Class 7 Operations on Rational Numbers Exercise 5.3

\begin{array}{l} \text {Q1. Multiply: } \\ \text {(i) } \quad \frac{7}{11} \text { by } \frac{5}{4} \quad \text {(ii) } \quad \frac{5}{7} \text { by } (\frac{-3}{4}) \quad \text {(iii) } \quad \frac{(-2)}{9} \text { by } \frac{5}{11} \quad \text {(iv) } \quad \frac{-3}{17} \text { by } \frac{-5}{-4} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad\frac{7}{11} \times \frac{5}{4} \\ =\frac{7 \times 5}{11 \times 4}=\frac{35}{44} \\ \\\text {(ii) } \quad \frac{5}{7} \times \frac{-3}{4} \\ =\frac{5 \times-3}{7 \times 4}=\frac{-15}{28} \\ \\\text {(iii) } \quad \frac{-2}{9} \times \frac{5}{11} \\ =\frac{-2 \times 5}{9 \times 11}=\frac{-10}{99} \\ \\\text {(iv) } \quad \frac{-3}{17} \times \frac{-5}{-4} \\ =\frac{-3 \times 5}{17 \times-4}=\frac{-15}{68} \\ \end{array}\begin{array}{l} \text {Q2. Multiply: } \\ \text {(i) } \quad \frac{-5}{17} \text { by } \frac{51}{-60} \quad \text {(ii) } \quad \frac{-6}{11} \text { by } \frac{-55}{36} \quad \text {(iii) } \quad \frac{-8}{25} \text { by } \frac{-5}{16} \quad \text {(iv) } \quad \frac{6}{7} \text { by } \frac{-49}{36} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \frac{-5}{17} \times \frac{51}{-60} \\ =\frac{-5}{17} \times \frac{17 \times 3}{-5 \times 3 \times 4}=\frac{1}{4} \\ \\\text {(ii) } \frac{-6}{11} \times \frac{-55}{36} \\ =\frac{-6}{11} \times \frac{-5 \times 11}{6 \times 6}=\frac{5}{6} \\ \\\text {(iii) } \frac{-8}{25} \times \frac{-5}{16} \\ =\frac{-8}{5 \times 5} \times \frac{-5}{8 \times 2}=\frac{1}{10} \\ \\\text {(iv) } \quad \frac{6}{7} \times \frac{-49}{36} \\ =\frac{6}{7} \times \frac{-7 \times 7}{6 \times 6}=\frac{-7}{6}\end{array}\begin{array}{l} \text {Q3. Simplify each of the following and express the result as a rational number in standard form: } \\ \text {(i) } \quad \frac{-16}{21} \times \frac{14}{5} \quad \text {(ii) } \quad \frac{7}{6} \times \frac{-3}{28} \quad \text {(iii) } \quad \frac{-19}{36} \times 16 \quad \text {(iv) } \quad \frac{-13}{9} \times \frac{27}{-26} \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad \frac{-16}{21} \times \frac{14}{5} \\ =\frac{-16 \times 14}{21 \times 5}=\frac{-32}{15} \\ \\\text {(ii) } \quad \frac{7}{6} \times \frac{-3}{28} \\ =\frac{7 \times-3}{6 \times 28}=\frac{-1}{8} \\ \\\text {(iii) } \quad \frac{-19}{36} \times 16 \\ =\frac{-19 \times 16}{36}=\frac{-76}{9} \\ \\\text {(iv) } \quad \frac{-13}{9} \times \frac{27}{-26} \\ =\frac{-13 \times 27}{9 \times-26}=\frac{3}{2}\end{array}\begin{array}{l} \text {Q4. Simplify: } \\ \text {(i) } \quad (-5 \times \frac{2}{15})-(-6 \times \frac{2}{9}) \quad \text {(ii) } \quad (\frac{-9}{4} \times \frac{5}{3})+(\frac{13}{2} \times \frac{5}{6}) \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (-5 \times \frac{2}{15})-(-6 \times \frac{2}{9}) \\ =(-5 \times \frac{2}{3 \times 5})-(-6 \times \frac{2}{3 \times 3}) \\=(\frac{-2}{3})-(\frac{-4}{3})=\frac{-2+4}{3}=\frac{2}{3} \\ \\\text {(ii) } \quad (\frac{-9}{4} \times \frac{5}{3})+(\frac{13}{2} \times \frac{5}{6}) \\ =(\frac{-3 \times 3}{4} \times \frac{5}{3})+(\frac{65}{12}) \\=(\frac{-15}{4})+(\frac{65}{12})=\frac{-15 \times 3}{4 \times 3}+\frac{65}{12}=\frac{-45+65}{12} \\ =\frac{20}{12}=\frac{5 \times 4}{3 \times 4} \\ =\frac{5}{3}\end{array}\begin{array}{l} \text {Q5. Simplify: } \\ \text {(i) } \quad (\frac{13}{9} \times \frac{-15}{2})+(\frac{7}{3} \times \frac{8}{5})+(\frac{3}{5} \times \frac{1}{2}) \\ \text {(ii) } \quad (\frac{3}{11} \times \frac{5}{6})-(\frac{9}{12} \times \frac{4}{3})+(\frac{5}{13} \times \frac{6}{15}) \end{array}\begin{array}{l} \text {Sol. } \\\text {(i) } \quad (\frac{13}{9} \times \frac{-15}{2})+(\frac{7}{3} \times \frac{8}{5})+(\frac{3}{5} \times \frac{1}{2}) \\=(\frac{13}{3 \times 3} \times \frac{-3 \times 5}{2})+(\frac{56}{15})+(\frac{3}{10})=\frac{-65}{6}+\frac{56}{15}+\frac{3}{10} \\=\frac{-65 \times 5}{6 \times 5}+\frac{56 \times 2}{15 \times 2}+\frac{3 \times 3}{10 \times 3} \\ =\frac{-325}{30}+\frac{112}{30}+\frac{9}{30}=\frac{-325+112+9}{30}=\frac{-204}{30} \\ =\frac{-34}{5} \\ \\\text {(ii) } \quad (\frac{3}{11} \times \frac{5}{6})-(\frac{9}{12} \times \frac{4}{3})+(\frac{5}{13} \times \frac{6}{15}) \\ =(\frac{3}{11} \times \frac{5}{2 \times 3})-(\frac{3 \times 3}{4 \times 3} \times \frac{4}{3})+(\frac{5}{13} \times \frac{3 \times 2}{5 \times 3}) \\ =\frac{5}{22}-\frac{1}{1}+\frac{2}{13} \\=\frac{5 \times 13}{22 \times 13}-\frac{1 \times 286}{1 \times 286}+\frac{2 \times 22}{13 \times 22} \\ =\frac{65}{286}-\frac{286}{286}+\frac{44}{286} \\ =\frac{65-286+44}{286}=\frac{-177}{286}\end{array}