Class 7 Rational Numbers Exercise 4.2

\begin{array}{l} \text {Q1. Express each of the following as a rational number with positive denominator: } \\ \text {(i) } \quad \frac{-15}{-28} \quad \text {(ii) } \quad \frac{6}{-9} \quad \text {(iii) } \quad \frac{-28}{-11} \quad \text {(iv) } \quad \frac{19}{-7} \end{array}\begin{array}{l} \text {Sol .}\text {(i) To make denominator positive we need to multiply both numerator and denominator of the number by -1, we get } \\ \frac{-15}{-28}=\frac{-15 \times -1}{-28 \times -1}=\frac{15}{28} \\ \\\text {(ii) To make denominator positive we need to multiply both numerator and denominator of the number by -1, we get } \\ \frac{6}{-9}=\frac{6 \times -1}{-9 \times -1}=\frac{-6}{9} \\ \\\text {(iii) To make denominator positive we need to multiply both numerator and denominator of the number by -1, we get } \\ \frac{-28}{-11}=\frac{-28 \times -1}{-11 \times -1}=\frac{28}{11} \\ \\\text {(iv) To make denominator positive we need to multiply both numerator and denominator of the number by -1, we get } \\ \frac{19}{-7}=\frac{19 \times -1}{-7 \times -1}=\frac{-19}{7} \\ \end{array}\begin{array}{l} \text {Q2. Express } \frac{3}{5} \text { as a rational number with numerator: } \\ \text {(i) } \quad 6 \quad \text {(ii) } \quad -15 \quad \text {(iii) } \quad 21 \quad \text {(iv) } \quad -27 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{3}{5} \text { as a rational number with numerator as 6, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{5} \text { by 2 } \\\Rightarrow \quad \frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \\ \\\text {(ii) In order to express } \frac{3}{5} \text { as a rational number with numerator as -15, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{5} \text { by -5 } \\\Rightarrow \quad \frac{3}{5} = \frac{3 \times -5}{5 \times -5} = \frac{-15}{-25} \\ \\\text {(iii) In order to express } \frac{3}{5} \text { as a rational number with numerator as 21, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{5} \text { by 7 } \\\Rightarrow \quad \frac{3}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35} \\ \\\text {(iv) In order to express } \frac{3}{5} \text { as a rational number with numerator as -27, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{5} \text { by -9 } \\\Rightarrow \quad \frac{3}{5} = \frac{3 \times -9}{5 \times -9} = \frac{-27}{-45} \\ \\\end{array}\begin{array}{l} \text {Q3. Express } \frac {5}{7} \text { as a rational number with denominator: } \\ \text {(i) } \quad -14 \quad \text {(ii) } \quad 70 \quad \text {(iii) } \quad -28 \quad \text {(iv) } \quad -84 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac {5}{7} \text { as a rational number with denominator as -14, } \\ \text {we need to mulitply both numerator and denominator of } \frac {5}{7} \text { by -2 } \\\Rightarrow \quad \frac {5}{7} = \frac {5 \times -2}{7 \times -2} = \frac {-10}{-14} \\ \\\text {(ii) In order to express } \frac {5}{7} \text { as a rational number with denominator as 70, } \\ \text {we need to mulitply both numerator and denominator of } \frac {5}{7} \text { by 10 } \\\Rightarrow \quad \frac {5}{7} = \frac {5 \times 10}{7 \times 10} = \frac {50}{70} \\ \\\text {(iii) In order to express } \frac {5}{7} \text { as a rational number with denominator as -28, } \\ \text {we need to mulitply both numerator