Exercise 4A
Question 1
Find five rational number equivalent to each of the following rational numbers:
(i) $\frac{3}{5}$
Sol :
$\frac{3 \times 2}{5 \times 2}=\frac{6}{10}$
$\frac{3 \times 3}{5 \times 3}=\frac{9}{15}$
$\frac{3 \times 4}{5 \times 4}=\frac{12}{20}$
$\frac{3 \times 5}{5 \times 5}=\frac{15}{25}$
$\frac{3 \times 6}{5 \times 6}=\frac{18}{30}$
(ii) $\frac{-6}{11}$
Sol :
$\frac{-6 \times 2}{11 \times 2}=\frac{-12}{22}$
$\frac{-6 \times 3}{11 \times 3}=\frac{-18}{33}$
$\frac{-6 \times 4}{11 \times 4}=\frac{-24}{44}$
$\frac{-6 \times 5}{11 \times 5}=\frac{-30}{55}$
$\frac{-6 \times 6}{11 \times 6}=\frac{-36}{66}$
(iii) $\frac{7}{-10}$
Sol :
$\frac{7\times 2}{-10 \times 2}=\frac{14}{-20}$
$\frac{7 \times 3}{-10 \times 3}=\frac{21}{-30}$
$\frac{7 \times 4}{-10 \times 4}=\frac{28}{-40}$
$\frac{7 \times 5}{-10 \times 5}=\frac{35}{-50}$
$\frac{7 \times 6}{-10 \times 6}=\frac{42}{-60}$
(iv) $\frac{8}{15}$
Sol :
$\frac{8\times 2}{15 \times 2}=\frac{16}{30}$
$\frac{8 \times 3}{15 \times 3}=\frac{24}{45}$
$\frac{8 \times 4}{15 \times 4}=\frac{32}{60}$
$\frac{8 \times 5}{15 \times 5}=\frac{40}{75}$
$\frac{8 \times 6}{15 \times 6}=\frac{48}{90}$
Question 2
Write each of the following rational numbers with a positive denominator:
(i) $\frac{3}{-5}$
Sol :
$\frac{3}{-5} \times \frac{-1}{-1}=\frac{-3}{5}$
(ii) $\frac{8}{-17}$
Sol :
$\frac{8}{-17} \times \frac{-1}{-1}=\frac{-8}{17}$
(iii) $\frac{-21}{-25}$
Sol :
$\frac{-21}{-25} \times \frac{-1}{-1}=\frac{21}{25}$
(iv) $\frac{-40}{-59}$
Sol :
$\frac{-40}{-59} \times \frac{-1}{-1}=\frac{40}{59}$
Question 3
Express $\frac{-15}{16}$ as a rational number with:
(i) numerator=-30
Sol :
$\frac{-15}{16} \times \frac{2}{2}=\frac{-30}{32}$
(ii) numerator=75
Sol :
$\frac{-15}{16} \times \frac{-5}{-5}=\frac{75}{-80}$
(iii) denominator=48
Sol :
$\frac{-15}{16} \times \frac{3}{3}=\frac{-45}{48}$
(iv) denominator=-96
Sol :
$\frac{-15}{16} \times \frac{-6}{-6}=\frac{90}{-96}$
Question 4
Express $\frac{19}{-5}$ as a rational number with:
(i) numerator=-38
Sol :
$\frac{19}{-5} \times \frac{-2}{-2}=\frac{-38}{10}$
(ii) numerator=-95
Sol :
$\frac{19}{-5} \times \frac{-5}{-5}=\frac{-95}{25}$
(iii) denominator=-35
Sol :
$\frac{19}{-5} \times \frac{7}{7}=\frac{133}{-35}$
(iv) denominator=20
Sol :
$\frac{19}{-5} \times \frac{-4}{-4}=\frac{76}{20}$
Question 5
Express $\frac{-192}{240}$ as a rational number with numerator:
(i) 96
Sol :
$\frac{-192 \div -2}{240 \div -2} = \frac{96}{-120}$
(ii) -32
Sol :
$\frac{-192 \div 6}{240 \div 6} =\frac{-32}{40}$
(iii) 16
Sol :
$\frac{-192 \div -12}{240 \div -12} =\frac{16}{-20}$
(iv) -8
Sol :
$\frac{-192 \div 24}{240 \div 24} =\frac{-8}{10}$
Question 6
Express the following rational numbers in standard form:
(i) $\frac{-6}{15}$
Sol : $=\frac{-2}{5}$
(ii) $\frac{-18}{-24}$
Sol : $\frac{3}{4}$
(iii) $\frac{21}{-35}$
Sol : $\frac{3}{-5}\times=\frac{-3}{5}$
(iv) $\frac{-36}{84}$
Sol : $\frac{-3}{7}$
(v) $\frac{-1302}{-1953}$
Sol : $\frac{2}{3}$
Question 7
Fill in the blanks :
(i) $\frac{-3}{4}=\frac{}{12}=\frac{}{28}$
Sol :
$\frac{-3\times 3}{4 \times 3}=\frac{-9}{12}$
$\frac{-3 \times 7}{4 \times 7}=\frac{-21}{28}$
(ii) $\frac{-5}{-8}=\frac{}{16}=\frac{25}{}$
Sol :
$\frac{-5 \times -2}{-8 \times -2}=\frac{10}{16}$
$\frac{-5 \times -5}{-8 \times -5}=\frac{25}{40}$
(iii) $\frac{7}{-9}=\frac{-14}{}=\frac{35}{}$
Sol :
$\frac{7 \times -2}{-9 \times -2}=\frac{-14}{18}$
$\frac{7 \times 5}{-9 \times 5}=\frac{35}{-45}$
(iv) $\frac{-8}{}=\frac{4}{13}=\frac{}{-65}$
Sol :
$\frac{-8 \div 2}{13 \times -2 =-26}=\frac{4}{13}$
$=\frac{4 \times -5}{13 \times -5}=\frac{-20}{-65}$
Question 8
Which of the two rational number is greater?
