RD Sharma solution class 8 chapter 1 Rational Numbers Exercise 1.4

Exercise 1.4

Page-1.22

Question 1:

Simplify each of the following and write as a rational number of the form $\frac{p}{q}:$
(i) $\frac{3}{4}+\frac{5}{6}+\frac{-7}{8}$
(ii) $\frac{2}{3}+\frac{-5}{6}+\frac{-7}{9}$
(iii) $\frac{-11}{2}+\frac{7}{6}+\frac{-5}{8}$
(iv) $\frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15}$
(v) $\frac{-9}{10}+\frac{22}{15}+\frac{13}{-20}$
(vi) $\frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3$

Answer 1:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{3}{4}+\frac{5}{6}+\frac{-7}{8} \\ =\frac{18}{24}+\frac{20}{24}+\frac{-21}{24} \\ =\frac{18+20+(-21)}{24} \\ =\frac{18+20-21}{24} \\ =\frac{17}{24} \\ \left(\mathrm{ii}\right) \\ \frac{2}{3}+\frac{-5}{6}+\frac{-7}{9} \\ =\frac{12}{18}+\frac{-15}{18}+\frac{-14}{18} \\ =\frac{12+(-15)+(-14)}{18} \\ =\frac{12-15-14}{18} \\ =\frac{-17}{18} \\ \left(\mathrm{iii}\right) \\ \frac{-11}{2}+\frac{7}{6}+\frac{-5}{8} \\ =\frac{-132}{24}+\frac{28}{24}+\frac{-15}{24} \\ =\frac{(-132)+28+(-15)}{24} \\ =\frac{-132+28-15}{24} \\ =\frac{-119}{24} \\ \left(\mathrm{iv}\right) \\ \frac{-4}{5}+\frac{-7}{10}+\frac{-8}{15} \\ =\frac{-24}{30}+\frac{-21}{30}+\frac{-16}{30} \\ =\frac{(-24)+(-21)+(-16)}{30} \\ =\frac{-24-21-16}{30} \\ =\frac{-61}{30} \\ \left(\mathrm{v}\right) \\ \frac{-9}{10}+\frac{22}{15}+\frac{13}{-20} \\ =\frac{-54}{60}+\frac{88}{60}+\frac{-39}{60} \\ =\frac{(-54)+88+(-39)}{60} \\ =\frac{-54+88-39}{60} \\ =\frac{-5}{60} \\ =\frac{-1}{12} \\ \left(\mathrm{vi}\right) \\ \frac{5}{3}+\frac{3}{-2}+\frac{-7}{3}+3 \\ =\frac{10}{6}+\frac{-9}{6}+\frac{-14}{6}+\frac{18}{6} \\ =\frac{10+(-9)+(-14)+18}{6} \\ =\frac{10-9-14+18}{6} \\ =\frac{5}{6} \end{gathered}$$

Page-1.23

Question 2:

Express each of the following as a rational number of the form $\frac{p}{q}:$
(i) $\frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3$
(ii) $\frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7}$
(iii) $\frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6}$
(iv) $\frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14}$
(v) $\frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2$

