RD Sharma solution class 8 chapter 1 Rational Numbers Exercise 1.3

Exercise 1.3

Page-1.18

Question 1:

Subtract the first rational number from the second in each of the following:
(i) $\frac{3}{8}, \frac{5}{8}$
(ii) $\frac{-7}{9}, \frac{4}{9}$
(iii) $\frac{-2}{11}, \frac{-9}{11}$
(iv) $\frac{11}{13}, \frac{-4}{13}$
(v) $\frac{1}{4}, \frac{-3}{8}$
(vi) $\frac{-2}{3}, \frac{5}{6}$
(vii) $\frac{-6}{7}, \frac{-13}{14}$
(viii) $\frac{-8}{33}, \frac{-7}{22}$

Answer 1:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{5}{8}-\frac{3}{8}=\frac{5-3}{8}=\frac{2}{8}=\frac{1}{4} \\ \left(\mathrm{ii}\right) \\ \frac{4}{9}-\frac{-7}{9}=\frac{4-(-7)}{9}=\frac{4+7}{9}=\frac{11}{9} \\ \left(\mathrm{iii}\right) \\ \frac{-9}{11}-\frac{-2}{11}=\frac{(-9)-(-2)}{11}=\frac{-9+2}{11}=\frac{-7}{11} \\ \left(\mathrm{iv}\right) \\ \frac{-4}{13}-\frac{11}{13}=\frac{-4-11}{13}=\frac{-15}{13} \\ \left(\mathrm{v}\right) \\ \frac{-3}{8}-\frac{1}{4}=\frac{-3-2}{8}=\frac{-5}{8} \\ \left(\mathrm{vi}\right) \\ \frac{5}{6}-\frac{-2}{3}=\frac{5}{6}-\frac{-4}{6}=\frac{5-(-4)}{6}=\frac{5+4}{6}=\frac{9}{6}=\frac{3}{2} \\ \left(\mathrm{vii}\right) \\ \frac{-13}{14}-\frac{-6}{7}=\frac{-13-(-12)}{14}=\frac{-13+12}{14}=\frac{-1}{14} \\ \left(\mathrm{viii}\right) \\ \frac{-7}{22}-\frac{-8}{33}=\frac{-21}{66}-\frac{-16}{66}=\frac{(-21)-(-16)}{66}=\frac{-21+16}{66}=\frac{-5}{66} \end{gathered}$$

Question 2:

Evaluate each of the following:
(i) $\frac{2}{3}-\frac{3}{5}$
(ii) $-\frac{4}{7}-\frac{2}{-3}$
(iii) $\frac{4}{7}-\frac{-5}{-7}$
(iv) $-2-\frac{5}{9}$
(v) $\frac{-3}{-8}-\frac{-2}{7}$
(vi) $\frac{-4}{13}-\frac{-5}{26}$
(vii) $\frac{-5}{14}-\frac{-2}{7}$
(viii) $\frac{13}{15}-\frac{12}{25}$
(ix) $\frac{-6}{13}-\frac{-7}{13}$
(x) $\frac{7}{24}-\frac{19}{36}$
(xi) $\frac{5}{63}-\frac{-8}{21}$

