Exercise: 4-H
Q1: Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal:
Step: A rational number \(\frac{p}{q}\) is expressible as a terminating decimal if the denominator \(q\) has no prime factor other than 2 and/or 5, i.e., the denominator must be of the form \(2^m \times 5^n\), where \(m, n\) are non-negative integers.
i. \(\frac{11}{16}\)
Step 1: Prime factorization of 16
\[
16 = 2^4
\]
Only prime factor is 2 → Terminating
Answer: \(\frac{11}{16}\) is a terminating decimal.
ii. \(\frac{17}{20}\)
Step 1: Prime factorization of 20
\[
20 = 2^2 \times 5
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{17}{20}\) is a terminating decimal.
iii. \(\frac{44}{125}\)
Step 1: Prime factorization of 125
\[
125 = 5^3
\]
Only prime factor is 5 → Terminating
Answer: \(\frac{44}{125}\) is a terminating decimal.
iv. \(\frac{9}{80}\)
Step 1: Prime factorization of 80
\[
80 = 2^4 \times 5
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{9}{80}\) is a terminating decimal.
v. \(\frac{123}{200}\)
Step 1: Prime factorization of 200
\[
200 = 2^3 \times 5^2
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{123}{200}\) is a terminating decimal.
vi. \(\frac{129}{320}\)
Step 1: Prime factorization of 320
\[
320 = 2^6 \times 5
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{129}{320}\) is a terminating decimal.
vii. \(\frac{431}{500}\)
Step 1: Prime factorization of 500
\[
500 = 2^2 \times 5^3
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{431}{500}\) is a terminating decimal.
viii. \(\frac{807}{1250}\)
Step 1: Prime factorization of 1250
\[
1250 = 2 \times 5^4
\]
Only prime factors are 2 and 5 → Terminating
Answer: \(\frac{807}{1250}\) is a terminating decimal.
Q2: By actual division, express each of the following rational numbers as a terminating decimal:
i. \(\frac{11}{8}\)
1.375
_______
8 | 11.000
- 8
-----
30
- 24
-----
60
- 56
-----
40
- 40
-----
0
Answer: 1.375
ii. \(\frac{23}{16}\)
1.4375
________
16 | 23.0000
-16
----
70
- 64
----
60
- 48
----
120
-112
----
80
- 80
----
0
Answer: 1.4375
iii. \(\frac{76}{125}\)
0.608
_______
125 | 76.000
-0
-----
760
- 750
-----
100
- 000
-----
1000
- 1000
----
0
Answer: 0.608
iv. \(\frac{103}{40}\)
2.575
_______
40 | 103.000
-80
-----
230
-200
-----
300
- 280
-----
200
- 200
-----
0
Answer: 2.575
v. \(\frac{17}{80}\)
0.2125
_______
80 | 17.0000
- 0
-----
170
-160
-----
100
- 80
-----
200
- 160
-----
400
- 400
-----
0
Answer: 0.2125
vi. \(\frac{2}{25}\)
0.08
______
25 | 2.00
- 0
----
200
- 200
----
0
Answer: 0.08
vii. \(\frac{1}{125}\)
0.008
_______
125 | 1.000
- 0
----
100
- 000
----
1000
- 1000
----
0
Answer: 0.008
viii. \(\frac{309}{1250}\)
0.2472
__________
1250 | 309.0000
-0
-----
3090
- 2500
-----
5900
- 5000
-----
9000
- 8750
-----
2500
- 2500
-----
0
Answer: 0.2472
Q3: Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal:
i. \(\frac{23}{24}\)
Step 1: Prime factorisation of denominator: \(24 = 2^3 \times 3\)
Step 2: The denominator contains 3 (a prime other than 2 or 5).
Answer: Repeating decimal
ii. \(\frac{79}{30}\)
Step 1: Prime factorisation of 30 = \(2 \times 3 \times 5\)
Step 2: The denominator contains 3 (other than 2, 5).
Answer: Repeating decimal
iii. \(\frac{100}{9}\)
Step 1: Prime factorisation of 9 = \(3^2\)
Step 2: Only prime factor is 3 (not 2 or 5).
Answer: Repeating decimal
iv. \(\frac{205}{27}\)
Step 1: Prime factorisation of 27 = \(3^3\)
Step 2: Contains 3 only, so repeating decimal.
