ICSE Class 9 Maths Chapter 01 Rational and Irrational Numbers

Unit 1 - Pure Arithmetic

Rational and Irrational Numbers

Exercise 1(A)

Question 1

Without actual division find which of the following fractions are terminating decimal:

(i) \(\frac{9}{25}\) (ii) \(\frac{7}{12}\)

(iii) \(\frac{13}{16}\) (iv) \(\frac{25}{128}\)

(v) \(\frac{9}{50}\) (vi) \(\frac{121}{125}\)

(vii) \(\frac{19}{55}\) (viii) \(\frac{37}{78}\)

(ix) \(\frac{23}{80}\) (x) \(\frac{19}{30}\)

Solution

A rational number is terminating if its denominator has prime factors only 2 or 5 or both.

(i) In \(\frac{9}{25}\), the prime factors of 25 are 5, 5

Therefore, it is terminating decimal.

(ii) In \(\frac{7}{12}\), the prime factors of 12 are 2, 2, 3

Therefore, it is not terminating decimal.

(iii) In \(\frac{13}{16}\), the prime factors of 16 are 2, 2, 2, 2

Therefore, it is terminating decimal.

(iv) In \(\frac{25}{128}\), the prime factors of 128 are 2, 2, 2, 2, 2, 2, 2.

Therefore, it is terminating decimal.

(v) In \(\frac{9}{50}\), the prime factors of 50 are 2, 5, 5.

Therefore, it is terminating decimal.

(vi) In \(\frac{121}{125}\), the prime factors of 125 are 5, 5, 5

Therefore, it is terminating decimal.

(vii) In \(\frac{19}{55}\), the prime factors of 55 are 5, 11

Therefore, it is not terminating decimal.

(viii) In \(\frac{37}{78}\), the prime factors of 78 are 2, 3, 13

Therefore, it is not terminating decimal.

(ix) In \(\frac{23}{80}\), the prime factors of 80 are 2, 2, 2, 2, 5

Therefore, it is terminating decimal.

(x) In \(\frac{19}{30}\), the prime factors of 30 are 2, 3, 5

Therefore, it is not terminating decimal.

Teacher's Note

Understanding terminating and non-terminating decimals helps in finance when calculating exact prices or currency conversions that must end after a certain number of decimal places.

Question 2

Convert each of the following decimals into a vulgar fraction in its lowest terms:

(i) 0-65 (ii) 1-08

(iii) 0-075 (iv) 2-016

(v) 1-732

Solution

(i) \(0.65 = \frac{65}{100} = \frac{65 \div 5}{100 \div 5} = \frac{13}{20}\)

(ii) \(1.08 = \frac{108}{100} = \frac{108 \div 4}{100 \div 4} = \frac{27}{25}\)

(iii) \(0.075 = \frac{75}{1000} = \frac{75 \div 25}{1000 \div 25} = \frac{3}{40}\)

(iv) \(2.016 = \frac{2016}{1000} = \frac{2016 \div 8}{1000 \div 8} = \frac{252}{125}\)

(v) \(1.732 = \frac{1732}{1000} = \frac{1732 \div 4}{1000 \div 4} = \frac{433}{250}\)

Teacher's Note

Converting decimals to fractions is essential in cooking and baking where recipes often require measurements as fractions rather than decimals.

Question 3

Convert each of the following into a decimal:

(i) \(\frac{1}{8}\) (ii) \(\frac{3}{32}\)

(iii) \(\frac{44}{9}\) (iv) \(\frac{11}{24}\)

(v) \(\frac{12}{13}\) (vi) \(\frac{27}{44}\)

(vii) \(2\frac{5}{12}\) (viii) \(1\frac{31}{55}\)

Solution

(i) \(\frac{1}{8} = 0.125\)

By long division: 8 ) 1.000 yields 0.125

(ii) \(\frac{3}{32} = 0.09375\)

By long division: 32 ) 3.00000 yields 0.09375

(iii) \(\frac{44}{9} = 4.888...\) = \(4.\overline{8}\)

By long division: 9 ) 44.000 yields 4.888...

(iv) \(\frac{11}{24} = 0.45833...\) = \(0.45\overline{8}\)

By long division: 24 ) 11.00000 yields 0.45833...

(v) \(\frac{12}{13} = 0.923076923076...\) = \(0.\overline{923076}\)

(vi) \(\frac{27}{44} = 0.6136363636...\) = \(0.6\overline{36}\)

(vii) \(2\frac{5}{12} = \frac{29}{12} = 2.41666...\) = \(2.41\overline{6}\)

(viii) \(1\frac{31}{55} = \frac{86}{55} = 1.5636363...\) = \(1.5\overline{63}\)

Teacher's Note

Understanding decimal representations helps when reading digital displays on calculators, scales, and meters in everyday situations.

Question 4

Express \(\frac{15}{56}\) as a decimal correct to four decimal places.

Solution

\(\frac{15}{56} = 0.26785\)

= 0.2679 (correct to four decimal places)

By long division: 56 ) 15.00000 yields 0.26785

Teacher's Note

Rounding decimals to specific places is crucial in scientific measurements and financial calculations where precision is required.

Question 5

Express \(\frac{13}{34}\) as a decimal correct to three decimal places.

Solution

\(\frac{13}{34} = 0.3823\)

= 0.382 (correct to three decimal places)

By long division: 34 ) 13.0000 yields 0.3823

Question 6

Express each of the following as a vulgar fraction:

(i) 0-6 (ii) 0-43

(iii) 1-3 (iv) 0-12

(v) 0-136 (vi) 0-57

Solution

(i) Let \(x = 0.\overline{6} = 0.6666\) ...(i)

then \(10x = 6.\overline{6}666\) ...(ii)

Subtracting (i) from (ii)

\(9x = 6 \Rightarrow x = \frac{6}{9} = \frac{2}{3}\)

Therefore, Required fraction = \(\frac{2}{3}\)

(ii) Let \(x = 0.\overline{43} = 0.4343\)4343 ...(i)

then \(100x = 43.434\)343 ...(ii)

Subtracting (i) from (ii)

\(99x = 43 \Rightarrow x = \frac{43}{99}\)

Therefore, Required fraction = \(\frac{43}{99}\)

(iii) Let \(x = 1.\overline{3} = 1.3333\) ...(i)

then \(10x = 13.\)3333 ...(ii)

Subtracting (i) from (ii)

\(9x = 12 \Rightarrow x = \frac{12}{9} = \frac{4}{3}\)

Therefore, Required fraction = \(\frac{4}{3}\)

(iv) Let \(x = 0.1\overline{2} = 0.12222\)...(i)

then \(10x = 1.\overline{2} = 1.2222\) ...(ii)

and \(100x = 12.\)2222 ...(iii)

Subtracting (ii) from (iii)

\(90x = 11 \Rightarrow x = \frac{11}{90}\)

Therefore, Required fraction = \(\frac{11}{90}\)

(v) Let \(x = 0.1\overline{36} = 0.1363\)63636 ...(i)

then \(10x = 1.\)363636 ...(ii)

and \(1000x = 136.\)363636 ...(iii)

Subtracting (ii) from (iii)

\(990x = 135 \Rightarrow x = \frac{135}{990} = \frac{3}{22}\)

Therefore, Required fraction = \(\frac{3}{22}\)

(vi) Let \(x = 0.\overline{57} = 0.57777\) ...(i)

then \(10x = 5.\)7777 ...(ii)

and \(100x = 57.\)7777 ...(iii)

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