Answer 2:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{2}{3}-\frac{3}{5}=\frac{10}{15}-\frac{9}{15}=\frac{10-9}{15}=\frac{1}{15} \\ \left(\mathrm{ii}\right) \\ \frac{-4}{7}-\frac{2}{-3}=\frac{-12}{21}-\frac{-14}{21}=\frac{(-12)-(-14)}{21}=\frac{-12+14}{21}=\frac{2}{21} \\ \left(\mathrm{iii}\right) \\ \frac{4}{7}-\frac{-5}{-7}=\frac{4-5}{7}=\frac{-1}{7} \\ \left(\mathrm{iv}\right) \\ -2-\frac{5}{9}=\frac{-18}{9}-\frac{5}{9}=\frac{-18-5}{9}=\frac{-23}{9} \\ \left(\mathrm{v}\right) \\ \frac{-3}{-8}-\frac{-2}{7}=\frac{21}{56}-\frac{-16}{56}=\frac{21-(-16)}{56}=\frac{21+16}{56}=\frac{37}{56} \\ \left(\mathrm{vi}\right) \\ \frac{-4}{13}-\frac{-5}{26}=\frac{-8}{26}-\frac{-5}{26}=\frac{-8-(-5)}{26}=\frac{-8+5}{26}=\frac{-3}{26} \\ \left(\mathrm{vii}\right) \\ \frac{-5}{14}-\frac{-2}{7}=\frac{-5}{14}-\frac{-4}{14}=\frac{-5-(-4)}{14}=\frac{-5+4}{14}=\frac{-1}{14} \\ \left(\mathrm{viii}\right) \\ \frac{13}{15}-\frac{12}{25}=\frac{65}{75}-\frac{36}{75}=\frac{65-36}{75}=\frac{29}{75} \\ \left(\mathrm{ix}\right) \\ \frac{-6}{13}-\frac{-7}{13}=\frac{-6-(-7)}{13}=\frac{-6+7}{13}=\frac{1}{13} \\ \left(\mathrm{x}\right) \\ \frac{7}{24}-\frac{19}{36}=\frac{21}{72}-\frac{38}{72}=\frac{21-38}{72}=\frac{-17}{72} \\ \left(\mathrm{xi}\right) \\ \frac{5}{63}-\frac{-8}{21}=\frac{5}{63}-\frac{-24}{63}=\frac{5-(-24)}{63}=\frac{5+24}{63}=\frac{29}{63} \end{gathered}$$

Question 3:

The sum of the two numbers is $\frac{5}{9}.$ If one of the numbers is $\frac{1}{3},$ find the other.

Answer 3:

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} the \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{numbers} \mathrm{is} \frac{5}{9}, \mathrm{where} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{number}s \mathrm{is} \frac{1}{3}. \\ \mathrm{Let} \mathrm{the} \mathrm{other} \mathrm{number} \mathrm{be} x. \\ \therefore x+\frac{1}{3}=\frac{5}{9} \\ \Rightarrow x=\frac{5}{9}-\frac{1}{3} \\ \Rightarrow x=\frac{5}{9}-\frac{3}{9} \\ \Rightarrow x=\frac{5-3}{9} \\ \Rightarrow x=\frac{2}{9} \end{gathered}$$

Question 4:

The sum of two numbers is $\frac{-1}{3}.$ If one of the numbers is $\frac{-12}{3},$ find the other.

Answer 4:

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} the \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{numbers} \mathrm{is} \frac{-1}{3}, \mathrm{where} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{number}s \mathrm{is} \frac{-12}{3}. \\ \mathrm{Let} \mathrm{the} \mathrm{other} \mathrm{number} be\mathit{ }x\mathit{.} \\ \therefore x+\frac{-12}{3}=\frac{-1}{3} \\ \Rightarrow x=\frac{-1}{3}-\frac{-12}{3} \\ \Rightarrow x=\frac{-1-(-12)}{3}=\frac{-1+12}{3}=\frac{11}{3} \end{gathered}$$

Question 5:

The sum of two numbers is $\frac{-4}{3}.$ If one of the numbers is −5, find the other.

Answer 5:

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{numbers} \mathrm{is} \frac{-4}{3}, \mathrm{where} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{number}s \mathrm{is} -5. \\ \mathrm{Let} \mathrm{the} \mathrm{other} \mathrm{number} \mathrm{be} x. \\ \therefore x+(-5)=\frac{-4}{3} \\ \Rightarrow x=\frac{-4}{3}-\left(\frac{-5}{1}\right) \\ \Rightarrow x=\frac{-4}{3}-\frac{-15}{3} \\ \Rightarrow x=\frac{-4-(-15)}{3} \\ \Rightarrow x=\frac{-4+15}{3}=\frac{11}{3} \end{gathered}$$

Question 6:

The sum of two rational numbers is −8. If one of the numbers is $\frac{-15}{7},$ find the other.

