Exercise: 1-A
Q1: Without actual division, find which of the following fractions are terminating decimals:
i. \(\frac{9}{12}\)
Step 1: Simplify the fraction to its lowest terms: \(\frac{9}{12} = \frac{3}{4}\).
Step 2: Prime factorise the denominator: \(4 = 2^2\).
Step 3: Since the denominator has only 2 as a prime factor, it is a terminating decimal.
Answer: Terminating
ii. \(\frac{7}{12}\)
Step 1: The fraction \(\frac{7}{12}\) is already in its simplest form.
Step 2: Prime factorise the denominator: \(12 = 2^2 \times 3\).
Step 3: Since the denominator contains a factor 3 (other than 2 or 5), it is a non-terminating repeating decimal.
Answer: Non-terminating
iii. \(\frac{13}{16}\)
Step 1: Prime factorise the denominator: \(16 = 2^4\).
Step 2: The denominator contains only powers of 2.
Answer: Terminating
iv. \(\frac{25}{128}\)
Step 1: Prime factorise the denominator: \(128 = 2^7\).
Step 2: The denominator contains only powers of 2.
Answer: Terminating
v. \(\frac{9}{50}\)
Step 1: Prime factorise the denominator: \(50 = 2 \times 5^2\).
Step 2: The prime factors are only 2 and 5.
Answer: Terminating
vi. \(\frac{121}{125}\)
Step 1: Prime factorise the denominator: \(125 = 5^3\).
Step 2: The denominator contains only powers of 5.
Answer: Terminating
vii. \(\frac{19}{55}\)
Step 1: Prime factorise the denominator: \(55 = 5 \times 11\).
Step 2: The presence of prime factor 11 (other than 2 or 5) makes it non-terminating.
Answer: Non-terminating
viii. \(\frac{37}{78}\)
Step 1: Prime factorise the denominator: \(78 = 2 \times 3 \times 13\).
Step 2: The presence of prime factors 3 and 13 makes it non-terminating.
Answer: Non-terminating
ix. \(\frac{23}{80}\)
Step 1: Prime factorise the denominator: \(80 = 2^4 \times 5\).
Step 2: The prime factors are only 2 and 5.
Answer: Terminating
x. \(\frac{19}{30}\)
Step 1: Prime factorise the denominator: \(30 = 2 \times 3 \times 5\).
Step 2: The presence of prime factor 3 makes it non-terminating.
Answer: Non-terminating
Q2: Convert each of the following decimal into vulgar fraction in its lowest terms:
i. 0.65
Step 1: Write 0.65 as a fraction with a denominator of 100: \(0.65 = \frac{65}{100}\).
Step 2: Find the H.C.F. of 65 and 100, which is 5.
Step 3: Divide both the numerator and denominator by 5: \(\frac{65 \div 5}{100 \div 5} = \frac{13}{20}\).
Answer: \(\frac{13}{20}\)
ii. 1.08
Step 1: Write 1.08 as a fraction: \(1.08 = \frac{108}{100}\).
Step 2: Find the H.C.F. of 108 and 100, which is 4.
Step 3: Divide both by 4: \(\frac{108 \div 4}{100 \div 4} = \frac{27}{25}\).
Answer: \(\frac{27}{25}\)
iii. 0.075
Step 1: Write 0.075 as a fraction with a denominator of 1000: \(0.075 = \frac{75}{1000}\).
Step 2: Find the H.C.F. of 75 and 1000, which is 25.
Step 3: Divide both by 25: \(\frac{75 \div 25}{1000 \div 25} = \frac{3}{40}\).
Answer: \(\frac{3}{40}\)
iv. 2.016
Step 1: Write 2.016 as a fraction: \(2.016 = \frac{2016}{1000}\).
Step 2: Find the H.C.F. of 2016 and 1000, which is 8.
Step 3: Divide both by 8: \(\frac{2016 \div 8}{1000 \div 8} = \frac{252}{125}\).
Answer: \(\frac{252}{125}\)
v. 1.732
Step 1: Write 1.732 as a fraction: \(1.732 = \frac{1732}{1000}\).