and denominator of } \frac {5}{7} \text { by -4 } \\\Rightarrow \quad \frac {5}{7} = \frac {5 \times -4}{7 \times -4} = \frac {-20}{-28} \\ \\\text {(iv) In order to express } \frac {5}{7} \text { as a rational number with denominator as -84, } \\ \text {we need to mulitply both numerator and denominator of } \frac {5}{7} \text { by -12 } \\\Rightarrow \quad \frac {5}{7} = \frac {5 \times -12}{7 \times -12} = \frac {-60}{-84} \\\end{array}\begin{array}{l} \text {Q4. Express } \frac{3}{4} \text { as a rational number with denominator: } \\ \text {(i) } \quad 20 \quad \text {(ii) } \quad 36 \quad \text {(iii) } \quad 44 \quad \text {(iv) } \quad -80 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{3}{4} \text { as a rational number with denominator as 20, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{4} \text { by 5 } \\\Rightarrow \quad \frac{3}{4} = \frac{3 \times 5}{4 \times 5}= \frac{15}{20} \\ \\\text {(ii) In order to express } \frac{3}{4} \text { as a rational number with denominator as 36, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{4} \text { by 9 } \\\Rightarrow \quad \frac{3}{4} = \frac{3 \times 9}{4 \times 9}= \frac{27}{36} \\ \\\text {(iii) In order to express } \frac{3}{4} \text { as a rational number with denominator as 44, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{4} \text { by 11 } \\\Rightarrow \quad \frac{3}{4} = \frac{3 \times 11}{4 \times 11}= \frac{33}{44} \\ \\\text {(iv) In order to express } \frac{3}{4} \text { as a rational number with denominator as -80, } \\ \text {we need to mulitply both numerator and denominator of } \frac{3}{4} \text { by -20 } \\\Rightarrow \quad \frac{3}{4} = \frac{3 \times -20}{4 \times -20}= \frac{-60}{-80} \\ \\\end{array}\begin{array}{l} \text {Q5. Express } \frac{2}{5} \text { as a rational number with numerator } \\ \text {(i) } \quad -56 \quad \text {(ii) } \quad 154 \quad \text {(iii) } \quad -750 \quad \text {(iv) } \quad 500 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{2}{5} \text { as a rational number with numerator as -56, } \\ \text {we need to mulitply both numerator and denominator of } \frac{2}{5} \text { by -28 } \\\Rightarrow \quad \frac{2}{5} = \frac{2 \times -28}{5 \times -28} = \frac{-56}{-140} \\ \\\text {(ii) In order to express } \frac{2}{5} \text { as a rational number with numerator as 154, } \\ \text {we need to mulitply both numerator and denominator of } \frac{2}{5} \text { by 77 } \\\Rightarrow \quad \frac{2}{5} = \frac{2 \times 77}{5 \times 77} = \frac{154}{385} \\ \\\text {(iii) In order to express } \frac{2}{5} \text { as a rational number with numerator as -750, } \\ \text {we need to mulitply both numerator and denominator of } \frac{2}{5} \text { by -375 } \\\Rightarrow \quad \frac{2}{5} = \frac{2 \times -375}{5 \times -375} = \frac{-750}{-1875} \\ \\\text {(iv) In order to express } \frac{2}{5} \text { as a rational number with numerator as 500, } \\ \text {we need to mulitply both numerator and denominator of } \frac{2}{5} \text { by 250 } \\\Rightarrow \quad \frac{2}{5} = \frac{2 \times 250}{5 \times 250} = \frac{500}{1250} \\ \\\end{array}\begin{array}{l} \text {Q6. Express \frac{-192}{108} as a rational number with numerator:} \\ \text {(i) } \quad 64 \quad \text {(ii) } \quad -16 \quad \text {(iii) } \quad 32 \quad \text {(iv) } \quad -48 