Answer 6:

$$\begin{gathered} \mathrm{It} \mathrm{is} \mathrm{given} \mathrm{that} the \mathrm{sum} \mathrm{of} \mathrm{two} \mathrm{rational} \mathrm{numbers} \mathrm{is} -8, \mathrm{where} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{number}s \mathrm{is} \frac{-15}{7}. \\ \mathrm{Le}\mathrm{t} \mathrm{the} \mathrm{other} \mathrm{rational} \mathrm{number} \mathrm{be} x. \\ \therefore x + \left(\frac{-15}{7}\right)=-8 \\ \Rightarrow x=\frac{-8}{1}-\frac{-15}{7} \\ \Rightarrow x=\frac{-56}{7}-\frac{-15}{7} \\ =\frac{-56-(-15)}{7} \\ =\frac{-56+15}{7} \\ =\frac{-41}{7} \\ \mathrm{Therefore}, \mathrm{the} \mathrm{other} \mathrm{rational} \mathrm{number} \mathrm{is} \frac{-41}{7}. \end{gathered}$$

Question 7:

What should be added to $\frac{-7}{8}$ so as to get $\frac{5}{9}?$

Answer 7:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{added} \mathrm{to} \frac{-7}{8} \mathrm{so} \mathrm{as} \mathrm{to} \mathrm{get} \frac{5}{9}. \\ \therefore x+\frac{-7}{8}=\frac{5}{9} \\ \Rightarrow x=\frac{5}{9}-\frac{-7}{8} \\ \Rightarrow x=\frac{40}{72}-\frac{-63}{72} \\ \Rightarrow x=\frac{40-(-63)}{72} \\ \Rightarrow x=\frac{40+63}{72}=\frac{103}{72} \end{gathered}$$

Question 8:

What number should be added to $\frac{-5}{11}$ so as to get $\frac{26}{33}?$

Answer 8:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{added}. \\ \therefore x+\frac{-5}{11}=\frac{26}{33} \\ \Rightarrow x=\frac{26}{33}-\frac{-5}{11} \\ \Rightarrow x=\frac{26}{33}-\frac{-15}{33} \\ \Rightarrow x=\frac{26-(-15)}{33} \\ \Rightarrow x=\frac{26+15}{33} \\ \Rightarrow x=\frac{41}{33} \end{gathered}$$

Question 9:

What number should be added to $\frac{-5}{7}$ to get $\frac{-2}{3}?$

Answer 9:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{added}. \\ \therefore x+\frac{-5}{7}=\frac{-2}{3} \\ \Rightarrow x=\frac{-2}{3}-\frac{-5}{7} \\ \Rightarrow x=\frac{-14}{21}-\frac{-15}{21} \\ \Rightarrow x=\frac{-14-(-15)}{21} \\ \Rightarrow x=\frac{-14+15}{21} \\ \Rightarrow x=\frac{1}{21} \end{gathered}$$

Question 10:

What number should be subtracted from $\frac{-5}{3}$ to get $\frac{5}{6}?$

Answer 10:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{subtracted}. \\ \therefore \frac{-5}{3}-x=\frac{5}{6} \\ \Rightarrow x=\frac{-5}{3}-\frac{5}{6} \\ \Rightarrow x=\frac{-10}{6}-\frac{5}{6} \\ \Rightarrow x=\frac{-10-5}{6}=\frac{-15}{6}=\frac{-5}{2} \end{gathered}$$

Page-1.19

Question 11:

What number should be subtracted from $\frac{3}{7}$ to get $\frac{5}{4}?$

Answer 11:

$$\begin{gathered} \mathrm{Let}, x \mathrm{be} \mathrm{subtracted}. \\ \therefore \frac{3}{7}-x=\frac{5}{4} \\ \Rightarrow x=\frac{3}{7}-\frac{5}{4} \\ \Rightarrow x=\frac{12}{28}-\frac{35}{28} \\ \Rightarrow x=\frac{12-35}{28}=\frac{-23}{28} \end{gathered}$$

Question 12:

What should be added to $\left(\frac{2}{3}+\frac{3}{5}\right)$ to get $\frac{-2}{15}?$

