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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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?

4. To convert
fractions to
decimals
To convert a
recurring
decimal to a
fraction

5. Fraction to decimal
How do we change a
fraction to a
decimal?

6. Fraction to decimal
We divide the
numerator by the
denominator.

7. Fraction to decimal
e.g.
1
8
= 1 ÷ 8 =
8 1

8. Fraction to decimal
e.g.
1
8
= 1 ÷ 8 =
8 1. 0 0 0
0.1 2 5

9. Fraction to decimal
Sometimes the decimal will be a recurring decimal
3 1. 0 0 0
e.g.
1
3
= 1 ÷ 3 =

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

12. Fraction to decimal

13. Fraction to decimal
11 3. 0 0 0 0
0.2 7 2 7

14. Fraction to decimal
11 3. 0 0 0 0
0.2 7 2 7 3
11
= 0. 27

15. Fraction to decimal
Change
5
6
into a decimal.

16. Fraction to decimal
6 5. 0 0 0 0
0. 8 3 3 3 5
6
= 0. 83
Change
5
6
into a decimal.
Notice only the 3 has a dot over it
as only the 3 recurs.

17. Fraction to decimal
Change these fractions into decimals.
1.
7
11
2.
8
9
3.
12
9

18. Fraction to decimal
Change these fractions into decimals.
1.
7
11
2.
8
9
3.
12
9
= 0. 63

19. Fraction to decimal
Change these fractions into decimals.
1.
7
11
2.
8
9
3.
12
9
= 0. 63
= 0. 8

20. Fraction to decimal
Change these fractions into decimals.
1.
7
11
2.
8
9
3.
12
9
= 0. 63
= 0. 8
= 1. 3

21. To convert
fractions to
decimals
To convert a
recurring
decimal to a
fraction

22. To convert
fractions to
decimals
To convert a
recurring
decimal to a
fraction

23. The algebra of
recurring decimals
What is a recurring
decimal?

24. The algebra of
recurring decimals
A decimal where the
digit(s) repeat.

25. The algebra of
recurring decimals

26. The algebra of
recurring decimals

27. The algebra of
recurring decimals

28. The algebra of
recurring decimals

29. The algebra of
recurring decimals

30. The algebra of
recurring decimals

31. The algebra of
recurring decimals

32. The algebra of
recurring decimals

33. The algebra of
recurring decimals

34. The algebra of
recurring decimals

35. The algebra of
recurring decimals

36. The algebra of
recurring decimals
Check this on a calculator by dividing 4 by 9.

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

53. Recurring
Decimals

54. Change 0.39 into a fraction
. .
Let x= 0.39
100x= 39.3939393939393939...
x= 0.3939393939393939...
99x=
x= 39
99
. .
so 100x = 39.39
. .
Multiply!
Subtract!
Divide!
39
Recurring
Decimals

55. Change 0.39 into a fraction
. .
Let x= 0.39
100x= 39.3939393939393939...
x= 0.3939393939393939...
99x=
x= 39
99
. .
so 100x = 39.39
. .
Multiply!
Subtract!
Divide!
39
=13 as required
33
Recurring
Decimals

56. Challenge
0.45
.

57. Challenge
0.45
.
Let x= 0.45
.

58. Challenge
0.45
.
Let x= 0.45
.
so 100x = 45.5
.

59. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
so 100x = 45.5
.

60. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
so 100x = 45.5
.

61. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
so 100x = 45.5
.

62. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
99x= 45.1
so 100x = 45.5
.

63. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
99x= 45.1
990x= 451
so 100x = 45.5
.

64. 100x= 45.555555555555555…
x= 0.455555555555555…
Challenge
0.45
.
Let x= 0.45
.
99x= 45.1
990x= 451
x= 451
990
=41
90
so 100x = 45.5
.

65. Change 0.213 into a fraction
. .
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….)

75. Change these recurring decimals into fractions.
Recurring
Decimals
1. 0. 45
2. 0. 235
3. 0.28
4. 0.213
=
45
99

76. Change these recurring decimals into fractions.
Recurring
Decimals
1. 0. 45
2. 0. 235
3. 0.28
4. 0.213
=
45
99
=
235
999

77. Change these recurring decimals into fractions.
Recurring
Decimals
1. 0. 45
2. 0. 235
3. 0.28
4. 0.213
=
45
99
=
235
999
=
26
90

78. Change these recurring decimals into fractions.
Recurring
Decimals
1. 0. 45
2. 0. 235
3. 0.28
4. 0.213
=
45
99
=
235
999
=
26
90
=
211
990

79. To convert
fractions to
decimals and
vice versa
To convert a
recurring
decimal to a
fraction

80. To convert
fractions to
decimals and
vice versa
To convert a
recurring
decimal to a
fraction

81. Extension
Convert the recurring decimal 2.136 to a mixed number.
Give your answer in its simplest form.

82. Extension
Convert the recurring decimal 2.136 to a mixed number.
Give your answer in its simplest form.
x=2.136 1000x=2136.363636…
1000x=2136.36363636…
10x= 21.36363636…
990x= 2115
x=
2115
990
= 2
115
990
= 2
23
198

83. Extension
Convert the recurring decimal 2.06 to a mixed number.
Give your answer in its simplest form.

84. Extension
Convert the recurring decimal 2.06 to a mixed number.
Give your answer in its simplest form.
x=2. 06 100x=206.66666666…
100x=206.66666666…
10x= 20.66666666…
90x=186
x=
186
90
= 2
6
90
= 2
1
15

85. How do we convert
recurring decimals to
fractions?