RD Sharma solution class 8 chapter 1 Rational Numbers Exercise 1.5

Exercise 1.5

Page-1.25

Question 1:

Multiply:
(i) $\frac{7}{11} \mathrm{by} \frac{5}{4}$
(ii) $\frac{5}{7} \mathrm{by} \frac{-3}{4}$
(iii) $\frac{-2}{9} \mathrm{by} \frac{5}{11}$
(iv) $\frac{-3}{17} \mathrm{by} \frac{-5}{-4}$
(v) $\frac{9}{-7} \mathrm{by} \frac{36}{-11}$
(vi) $\frac{-11}{13} \mathrm{by} \frac{-21}{7}$
(vii) $-\frac{3}{5} \mathrm{by} -\frac{4}{7}$
(viii) $-\frac{15}{11} \mathrm{by} 7$

Answer 1:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{7}{11}\times \frac{5}{4}=\frac{7\times 5}{11\times 4}=\frac{35}{44} \\ \left(\mathrm{ii}\right) \\ \frac{5}{7}\times \frac{-3}{4}=\frac{5\times (-3)}{7\times 4}=\frac{-15}{28} \\ \left(\mathrm{iii}\right) \\ \frac{-2}{9}\times \frac{5}{11}=\frac{(-2)\times 5}{9\times 11}=\frac{-10}{99} \\ \left(\mathrm{iv}\right) \\ \frac{-3}{17}\times \frac{-5}{-4}=\frac{(-3)\times (-5)}{17\times (-4)}=\frac{15}{-68} \\ \left(\mathrm{v}\right) \\ \frac{9}{-7}\times \frac{36}{-11}=\frac{9\times 36}{(-7)\times (-11)}=\frac{324}{77} \\ \left(\mathrm{vi}\right) \\ \frac{-11}{13}\times \frac{-21}{7}=\frac{(-11)\times (-21)}{13\times 7}=\frac{231}{91}=\frac{33}{13} \\ \left(\mathrm{vii}\right) \\ \frac{-3}{5}\times \frac{-4}{7}=\frac{(-3)\times (-4)}{5\times 7}=\frac{12}{35} \\ \left(\mathrm{viii}\right) \\ \frac{-15}{11}\times 7=\frac{-15}{11}\times \frac{7}{1}=\frac{(-15)\times 7}{11\times 1}=\frac{-105}{11} \end{gathered}$$

Question 2:

Multiply:
(i) $\frac{-5}{17} \mathrm{by} \frac{51}{-60}$

(ii) $\frac{-6}{11} \mathrm{by} \frac{-55}{36}$

(iii) $\frac{-8}{25} \mathrm{by} \frac{-5}{16}$

(iv) $\frac{6}{7} \mathrm{by} \frac{-49}{36}$

(v) $\frac{8}{-9} \mathrm{by} \frac{-7}{-16}$

(vi) $\frac{-8}{9} \mathrm{by} \frac{3}{64}$

Answer 2:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{-5}{17}\times \frac{51}{-60}=\frac{-5}{17}\times \frac{3\times 17}{-5\times 3\times 4}=\frac{-5\times 3\times 17}{-17\times 5\times 3\times 4}=\frac{1}{4} \\ \left(\mathrm{ii}\right) \\ \frac{-6}{11}\times \frac{-55}{36}=\frac{-6}{11}\times \frac{-5\times 11}{6\times 6}=\frac{5}{6} \\ \left(\mathrm{iii}\right) \\ \frac{-8}{25}\times \frac{-5}{16}=\frac{-1}{5\times 5}\times \frac{-5}{2}=\frac{1}{10} \\ \left(\mathrm{iv}\right) \\ \frac{6}{7}\times \frac{-49}{36}=\frac{6}{7}\times \frac{-7\times 7}{6\times 6}=\frac{-7}{6} \\ \left(\mathrm{v}\right) \\ \frac{8}{-9}\times \frac{-7}{-16}=\frac{1}{-9}\times \frac{-7}{-2}=\frac{-7}{18} \\ \left(\mathrm{vi}\right) \\ \frac{-8}{9}\times \frac{3}{64}=\frac{-8}{3\times 3}\times \frac{3}{8\times 8}=\frac{-1}{24} \end{gathered}$$

Page-1.26

Question 3:

Simplify each of the following and express the result as a rational number in standard form:
(i) $\frac{-16}{21}\times \frac{14}{5}$