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{-192}{108} \text { as a rational number with numerator as 64, } \\ \text {we need to divide both numerator and denominator of } \frac{-192}{108} \text { by -3 } \\\Rightarrow \quad \frac{-192}{108} = \frac{-192 \div -3}{108 \div -3} = \frac{64}{-36} \\ \\\text {(ii) In order to express } \frac{-192}{108} \text { as a rational number with numerator as -16, } \\ \text {we need to divide both numerator and denominator of } \frac{-192}{108} \text { by 12 } \\\Rightarrow \quad \frac{-192}{108} = \frac{-192 \div 12}{108 \div 12} = \frac{-16}{9} \\ \\\text {(iii) In order to express } \frac{-192}{108} \text { as a rational number with numerator as 32, } \\ \text {we need to divide both numerator and denominator of } \frac{-192}{108} \text { by -6 } \\\Rightarrow \quad \frac{-192}{108} = \frac{-192 \div -6}{108 \div -6} = \frac{32}{-18} \\ \\\text {(iv) In order to express } \frac{-192}{108} \text { as a rational number with numerator as -48, } \\ \text {we need to divide both numerator and denominator of } \frac{-192}{108} \text { by 4 } \\\Rightarrow \quad \frac{-192}{108} = \frac{-192 \div 4}{108 \div 4} = \frac{-48}{27} \\ \\\end{array}\begin{array}{l} \text {Q7. Express } \frac{168}{-294} \text { as a rational number with denominator: } \\ \text {(i) } \quad 14 \quad \text {(ii) } \quad -7 \quad \text {(iii) } \quad -49 \quad \text {(iv) } \quad 1470 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{168}{-294} \text { as a rational number with denominator as 14, } \\ \text {we need to divide both numerator and denominator of } \frac{168}{-294} \text { by -21 } \\\Rightarrow \quad \frac{168}{-294} = \frac{168 \div -21}{-294 \div -21} = \frac{-8}{14} \\ \\\text {(ii) In order to express } \frac{168}{-294} \text { as a rational number with denominator as -7, } \\ \text {we need to divide both numerator and denominator of } \frac{168}{-294} \text { by 42 } \\\Rightarrow \quad \frac{168}{-294} = \frac{168 \div 42}{-294 \div 42} = \frac{4}{-7} \\ \\\text {(iii) In order to express } \frac{168}{-294} \text { as a rational number with denominator as -49, } \\ \text {we need to divide both numerator and denominator of } \frac{168}{-294} \text { by 6 } \\\Rightarrow \quad \frac{168}{-294} = \frac{168 \div 6}{-294 \div 6} = \frac{28}{-49} \\ \\\text {(iv) In order to express } \frac{168}{-294} \text { as a rational number with denominator as 1470, } \\ \text {we need to multiple both numerator and denominator of } \frac{168}{-294} \text { by -5 } \\\Rightarrow \quad \frac{168}{-294} = \frac{168 \times -5}{-294 \times -5} = \frac{-840}{1470} \\\end{array}\begin{array}{l} \text {Q8. Write } \frac{-14}{42} \text { in a form so that the numerator is equal to: } \\ \text {(i) } \quad -2 \quad \text {(ii) } \quad 7 \quad \text {(iii) } \quad 42 \quad \text {(iv) } \quad -70 \end{array}\begin{array}{l} \text {Sol .