Step 2: Find the H.C.F. of 1732 and 1000, which is 4.
Step 3: Divide both by 4: \(\frac{1732 \div 4}{1000 \div 4} = \frac{433}{250}\).
Answer: \(\frac{433}{250}\)
Q3: Convert each of the following fractions into a decimal:
i. \( \frac{1}{8} \)
Step: Divide 1 by 8
8 ) 1.000 ( 0.125
0
----
10
8
----
20
16
----
40
40
----
0
Answer: \( 0.125 \)
ii. \( \frac{3}{32} \)
Step: Divide 3 by 32
32 ) 3.0000 ( 0.09375
0
----
30
0
----
300
288
----
120
96
----
240
224
----
160
160
----
0
Answer: \( 0.09375 \)
iii. \( \frac{44}{9} \)
Step: Divide 44 by 9
9 ) 44.000 ( 4.88
36
----
80
72
----
80
72
----
80
Answer: \( 4.\overline{8} \)
iv. \( \frac{11}{24} \)
Step: Divide 11 by 24
24 ) 11.0000 ( 0.4583
0
----
110
96
----
140
120
----
200
192
----
80
72
---
80
Answer: \( 0.458\overline{3} \)
v. \( \frac{12}{13} \)
Step: Divide 12 by 13
13 ) 12.000000 ( 0.9230769
0
----
120
117
----
30
26
----
40
39
----
10
0
----
100
91
----
90
78
----
120
117
----
30
Answer: \( 0.\overline{923076} \)
vi. \( \frac{27}{44} \)
Step: Divide 27 by 44
44 ) 27.0000 ( 0.61363
0
----
270
264
----
60
44
----
160
132
----
280
264
----
160
132
----
28
Answer: \( 0.61\overline{36} \)
vii. \( 2\frac{5}{12} \)
Step 1: Convert to improper fraction:
\( = \frac{29}{12} \)
Step 2: Divide 29 by 12
12 ) 29.000 ( 2.4166
24
----
50
48
----
20
12
----
80
72
----
8
Answer: \( 2.41\overline{6} \)
viii. \( 1\frac{31}{55} \)
Step 1: Convert to improper fraction:
\( = \frac{86}{55} \)
Step 2: Divide 86 by 55
55 ) 86.000 ( 1.5636
55
----
310
275
----
350
330
----
200
165
----
35
Answer: \( 1.5\overline{63} \)
Q4: Express \( \frac{15}{56} \) as a decimal, correct to four decimal places.
Step 1: Divide 15 by 56 using long division
56 ) 15.000000 ( 0.267857
0
--------
150
112
--------
380
336
--------
440
392
--------
480
448
--------
320
280
--------
400
392
--------
8
Step 2: Decimal obtained:
\( \frac{15}{56} = 0.267857… \)
Step 3: Round to four decimal places:
5th digit = 5 → increase 4th digit by 1
Step 4:
\( 0.2679 \)
Answer: \( 0.2679 \)
Q5: Express \( \frac{13}{34} \) as a decimal, correct to three decimal places.
