit
pattern we multiply this expression by 100
100x = 101.454545…
x=
1.014545…
99x = 100.440000...
100.44 = 10044 = 2511
x=
9900 2475
99
0.01454545… =
279
275
© T Madas
Convert 2.9135135135… into a fraction
Let x = 2.9135135135…
Since the recurring decimal has a three-digit
pattern we multiply this expression by 1000
1000x = 2913.5135135…
x=
2.9135135…
999x = 2910.6000000...
2910.6 = 29106 = 539
x=
185
9990
999
[HCF:54]
539
2.9135135135… = 185
© T Madas
Convert 0.153846153846153846… into a fraction
Let x = 0.153846153846153846…
Since the recurring decimal has a six-digit pattern
we multiply this expression by 1000000
1000000x = 153846.153846153846…
x=
0.153846153846…
999999x = 153846. 000000000000 ...
153846 = 17094 = 5698 = 518 = 2
x=
999999 111111 37037 3367 13
÷9
÷3
÷11
÷259
2
0.153846153846153846… =
13
© T Madas
It is worth noting a pattern in some recurring decimals:
4
0.444… =
9
31
0.313131… =
99
107
0.107107… =
999
7
0.777… =
9
8
0.080808… =
99
23
0.023023… =
999
37
1.373737… = 1
99
3.163163… = 3
2.555… = 2
5
9
163
999
This might save a bit of work when converting:
“write
5
as
11
a decimal”
x9
45
5
=
=
11 99 0.454545…
x9
© T Madas
© T Madas
Write
17
as a recurring decimal
33
Method A
by division:
0 5 1 5 1
3 3
1 7 0 0 0 0
5 17 5 17
17
= 0.515151…
33
Method B
by recognising patterns:
x3
17 51
=
=
33 99 0.515151…
x3
© T Madas
© T Madas
Calculate the mean of 0.6 and 0.16 giving your final
answer as a recurring decimal
by immediate recognition:
Mean
2
0.6 = 0.666… = 3
•add them
1
•divide by 2
0.16 = 0.1666… = 6
Method 1
2x 2 1
+
3 x2 6
2
=
Method 2
4
1
+
6
6
2
0 4 1 6 6
1 2
5 0 0 0 0
2
8
8
=
5
6
2
1
5
= 12
0.6666…
+ 0.1666…
0.8333…
0 41 6 6
2
8
5
= 0.416
12
0 8333
0 1 1 1
5
= 0.416
12
© T Madas