} \text {(i) In order to express } \frac{-14}{42} \text { as a rational number with numerator as -2, } \\ \text {we need to divide both numerator and denominator of } \frac{-14}{42} \text { by 7 } \\\Rightarrow \quad \frac{-14}{42} = \frac{-14 \div 7}{42 \div 7} = \frac{-2}{6} \\ \\\text {(ii) In order to express } \frac{-14}{42} \text { as a rational number with numerator as 7, } \\ \text {we need to divide both numerator and denominator of } \frac{-14}{42} \text { by -2 } \\\Rightarrow \quad \frac{-14}{42} = \frac{-14 \div -2}{42 \div -2} = \frac{7}{-21} \\ \\\text {(iii) In order to express } \frac{-14}{42} \text { as a rational number with numerator as 42, } \\ \text {we need to multiply both numerator and denominator of } \frac{-14}{42} \text { by -3 } \\\Rightarrow \quad \frac{-14}{42} = \frac{-14 \times -3}{42 \times -3} = \frac{42}{-126} \\ \\\text {(iv) In order to express } \frac{-14}{42} \text { as a rational number with numerator as -70, } \\ \text {we need to multiply both numerator and denominator of } \frac{-14}{42} \text { by 5 } \\\Rightarrow \quad \frac{-14}{42} = \frac{-14 \times 5}{42 \times 5} = \frac{-70}{210} \\ \\\end{array}\begin{array}{l} \text {Q9. Select those rational numbers which can be written as a rational number with numerator 6 : }\\\frac{1}{22}, \frac{2}{3}, \frac{3}{4}, \frac{4}{-5}, \frac{5}{6}, \frac{-6}{7}, \frac{-7}{8} \end{array}\begin{array}{l} \text {Sol. Given rational numbers that can be written as a rational number with numerator as 6 are }\\\frac{1}{22}=\frac{1 \times 6}{22 \times 6} = \frac{6}{132} \\\frac{2}{3}=\frac{2 \times 3}{3 \times 3} = \frac{6}{9} \\\frac{3}{4}=\frac{3 \times 2}{4 \times 2} = \frac{6}{8} \\\frac{-6}{7}=\frac{-6 \times -1}{7 \times -1} = \frac{6}{-7} \\ \end{array}\begin{array}{l} \text {Q10. Select those rational numbers which can be written as a rational number with denominator 4:} \\ \frac{7}{8}, \frac{64}{16}, \frac{36}{-12}, \frac{-16}{17}, \frac{5}{-4}, \frac{140}{28} \end{array}\begin{array}{l} \text {Sol. Given rational numbers that can be written as a rational number with denominator as 4 are }\\\frac{64}{16}=\frac{64 \div 4}{16 \div 4} = \frac{16}{4} \\\frac{36}{-12}=\frac{36 \div -3}{-12 \div -3} = \frac{-12}{4} \\\frac{5}{-4}=\frac{5 \times -1}{-4 \times -1} = \frac{-5}{4} \\\frac{140}{28}=\frac{140 \div 7}{28 \div 7} = \frac{20}{4} \\ \end{array}\begin{array}{l} \text {Q11 . In each of the following, find an equivalent form of the rational number having a common denominator: } \\ \text {(i) } \quad \frac{3}{4} \text { and } \frac{5}{12} \\ \text {(ii) } \quad \frac{2}{3}, \frac{7}{6} \text { and } \frac{11}{12} \\ \text {(iii) } \quad \frac{5}{7}, \frac{3}{8}, \frac{9}{14} \text { and } \frac{20}{21} \end{array}\begin{array}{l} \text {Sol. } \\ \text {(i) LCM of denominator 4 and 12 is 12 } \\ \therefore \frac{3}{4} =\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\\ \text {Hence, the required fractions with common denominators are } \frac{9}{12} \text { and } \frac{5}{12} \\\text {(ii) LCM of denominator 3, 6 and 12 is 12 } \\ \therefore \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \\ \frac{7}{6} =\frac{7 \times 2}{6 \times 2} = \frac{14}{12} \\ \text {Hence, the required fractions with common denominators are } \frac{8}{12}, \frac{14}{12} \text { and } \frac{11}{12} \\\text {(iii) LCM of denominator 7, 8, 14 and 21 is 168 } \\ \therefore \frac{5}{7}=\frac{5 \times 24}{7 \times 24}=\frac{120}{168} \\ \frac{3}{8}=\frac{3 \times 21}{8 \times 21}=\frac{63}{168} \\ \frac{9}{14}=\frac{9 \times 12}{14 \times 12}=\frac{108}{168} \\ \frac{20}{21}=\frac{20 \times 8}{21 \times 8}=\frac{160}{168} \\\text {Hence, the required fractions with common denominators are } \frac{120}{168}, \frac{63}{168}, \frac{108}{168} \text { and } \frac{160}{168}\end{array}