Answer 12:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{added}. \\ \therefore x+(\frac{2}{3}+\frac{3}{5})=\frac{-2}{15} \\ \Rightarrow x+(\frac{10}{15}+\frac{9}{15})=\frac{-2}{15} \\ \Rightarrow x+\left(\frac{10+9}{15}\right)=\frac{-2}{15} \\ \Rightarrow x=\frac{-2}{15}-\frac{19}{15} \\ \Rightarrow x=\frac{-2-19}{15} \\ \Rightarrow x=\frac{-21}{15} \\ \Rightarrow x=\frac{-7}{5} \end{gathered}$$

Question 13:

What should be added to $\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{5}\right)$ to get 3?

Answer 13:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{added}. \\ \therefore x+(\frac{1}{2}+\frac{1}{3}+\frac{1}{5})=3 \\ \Rightarrow x+(\frac{15}{30}+\frac{10}{30}+\frac{6}{30})=3 \\ \Rightarrow x+\left(\frac{15+10+6}{30}\right)=3 \\ \Rightarrow x+\frac{31}{30}=3 \\ \Rightarrow x=\frac{3}{1}-\frac{31}{30} \\ \Rightarrow x=\frac{90}{30}-\frac{31}{30} \\ \Rightarrow x=\frac{90-31}{30} \\ \Rightarrow x=\frac{59}{30} \end{gathered}$$

Question 14:

What should be subtracted from $\left(\frac{3}{4}-\frac{2}{3}\right)$ to get $\frac{-1}{6}?$

Answer 14:

$$\begin{gathered} \mathrm{Let} x \mathrm{be} \mathrm{subtracted}. \\ \therefore (\frac{3}{4}-\frac{2}{3})-x=\frac{-1}{6} \\ \Rightarrow (\frac{9}{12}-\frac{8}{12})-x=\frac{-1}{6} \\ \Rightarrow \left(\frac{9-8}{12}\right)-x=\frac{-1}{6} \\ \Rightarrow x=\frac{1}{12}-\frac{-1}{6} \\ \Rightarrow x=\frac{1}{12}-\frac{-2}{12} \\ \Rightarrow x=\frac{1-(-2)}{12} \\ \Rightarrow x=\frac{1+2}{12} \\ \Rightarrow x=\frac{3}{12} \\ \Rightarrow x=\frac{1}{4} \end{gathered}$$

Question 15:

Fill in the blanks:
(i) $\frac{-4}{13}-\frac{-3}{26}=. . .$
(ii) $\frac{-9}{14}+. . .=-1$
(iii) $\frac{-7}{9}+. . .=3$
(iv) $. . .+\frac{15}{23}=4$

Answer 15:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{-4}{13}-\frac{-3}{26}=\frac{-8}{26}-\frac{-3}{26} =\frac{-8-(-3)}{26}=\frac{-8+3}{26}=\frac{-5}{26} \\ \left(\mathrm{ii}\right) \frac{-9}{14}+x=-1 \\ \Rightarrow x=-1-\frac{-9}{14} \\ \Rightarrow x=\frac{-14}{14}-\frac{-9}{14} \\ \Rightarrow x=\frac{-14-(-9)}{14} \\ \Rightarrow x=\frac{-14+9}{14} \\ \Rightarrow x=\frac{-5}{14} \\ \left(\mathrm{iii}\right) \frac{-7}{9}+x=3 \\ \Rightarrow x=3-\frac{-7}{9} \\ \Rightarrow x=\frac{27}{9}-\frac{-7}{9} \\ \Rightarrow x=\frac{27-(-7)}{9} \\ \Rightarrow x=\frac{27+7}{9} \\ \Rightarrow x= \frac{34}{9} \\ \left(\mathrm{iv}\right) x+\frac{15}{23}=4 \\ \Rightarrow x=4- \frac{15}{23} \\ \Rightarrow x=\frac{92}{23}- \frac{15}{23} \\ \Rightarrow x=\frac{92-15}{23} \\ \Rightarrow x=\frac{77}{23} \end{gathered}$$