Answer: Repeating decimal
v. \(\frac{461}{60}\)
Step 1: Prime factorisation of 60 = \(2^2 \times 3 \times 5\)
Step 2: Contains 3 → repeating decimal.
Answer: Repeating decimal
vi. \(\frac{1003}{112}\)
Step 1: Prime factorisation of 112 = \(2^4 \times 7\)
Step 2: Contains 7 → repeating decimal.
Answer: Repeating decimal
vii. \(\frac{127}{225}\)
Step 1: Prime factorisation of 225 = \(3^2 \times 5^2\)
Step 2: Contains 3 → repeating decimal.
Answer: Repeating decimal
viii. \(\frac{219}{440}\)
Step 1: Prime factorisation of 440 = \(2^3 \times 5 \times 11\)
Step 2: Contains 11 → repeating decimal.
Answer: Repeating decimal
Q4: By actual division, express each of the following as a repeating decimal:
i. \(\frac{103}{9}\)
Step 1: Divide 103 by 9.
11.4444...
_____________
9 | 103.000000
- 9
______
13
- 9
______
40
-36
______
40
Answer: \(\frac{103}{9} = 11.\overline{4}\)
ii. \(\frac{7}{12}\)
Step 1: Divide 7 by 12.
0.583333...
_______________
12 | 7.000000
-0
-----
70
- 60
-----
100
- 96
-----
40
- 36
-----
40
Answer: \(\frac{7}{12} = 0.58\overline{3}\)
iii. \(\frac{101}{15}\)
Step 1: Divide 101 by 15.
6.7333...
______________
15 | 101.000
- 90
-----
110
-105
-----
50
- 45
-----
50
Answer: \(\frac{101}{15} = 6.7\overline{3}\)
iv. \(\frac{303}{11}\)
Step 1: Divide 303 by 11.
27.545454...
________________
11 | 303.000000
-22
-----
83
-77
-----
60
-55
-----
50
-44
-----
60
Answer: \(\frac{303}{11} = 27.\overline{54}\)
v. \(\frac{212}{143}\)
Step 1: Divide 212 by 143.
1.482517482517...
________________
143 | 212.000000
-143
-----
690
-572
-----
1180
-1144
-----
360
-286
-----
740
-715
-----
250
-143
-----
1070
-1001
-----
690
Answer: \(\frac{212}{143} = 1.\overline{482517}\)
vi. \(\frac{16}{7}\)
Step 1: Divide 16 by 7.
2.285714285714...
______________________
7 | 16.000000000
-14
-----
20
-14
-----
60
-56
-----
40
-35
-----
50
-49
-----
10
-7
-----
30
-28
-----
20
Answer: \(\frac{16}{7} = 2.\overline{285714}\)
vii. \(\frac{227}{30}\)
Step 1: Divide 227 by 30.
7.56666...
_______________
30 | 227.0000
-210
-----
170
-150
-----
200
-180
-----
200
Answer: \(\frac{227}{30} = 7.5\overline{6}\)
viii. \(\frac{2000}{33}\)
Step 1: Divide 2000 by 33.
60.606060...
___________________
33 | 2000.0000
-198
-----
20
- 0
-----
200
-198
-----
20
- 0
-----
200
-198
-----
20
Answer: \(\frac{2000}{33} = 60.\overline{60}\)
Q5: Fill in the blanks:
i. \(\frac{2}{3} =\) ____
Step 1: Divide 2 by 3.
0.6666...
____________
3 | 2.000000
-0
-----
20
-18
-----
20
-18
-----
20
Answer: \(\frac{2}{3} = 0.\overline{6}\)
ii. \(\frac{11}{30} =\) ____
Step 1: Divide 11 by 30.
0.3666...
__________
30 | 11.0000
- 0
-----
110
-90
-----
200
-180
-----
200
Answer: \(\frac{11}{30} = 0.3\overline{6}\)
iii. \(\frac{13}{11} =\) ____
Step 1: Divide 13 by 11.
1.181818...
___________
11 | 13.00000
-11
-----
20
- 11
-----
90
- 88
-----
20
Answer: \(\frac{13}{11} = 1.\overline{18}\)
iv. \(\frac{23}{55} =\) ____
Step 1: Divide 23 by 55.
0.418181...
_______________
55 | 23.00000
- 0
-----
230
-220
-----
100
- 55
-----
450
-440
-----
100
Answer: \(\frac{23}{55} = 0.4\overline{18}\)
