12
enrichme
Recurring decimals
nt
Identity cards
The people of Decimal City are applying for identity cards.
1 ∏ 3 = 0.333… ?
Application counter
1 ÷ 2 equals 0.5.
Your identity card
number is 0.5.
1∏2
1∏4
1∏3
The officer uses a calculator to find the identity card numbers for their people.
1
1∏4=
1
4
0.285888888
888
1 ∏ 4 = 0.
2
1∏3=
1
3
0. 383888888
888
333333
1 ∏ 3 = 0.
What is the difference between the two answers above?
1 is divisible by 2 or 4.
1 ÷ 2 = 0.5 ; 1 ÷ 4 = 0.25
1 is not divisible by 3.
1 ÷ 3 = 0.33333...
62
3
Finish the table.
Fraction
Division form
Quotient
1
2
1
3
1
4
1
5
1
6
1
8
1
9
1
11
1∏2
0.5
1∏3
0.333…
1∏4
0.25
Divisible Indivisible
Number(s) repeated
in the quotient
✓
✓
3
✓
1∏5
1∏6
1∏8
1∏9
1 ∏ 11
A recurring decimal repeats a pattern of digits continuously. The repeated pattern is
called a recurring period which is shown with a recurring point or a pair of recurring
points.
A
(a)
0.333… = 0.3
Recurring point
Recurring period
Recurring points
(b)
A
A
0.126126126… = 0.126
Recurring period
(c)
0.16666… =
63
4
Use a calculator to do these sums. Write each answer as a recurring decimal.
2∏3
(a)
5 ∏ 37
(b)
=
=
=
=
23 ∏ 15
(c)
6∏7
(d)
=
=
=
=
Maths window
The answers to the sums below are recurring decimals. Their recurring periods are very
long, so they are not shown on an ordinary calculator.
A
A
1
1 ∏ 17 = 0.0588235294117647
2
1 ∏ 23 = 0.0434782608695652173913
3
1 ∏ 29 = 0.0344827586206896551724137931
4
1 ∏ 31 = 0.032258064516129
A
A
A
A
A
A
Exercise 12A
A.
Write each decimal as a recurring decimal.
1.
0.4444…
2.
0.2828…
3.
11.6111…
4.
4.108108…
B.
Do these sums. Write the answer in a recurring decimal.
5.
8∏9
6.
9 ∏ 11
7.
7 ∏ 18
8.
65 ∏ 54
64
Finding out identity card numbers
We also want to apply for the Decimal City identity cards.
7
22
1
8
11
1
37
Use a calculator to find the answers.
1
7
= 7 ∏ 22
22
Change fractions to
decimals using division.
= 0.31818...
7
22
=
2
1
8
8
=1+
11
11
= 1.
3
1
8
11
1
= 1 ∏ 37
37
1
37
=
4
I’ve found that when changing the following fractions to decimals,
the decimals are terminating.
(a)
1
5
=
= 0.5
2 10
(b)
5
625
= 0.625
=
8 1000
(c)
3
75
= 0.75
=
4 100
(d)
12
48
=
= 0.48
25 100
Talk about it
Can you give more examples?
65
5
I know when the recurring period ends.
5
27
(a)
5
= 5 ∏ 27
27
A
A
= 0.185
(b)
18
= 18 ∏ 55
55
=
0.1 8 5
27
)
5.0
27
23
21
1
1
0.3 2 7
55
0
6
40
35
5
)
1 8.0
165
15
11
4
3
The number
repeats itself.
0
0
00
85
15
The number
repeats itself.
Maths window
Look at the picture on the right. Then find the answers below.
1
2
1
7
2
7
3
7
4
7
5
7
6
7
A
A
= 0.142857
1
A
= 0.2
A
= 0.4
7
4
5
2
A
= 0.5
A
= 0.7
A
8
= 0.8
1
19
2
19
3
19
10
19
17
19
A
A
= 0.052631578947368421
A
2
A
= 0.105263157894736842
4
5
12
6
6
4
3
3
11
9 18 17 15
7
66
2
1 10
7
14
3
=
5
13
6
=
4 2
0
8
16
8
=
1
9
8
1
5
7
Exercise 12B
A.
Change each fraction to a recurring decimal.
1.
2
9
B.
Which of the fractions below cannot be changed to a recurring decimal?
6.
3
25
2.
7.
13
99
3.
9
40
8.
9
1
11
4.
6
11
9.
11
18
7
13
5.
10.
7
999
8
9
Recurring decimals
1
A recurring decimal repeats a pattern of digits continuously.
A
0.2222... = 0.2
e.g. (a)
Recurring points
A A
(b)
0.1313... = 0.13
(c)
0.0333... = 0.03
(d)
0.126126... = 0.126
A
A
A
A
0.126
A
Recurring period
2
0.8 3
Recurring period
A
5 ∏ 6 = 0.83
6
)
5.0
48
20
18
2
3
Changing fractions to recurring decimals
1
=1∏3
3
= 0.333...
Using division
A
= 0.3
4
Terminating decimals
16
4
=
= 0.16
e.g.
25 100
67
The recurring period
ends when the
number repeats itself.