Exercise 4B
Question 1
Question 2
Question 3
(ii) $\frac{-13}{9}+\frac{7}{9}+\frac{2}{-9}$
Sol :
$=\frac{-13+7+(-2)}{9}=\frac{-13+7-2}{9}$
$=\frac{-15+7}{9}=\frac{-8}{9}$
(iii) $\frac{-1}{2}+\frac{2}{3}+\frac{4}{5}$
Sol :
L.C.M of 2, 3, 5 is 30
$\frac{-1 \times 15}{2 \times 15}+\frac{2 \times 10}{3 \times 10}+\frac{4 \times 6}{5 \times 6}$
$=\frac{-15}{30}+\frac{20}{30}+\frac{24}{30}$
$=\frac{-15+20+24}{30}=\frac{29}{30}$
(iv) $8+\left(\frac{-5}{6}\right)+1\frac{1}{5}$
Sol :
$=\frac{8}{1}-\frac{5}{6}+\frac{6}{5}$
L.C.M of 1, 6 , 5 is 30
$=\frac{8 \times 30}{1 \times 30}-\frac{5 \times 5}{6 \times 5}+\frac{6 \times 6}{5 \times 6}$
$=\frac{240-25+36}{30}=\frac{251}{30}=8\frac{11}{30}$
(v) $\frac{-12}{5}+\frac{31}{10}+\frac{11}{-15}+\frac{-7}{-18}$
Sol :
L.C.M of 5, 10 ,15 , 18 is 90
$=\frac{-12 \times 18}{5 \times 18}+\frac{31 \times 9}{10 \times 9}+\frac{11 \times 6}{-15 \times 6}+\frac{-7 \times 5}{-18 \times 5}$
$=\frac{-216}{90}+\frac{279}{90}-\frac{66}{90}+\frac{35}{90}$
$=\frac{-216+279-66+35}{90}=\frac{32}{90}=\frac{16}{45}$
(vi) $\frac{-7}{12}+\frac{13}{-24}+\frac{-29}{-36}+5$
Sol :
L.C.M of 12 , 24 , 36 is 72
$=\frac{-7 \times 6}{12 \times 6}+\frac{13 \times 3}{-24 \times 3}+\frac{-29 \times 2}{-36 \times }+\frac{5 \times 72}{1 \times 72}$
$=\frac{-42}{72}+\frac{39}{-72}+\frac{58}{72}+\frac{360}{72}$
$=\frac{-42-39+58+360}{72}=\frac{337}{72}$
$=4\frac{49}{72}$
Question 4
Express each of the following rational numbers as sum of an integer and a rational number
(i) $\frac{33}{8}$
Sol :
$=4\frac{1}{8}$
$=4+\frac{1}{8}$
(ii) $\frac{-25}{8}$
Sol :
$=-\left(3+\frac{1}{8}\right)$
$=-3-\frac{1}{8}$
(iii) $\frac{-27}{5}$
Sol :
$=-\left(5+\frac{2}{5}\right)$
$=-5-\frac{2}{5}$
(iv) $\frac{107}{-11}$
Sol :
$=-\left(9+\frac{8}{11}\right)$
$=-9-\frac{8}{11}$
Question 5
Subtract the first rational number from the second in each of the following:
(i) $\frac{2}{7} , \frac{5}{7}$
Sol :
$=\frac{5}{7}-\frac{2}{7}=\frac{5-2}{7}=\frac{3}{7}$
(ii) $\frac{-5}{11}, \frac{8}{11}$
Sol :
$=\frac{8}{11}-\left(\frac{-5}{-11}\right)$
$=\frac{8+5}{11}=\frac{13}{11}$
(iii) $\frac{-7}{25} , \frac{-14}{25}$
Sol :
$=\frac{-14}{25}-\left(\frac{-7}{25}\right)$
$=\frac{-14+7}{25}=\frac{-7}{25}$
(iv) $\frac{-12}{-19} , \frac{-7}{19}$
Sol :
$\frac{-7}{19}-\frac{12}{19}=\frac{-7-12}{19}=\frac{-19}{19}$
=-1
(v) $\frac{5}{6} , \frac{-7}{8}$
Sol :
L.C.M of 6 and 8 is 24
$=\frac{-7 \times 3}{8 \times 3} -\frac{5 \times 4}{6 \times 4}$
$=\frac{-21}{24}-\frac{20}{24}=\frac{-21-20}{24}$
$=\frac{-41}{24}$
(vi) $\frac{-3}{7} , \frac{5}{14}$
Sol :
L.C.M of 7 and 14 is 14
$\frac{-3 \times 2}{7 \times 2}$
$=\frac{5}{14}-\left(\frac{-6}{14}\right)$
$=\frac{5+6}{14}=\frac{11}{14}$
(vii) $\frac{-8}{15} , \frac{17}{-20}$
Sol :
L.C.M of 15 and 20 is 60