(i) $\frac{-7}{12} \text{ or } \frac{5}{12}$
Sol : <
(ii) $\frac{-3}{14} \text{ or } \frac{-6}{7}$
Sol : >
(iii) $\frac{-7}{13} \text{ or } 0$
Sol : <
(iv) $\frac{-22}{-33} \text{ or } \frac{45}{-65}$
Sol : >
Question 9
Which of the two rational number is smaller?
(i) $\frac{-5}{8} \text{ or } \frac{3}{-8}$
Sol : <
(ii) $\frac{8}{17} \text{ or } \frac{-9}{-17}$
Sol : <
(iii) $\frac{19}{-5} \text{ or } 1$
Sol : <
(iv) $\frac{-11}{1111} \text{ or } \frac{1}{-103}$
Sol : <
Question 10
Which of the symbols '=' , < or > . Should replace the blank space.
(i) $\dfrac{5}{4} \Box{} 1$
Sol : >
(ii) $\dfrac{-1}{2} \Box \dfrac{-3}{5}$
Sol : >
(iii) $\dfrac{-3}{4} \Box{} \dfrac{-2}{3}$
Sol : <
(iv) $\left(-2\dfrac{3}{7}\right) \Box{} \left(-2\dfrac{3}{5}\right)$
Sol : >
Question 11
Arrange in order from least to greatest.
(i) $\frac{1}{3} , \frac{3}{7} , \frac{2}{5}$
Sol :
LCM of 3, 7 , 5 is 105
$\frac{1 \times 35}{3 \times 35} =\frac{35}{105}$
$\frac{3 \times 15}{7 \times 15} =\frac{45}{105}$
$\frac{2 \times 21}{5 \times 21} =\frac{42}{105}$
$\frac{35}{105}< \frac{42}{105}< \frac{45}{105}$
$\frac{1}{3} < \frac{2}{5} < \frac{3}{7}$
(ii) $\frac{-9}{6} , \frac{-4}{3} , \frac{-17}{12}$
Sol :
L.C.M of 6 , 3 , 12 is 12
$\frac{-9 \times 2}{6 \times 2} =\frac{-18}{12}$
$\frac{-4 \times 4}{3 \times 4} =\frac{-16}{12}$
$\frac{-17 \times 1}{12 \times 1} =\frac{-17}{12}$
$\frac{-18}{12} < \frac{-17}{12} < \frac{-16}{12}$
$\frac{-9}{6} < \frac{-17}{12} < \frac{-4}{3} $
(iii) $\frac{-3}{5} , \frac{7}{-10} , \frac{-5}{6}$
Sol :
L.C.M of 5 , 10 , 6 is 30
$\frac{-3 \times 6}{5 \times 6} =\frac{-18}{30}$
$\frac{7 \times -3}{-10 \times -3} =\frac{-21}{30}$
$\frac{-5 \times 5}{6 \times 5} =\frac{-25}{30}$
$\frac{-25}{30} < \frac{-21}{30} < \frac{-18}{30}$
$\frac{-5}{6} < \frac{7}{-10} < \frac{-3}{5}$
Question 12
Arrange in descending order.
(i) $\frac{3}{4} , \frac{11}{8} , \frac{9}{16} , \frac{33}{16}$
Sol :
L.C.M of 4 , 16 , 16 is 16
$\frac{3 \times 4}{4 \times 4} =\frac{12}{16}$
$\frac{11 \times 2}{8 \times 2}=\frac{22}{16}$
$\frac{9 \times 1}{16 \times 1}=\frac{9}{16}$
$\frac{33 \times 1}{16 \times 1} =\frac{33}{16}$
$\frac{33}{16} > \frac{22}{16} > \frac{12}{16} > \frac{9}{16}$
$\frac{33}{16} > \frac{11}{8} > \frac{3}{4} > \frac{9}{16}$
(ii) $\frac{1}{7} , \frac{-1}{3} , \frac{-2}{8}$
Sol :
L.C.M of 7 , 3 , 8 is 168
$\frac{1 \times 24}{7 \times 24} =\frac{24}{168}$
$\frac{-1 \times 56}{3 \times 56} =\frac{-56}{168}$
$\frac{-2 \times 21}{8 \times 21}=\frac{-42}{168}$
$\frac{24}{168} > \frac{-42}{168} >\frac{-56}{168}$
$\frac{1}{7}> \frac{-2}{8} > \frac{-1}{3}$
(iii) $\frac{3}{-5} , \frac{-7}{10} , \frac{8}{-10} , \frac{-17}{15}$
Sol :
L.C.M of 5 , 10 , 15 is 30
$\frac{3 \times -6}{-5 \times -6}=\frac{-18}{30}$
$\frac{-7 \times 3}{10 \times 3}=\frac{-21}{30}$
$\frac{8 \times -3}{-10 \times -3}=\frac{-24}{30}$
$\frac{-17 \times 2}{15 \times 2}=\frac{-34}{30}$
$\frac{-18}{30} >\frac{-21}{30}>\frac{-24}{30}>\frac{-34}{30}$
$\frac{3}{-5} > \frac{-7}{10} > \frac{8}{-10} > \frac{-17}{15}$
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