(ii) $\frac{7}{6}\times \frac{-3}{28}$

(iii) $\frac{-19}{36}\times 16$

(iv) $\frac{-13}{9}\times \frac{27}{-26}$

(v) $\frac{-9}{16}\times \frac{-64}{-27}$

(vi) $\frac{-50}{7}\times \frac{14}{3}$

(vii) $\frac{-11}{9}\times \frac{-81}{-88}$

(viii) $\frac{-5}{9}\times \frac{72}{-25}$

Answer 3:

$$\begin{gathered} \left(\mathrm{i}\right)\frac{-16}{21}\times \frac{14}{5}=\frac{-2\times 2\times 2\times 2}{3\times 7}\times \frac{2\times 7}{5}=\frac{-32}{15} \\ \left(\mathrm{ii}\right) \frac{7}{6}\times \frac{-3}{28}=\frac{7}{2\times 3}\times \frac{-3}{2\times 2\times 7}=\frac{-1}{8} \\ \left(\mathrm{iii}\right) \frac{-19}{36}\times 16=\frac{-19}{2\times 2\times 3\times 3}\times 2\times 2\times 2\times 2=\frac{-76}{9} \\ \left(\mathrm{iv}\right) \frac{-13}{9}\times \frac{27}{-26}=\frac{-13}{3\times 3}\times \frac{3\times 3\times 3}{-2\times 13}=\frac{3}{2} \\ \left(\mathrm{v}\right) \frac{-9}{16}\times \frac{-64}{-27}=\frac{-9}{16}\times \frac{-4\times 16}{-3\times 9}=\frac{-4}{3} \\ \left(\mathrm{vi}\right) \frac{-50}{7}\times \frac{14}{3}=\frac{-50}{7}\times \frac{2\times 7}{3}=\frac{-100}{3} \\ \left(\mathrm{vii}\right) \frac{-11}{9}\times \frac{-81}{-88}=\frac{-11}{9}\times \frac{-9\times 9}{-8\times 11}=\frac{-9}{8} \\ \left(\mathrm{viii}\right) \frac{-5}{9}\times \frac{72}{-25}=\frac{-5}{9}\times \frac{8\times 9}{-5\times 5}=\frac{8}{5} \end{gathered}$$

Question 4:

Simplify:
(i) $\left(\frac{25}{8}\times \frac{2}{5}\right)-\left(\frac{3}{5}\times \frac{-10}{9}\right)$

(ii) $\left(\frac{1}{2}\times \frac{1}{4}\right)+\left(\frac{1}{2}\times 6\right)$

(iii) $\left(-5\times \frac{2}{15}\right)-\left(-6\times \frac{2}{9}\right)$

(iv) $\left(\frac{-9}{4}\times \frac{5}{3}\right)+\left(\frac{13}{2}\times \frac{5}{6}\right)$

(v) $\left(\frac{-4}{3}\times \frac{12}{-5}\right)+\left(\frac{3}{7}\times \frac{21}{15}\right)$

(vi) $\left(\frac{13}{5}\times \frac{8}{3}\right)-\left(\frac{-5}{2}\times \frac{11}{3}\right)$

(vii) $\left(\frac{13}{7}\times \frac{11}{26}\right)-\left(\frac{-4}{3}\times \frac{5}{6}\right)$

(viii) $\left(\frac{8}{5}\times \frac{-3}{2}\right)+\left(\frac{-3}{10}\times \frac{11}{16}\right)$

Answer 4:

$$\begin{gathered} \left(\mathrm{i}\right) (\frac{25}{8}\times \frac{2}{5})-(\frac{3}{5}\times \frac{-10}{9}) \\ =\frac{5}{4}-\frac{-2}{3} \\ =\frac{5\times 3+2\times 4}{12} \\ =\frac{23}{12} \\ \left(\mathrm{ii}\right) (\frac{1}{2}\times \frac{1}{4})+(\frac{1}{2}\times 6) \\ =\frac{1}{8}+3 \\ =\frac{1+3\times 8}{8} \\ =\frac{25}{8} \\ \left(\mathrm{iii}\right) (-5\times \frac{2}{15})-(-6\times \frac{2}{9}) \\ =\frac{-2}{3}-\frac{-4}{3} \\ =\frac{-2+4}{3} \\ =\frac{2}{3} \\ \left(\mathrm{iv}\right) (\frac{-9}{4}\times \frac{5}{3})+(\frac{13}{2}\times \frac{5}{6}) \\ =\frac{-3\times 5}{4}+\frac{13\times 5}{12} \\ =\frac{-15}{4}+\frac{65}{12} \\ =\frac{-15\times 3+65}{12} \\ =\frac{20}{12}=\frac{5}{3} \\ \left(\mathrm{v}\right) (\frac{-4}{3}\times \frac{12}{-5})+(\frac{3}{7}\times \frac{21}{15}) \\ =\frac{4\times 4}{5}+\frac{1\times 3}{5} \\ =\frac{16}{5}+\frac{3}{5} \\ =\frac{16+3}{5} \\ =\frac{19}{5} \\ \left(\mathrm{vi}\right) (\frac{13}{5}\times \frac{8}{3})-(\frac{-5}{2}\times \frac{11}{3}) \\ =\frac{13\times 8}{15}-\frac{-5\times 11}{6} \\ =\frac{104}{15}-\frac{-55}{6} \\ =\frac{104\times 2+55\times 5}{30} \\ =\frac{483}{30} \\ \left(\mathrm{vii}\right) (\frac{13}{7}\times \frac{11}{26})-(\frac{-4}{3}\times \frac{5}{6}) \\ =\frac{1\times 11}{7\times 2}-\frac{-2\times 5}{3\times 3} \\ =\frac{11}{14}-\frac{-10}{9} \\ =\frac{11\times 9+10\times 14}{126} \\ =\frac{239}{126} \\ \left(\mathrm{viii}\right) (\frac{8}{5}\times \frac{-3}{2})+(\frac{-3}{10}\times \frac{11}{16}) \\ =\frac{4\times (-3)}{5}+\frac{-3\times 11}{10\times 16} \\ =\frac{-12}{5}+\frac{-33}{160} \\ =\frac{-12\times 32-33}{160} \\ =\frac{-384-33}{160} \\ =\frac{-417}{160} \end{gathered}$$

Question 5:

Simplify:
(i) $\left(\frac{3}{2}\times \frac{1}{6}\right)+\left(\frac{5}{3}\times \frac{7}{2}\right)-\left(\frac{13}{8}\times \frac{4}{3}\right)$

(ii) $\left(\frac{1}{4}\times \frac{2}{7}\right)-\left(\frac{5}{14}\times \frac{-2}{3}\right)+\left(\frac{3}{7}\times \frac{9}{2}\right)$

(iii) $\left(\frac{13}{9}\times \frac{-15}{2}\right)+\left(\frac{7}{3}\times \frac{8}{5}\right)+\left(\frac{3}{5}\times \frac{1}{2}\right)$

(iv) $\left(\frac{3}{11}\times \frac{5}{6}\right)-\left(\frac{9}{12}\times \frac{4}{3}\right)+\left(\frac{5}{13}\times \frac{6}{15}\right)$

Answer 5:

$$\begin{gathered} \left(\mathrm{i}\right) (\frac{3}{2}\times \frac{1}{6})+(\frac{5}{3}\times \frac{7}{2})-(\frac{13}{8}\times \frac{4}{3}) \\ =\frac{1}{4}+\frac{35}{6}-\frac{13}{6} \\ =\frac{1\times 3+35\times 2-13\times 2}{12} \\ =\frac{3+70-26}{12} \\ =\frac{47}{12} \\ \left(\mathrm{ii}\right)(\frac{1}{4}\times \frac{2}{7})-(\frac{5}{14}\times \frac{-2}{3})+(\frac{3}{7}\times \frac{9}{2}) \\ =\frac{1}{14}-\frac{-5}{21}+\frac{27}{14} \\ =\frac{1\times 3+5\times 2+27\times 3}{42} \\ =\frac{94}{42}=\frac{47}{21} \\ \left(\mathrm{iii}\right) (\frac{13}{9}\times \frac{-15}{2})+(\frac{7}{3}\times \frac{8}{5})+(\frac{3}{5}\times \frac{1}{2}) \\ =\frac{-13\times 5}{6}+\frac{7\times 8}{15}+\frac{3}{10} \\ =\frac{-65}{6}+\frac{56}{15}+\frac{3}{10} \\ =\frac{-65\times 5+56\times 2+3\times 3}{30} \\ =\frac{-204}{30}=\frac{-34}{5} \\ \left(\mathrm{iv}\right) (\frac{3}{11}\times \frac{5}{6})-(\frac{9}{12}\times \frac{4}{3})+(\frac{5}{13}\times \frac{6}{15}) \\ =\frac{5}{22}-1+\frac{2}{13} \\ =\frac{5\times 13-286+2\times 22}{286} \\ =\frac{65-286+44}{286} \\ =\frac{-177}{286} \end{gathered}$$