$\frac{-8 \times 4}{15 \times 4}=\frac{-32}{60} ,\frac{17 \times 3}{-20 \times 3}\frac{51}{-60} $
$=\frac{51}{-60}-\left(\frac{-32}{60}\right)$
$=\frac{-51+32}{60}=\frac{-19}{60}$
(viii) $\frac{-11}{26} , \frac{17}{-39}$
Sol :
L.C.M of 26 and 39 is 78
$\frac{-11 \times 3}{26 \times 3}, \frac{17 \times 2}{-39 \times 2}$
$=\frac{-34}{78}-\left(\frac{-33}{78}\right)$
$=\frac{-34+33}{78}=\frac{-1}{78}$
Question 6
Evaluate each of the following:
(i) $\frac{2}{7}-\frac{12}{-35}$
Sol :
$\frac{2 \times 5}{7 \times 5}-\frac{12}{-35}$
$=\frac{10}{35}+\frac{12}{35}=\frac{10+12}{35}=\frac{22}{35}$
(ii) $\frac{-6}{13}-5-\frac{1}{-26}$
Sol :
$=\frac{-6\times 2}{13 \times 2}-\frac{5 \times 16}{1 \times 16}-\frac{1 \times 1}{-26 \times 1}$
$=\frac{-12}{26}-\frac{130}{26}+\frac{1}{26}$
$=\frac{-12-130+1}{26}=\frac{-141}{26}=-5\frac{11}{26}$
(iii) $\frac{7}{57}-\frac{-9}{19}+\frac{5}{-38}$
Sol :
L.C.M of 57 , 19 , 38 is 114
$=\frac{7 \times 2}{57 \times 2}-\frac{-9 \times 6}{15 \times 6}+\frac{5 \times 3}{-38 \times 3}$
$=\frac{14}{114}+\frac{54}{114}-\frac{15}{114}$
$=\frac{14+54-15}{114}=\frac{53}{114}$
Question 7
(i) The sum of two rational numbers is $\frac{8}{7}$. If one of the numbers is $\frac{1}{14}$, find the other.
Sol :
Let x be the other number.
$x+\frac{1}{14}=\frac{8}{7}$
$x=\frac{8 \times 2}{7 \times 2}=\frac{1}{14}$
$x=\frac{16}{14}-\frac{1}{14}=\frac{16-1}{14}$
$=\frac{15}{14}=1\frac{1}{14}$
(ii) What number should be added to $\frac{-6}{11}$ so as to get $\frac{4}{9}$ ?
Sol :
$\frac{-6}{11}+x=\frac{4}{9}$
$x=\frac{4 \times 11}{9 \times 11}+\frac{6 \times 9}{11 \times 9}$
$x=\frac{44}{99}+\frac{54}{99}=\frac{44+54}{99}=\frac{98}{99}$
(iii) What number should be subtracted from $\frac{-8}{15}$ so as to get $\frac{-5}{6}$ ?
Sol :
$\frac{-8}{15}-x=\frac{-5}{6}$
$-x=\frac{-5 \times 5}{6 \times 5}+\frac{8 \times 2}{15 \times 2}$
$=\frac{-25}{30}+\frac{16}{30}=\frac{-25+16}{30}=\frac{-9}{30}$
$x=\frac{9}{30}=\frac{3}{10}$
Question 8
What should be added to $\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{5}\right)$ to get 4 ?
Sol :
$\left(\frac{1 \times 15}{2 \times 15}+\frac{1 \times 10}{3 \times 10}+\frac{1 \times 6}{5 \times 6}\right)+x=4$
$\left(\frac{15}{30}+\frac{10}{30}+\frac{6}{30}\right)+x=4$
$\left(\frac{15+10+6}{30}\right)+x=4$
$x=\frac{4}{1}-\frac{31}{30}$
$x=\frac{120-31}{30}=\frac{89}{30}=2\frac{29}{30}$
Question 9
Simplify : $\frac{-5}{6}-\frac{7}{-8}+\frac{11}{12}-\frac{-1}{6}+0-\frac{-7}{-16}$
Sol :
L.C.M of 6, 8, 12 , 6, 16 is 48
$=\frac{-5}{6}+\frac{7}{8}+\frac{11}{12}+\frac{1}{6}-\frac{7}{16}$
$=\frac{-5\times 8+7\times 6+11\times 4+1\times 8-7\times 3}{48}$
$=\frac{-40+42+44+8-21}{48}$
$=\frac{33}{48}=\frac{11}{16}$
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