Step 1: Divide 13 by 34 using long division
34 ) 13.00000 ( 0.38235
0
--------
130
102
--------
280
272
--------
80
68
--------
120
102
--------
180
170
-------
10
Step 2: Decimal obtained:
\( \frac{13}{34} = 0.38235… \)
Step 3: Round to three decimal places:
4th digit = 3 (less than 5) → no change
Step 4:
\( 0.382 \)
Answer: \( 0.382 \)
Q6: By actual division, show that:
i. \(\frac{11}{9}=1.\overline{2}\)
Step 1: Divide 11 by 9:
9 ) 11.000 ( 1.22
9
----
20
18
----
20
18
----
20
Step 2: Remainder repeats → digit 2 repeats
Answer: \(1.222… = 1.\overline{2}\)
ii. \(\frac{43}{11}=3.\overline{90}\)
Step 1: Divide 43 by 11:
11 ) 43.000 ( 3.909
33
----
100
99
----
10
0
----
100
99
----
10
Step 2: Pattern 90 repeats
Answer: \(3.9090… = 3.\overline{90}\)
iii. \(\frac{107}{45}=2.3\overline{7}\)
Step 1: Divide 107 by 45:
45 ) 107.000 ( 2.377
90
----
170
135
----
350
315
----
350
315
----
350
Step 2: Digit 7 repeats
Answer: \(2.3777… = 2.3\overline{7}\)
iv. \(\frac{21}{55}=1.3\overline{81}\)
Step 1: Divide 21 by 55:
55 ) 21.0000 ( 0.381
0
----
210
165
----
450
440
----
100
55
----
450
440
----
100
Step 2: Pattern 81 repeats
Answer: \(0.3818… = 0.3\overline{81}\)
Q7: Express each of the following as a vulgar fraction in simplest form:
i. \(0.\overline{5}\)
Step 1: Let \(x = 0.555…\) (Eq. 1)
Step 2: Multiply by 10: \(10x = 5.555…\) (Eq. 2)
Step 3: Subtract Eq. 1 from Eq. 2: \(9x = 5\).
Answer: \(x = \frac{5}{9}\)
ii. \(0.\overline{43}\)
Step 1: Let \(x = 0.4343…\) (Eq. 1)
Step 2: Multiply by 100: \(100x = 43.4343…\) (Eq. 2)
Step 3: Subtract Eq. 1 from Eq. 2: \(99x = 43\).
Answer: \(x = \frac{43}{99}\)
iii. \(0.\overline{158}\)
Step 1: Let \(x = 0.158158…\) (Eq. 1)
Step 2: Multiply by 1000: \(1000x = 158.158158…\) (Eq. 2)
Step 3: Subtract Eq. 1 from Eq. 2: \(999x = 158\).
Answer: \(x = \frac{158}{999}\)
iv. \(1.\overline{3}\)
Step 1: Let \(x = 1.333…\) (Eq. 1)
Step 2: Multiply by 10: \(10x = 13.333…\) (Eq. 2)
Step 3: Subtract Eq. 1 from Eq. 2: \(9x = 12\).
Step 4: Simplify: \(x = \frac{12}{9} = \frac{4}{3}\).
Answer: \(\frac{4}{3}\)
v. \(4.\overline{17}\)
Step 1: Let \(x = 4.1717…\) (Eq. 1)
Step 2: Multiply by 100: \(100x = 417.1717…\) (Eq. 2)
Step 3: Subtract Eq. 1 from Eq. 2: \(99x = 413\).
Answer: \(x = \frac{413}{99}\)
vi. \(0.1\overline{2}\)
Step 1: Let \(x = 0.1222…\) (Eq. 1)
Step 2: Multiply by 10: \(10x = 1.222…\) (Eq. 2)
Step 3: Multiply by 100: \(100x = 12.222…\) (Eq. 3)
Step 4: Subtract Eq. 2 from Eq. 3: \(90x = 11\).
Answer: \(x = \frac{11}{90}\)
vii. \(0.1\overline{36}\)
Step 1: Let \(x = 0.13636…\) (Eq. 1)
Step 2: Multiply by 10: \(10x = 1.3636…\) (Eq. 2)
Step 3: Multiply by 1000: \(1000x = 136.3636…\) (Eq. 3)
Step 4: Subtract Eq. 2 from Eq. 3: \(990x = 135\).
Step 5: Simplify: \(x = \frac{135}{990} = \frac{3}{22}\).
Answer: \(x = \frac{3}{22}\)
viii. \(1.5\overline{7}\)
Step 1: Let \(x = 1.5777…\) (Eq. 1)
Step 2: Multiply by 10: \(10x = 15.777…\) (Eq. 2)
Step 3: Multiply by 100: \(100x = 157.777…\) (Eq. 3)
Step 4: Subtract Eq. 2 from Eq. 3: \(90x = 142\).
Step 5: Simplify: \(x = \frac{142}{90} = \frac{71}{45}\).
Answer: \(\frac{71}{45}\)
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