Answer 2:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{-8}{3}+\frac{-1}{4}+\frac{-11}{6}+\frac{3}{8}-3 \\ =\frac{-64}{24}+\frac{-6}{24}+\frac{-44}{24}+\frac{9}{24}-\frac{72}{24} \\ =\frac{(-64)+(-6)+(-44)+9+(-72)}{24} \\ =\frac{-64-6-44+9-72}{24} \\ =\frac{-177}{24} \\ =\frac{-59}{8} \\ \left(\mathrm{ii}\right) \\ \frac{6}{7}+1+\frac{-7}{9}+\frac{19}{21}+\frac{-12}{7} \\ =\frac{54}{63}+\frac{63}{63}+\frac{-49}{63}+\frac{57}{63}+\frac{-108}{63} \\ =\frac{54+63+(-49)+57+(-108)}{63} \\ =\frac{54+63-49+57-108}{63} \\ =\frac{17}{63} \\ \left(\mathrm{iii}\right) \\ \frac{15}{2}+\frac{9}{8}+\frac{-11}{3}+6+\frac{-7}{6} \\ =\frac{180}{24}+\frac{27}{24}+\frac{-88}{24}+\frac{144}{24}+\frac{-28}{24} \\ =\frac{180+27+(-88)+144+(-28)}{24} \\ =\frac{180+27-88+144-28}{24} \\ =\frac{235}{24} \\ \left(\mathrm{iv}\right) \\ \frac{-7}{4}+0+\frac{-9}{5}+\frac{19}{10}+\frac{11}{14} \\ =\frac{-245}{140}+\frac{-252}{140}+\frac{266}{140}+\frac{110}{140} \\ =\frac{(-245)+(-252)+266+110}{140} \\ =\frac{-245-252+266+110}{140} \\ =\frac{-121}{140} \\ \left(\mathrm{v}\right) \\ \frac{-7}{4}+\frac{5}{3}+\frac{-1}{2}+\frac{-5}{6}+2 \\ =\frac{-21}{12}+\frac{20}{12}+\frac{-6}{12}+\frac{-10}{12}+\frac{24}{12} \\ =\frac{(-21)+20+(-6)+(-10)+24}{12} \\ =\frac{-21+20-6-10+24}{12} \\ =\frac{7}{12} \end{gathered}$$

Question 3:

Simplify:
(i) $\frac{-3}{2}+\frac{5}{4}-\frac{7}{4}$

(ii) $\frac{5}{3}-\frac{7}{6}+\frac{-2}{3}$

(iii) $\frac{5}{4}-\frac{7}{6}-\frac{-2}{3}$

(iv) $\frac{-2}{5}-\frac{-3}{10}-\frac{-4}{7}$

(v) $\frac{5}{6}+\frac{-2}{5}-\frac{-2}{15}$

(vi) $\frac{3}{8}-\frac{-2}{9}+\frac{-5}{36}$

Answer 3:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{-3}{2}+\frac{5}{4}-\frac{7}{4} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{-6}{4}+\frac{5}{4}-\frac{7}{4} \\ =\frac{-6+5-7}{4} \\ =\frac{-8}{4} \\ =- 2 \\ \left(\mathrm{ii}\right) \\ \frac{5}{3}-\frac{7}{6}+\frac{-2}{3} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{10}{6}-\frac{7}{6}+\frac{-4}{6} \\ =\frac{10-7+(-4)}{6} \\ =\frac{10-7-4}{6} \\ =\frac{-1}{6} \\ \left(\mathrm{iii}\right) \\ \frac{5}{4}-\frac{7}{6}-\frac{-2}{3} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{15}{12}-\frac{14}{12}-\frac{-8}{12} \\ =\frac{15-14-(-8)}{12} \\ =\frac{15-14+8}{12} \\ =\frac{9}{12} \\ =\frac{3}{4} \\ \left(\mathrm{iv}\right) \\ \frac{-2}{5}-\frac{-3}{10}-\frac{-4}{7} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{-28}{70}-\frac{-21}{70}-\frac{-40}{70} \\ =\frac{(-28)-(-21)-(-40)}{70} \\ =\frac{-28+21+40}{70} \\ =\frac{33}{70} \\ \left(\mathrm{v}\right) \\ \frac{5}{6}+\frac{-2}{5}-\frac{-2}{15} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{25}{30}+\frac{-12}{30}-\frac{-4}{30} \\ =\frac{25+(-12)-(-4)}{30} \\ =\frac{25-12+4}{30} \\ =\frac{17}{30} \\ \left(\mathrm{vi}\right) \\ \frac{3}{8}-\frac{-2}{9}+\frac{-5}{36} \\ \mathrm{Taking} the L.C.M.\hspace{0.17em}\mathrm{of} \mathrm{the} \mathrm{denominators}: \\ \frac{27}{72}-\frac{-16}{72}+\frac{-10}{72} \\ =\frac{27-(-16)+(-10)}{72} \\ =\frac{27+16-10}{72} \\ =\frac{33}{72} \\ =\frac{11}{24} \end{gathered}$$