RD Sharma solution class 8 chapter 1 Rational Numbers Exercise 1.1

Exercise 1.1

Page-1.5

Question 1:

Add the following rational numbers.
(i) $\frac{-5}{7} \mathrm{and} \frac{3}{7}$
(ii) $\frac{-15}{4} \mathrm{and} \frac{7}{4}$
(iii) $\frac{-8}{11} \mathrm{and} \frac{-4}{11}$
(iv) $\frac{6}{13} \mathrm{and} \frac{-9}{13}$

Answer 1:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{-5}{7} + \frac{3}{7} = \frac{-5+3}{7} = \frac{-2}{7} \\ \left(\mathrm{ii}\right) \\ \frac{-15}{4} + \frac{7}{4} = \frac{-15+7}{4} = \frac{-8}{4} = -2 \\ \left(\mathrm{iii}\right) \\ \frac{-8}{11} + \frac{-4}{11} = \frac{-8-4}{11} = \frac{-12}{11} \\ \left(\mathrm{iv}\right) \\ \frac{6}{13} + \frac{-9}{13} = \frac{6-9}{13} = \frac{-3}{13} \end{gathered}$$

Page-1.6

Question 2:

Add the following rational numbers:
(i) $\frac{3}{4} \mathrm{and} \frac{-5}{8}$
(ii) $\frac{5}{-9} \mathrm{and} \frac{7}{3}$
(iii) $-3 \mathrm{and} \frac{3}{5}$
(iv) $\frac{-7}{27} \mathrm{and} \frac{11}{18}$
(v) $\frac{31}{-4} \mathrm{and} \frac{-5}{8}$
(vi) $\frac{5}{36} \mathrm{and} \frac{-7}{12}$
(vii) $\frac{-5}{16} \mathrm{and} \frac{7}{24}$
(viii) $\frac{7}{-18} \mathrm{and} \frac{8}{27}$

Answer 2:

$$\begin{gathered} \text{(i) } \\ \text{Clearly, denominators of the given numbers are positive. } \\ \text{The L.C.M. of denominators 4 and 8 is 8. } \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{3}{4}\mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{it} \mathrm{takes} \mathrm{the} \mathrm{denominator} \mathrm{is} 8. \\ \frac{3\times 2}{4\times 2}=\frac{6}{8} \\ \frac{3}{4} + \frac{-5}{8}\mathrm{=}\frac{6}{8}\mathrm{+}\frac{-5}{8} \\ =\frac{6+(-5)}{8} \\ =\frac{6-5}{8} \\ = \frac{1}{8} \\ \left(\mathrm{ii}\right) \\ \frac{5}{-9} + \frac{7}{3}=\frac{-5}{9} + \frac{7}{3} \\ \mathrm{The L.C.M. of} \mathrm{denominators} 9 \mathrm{and} 3 \mathrm{is} 9. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{7}{3} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{it} \mathrm{takes} \mathrm{the} \mathrm{denominator} \mathrm{is} 9. \\ \frac{7\times 3}{3\times 3}=\frac{21}{9} \\ \frac{-5}{9} + \frac{7}{3}\mathrm{=}\frac{-5}{9} + \frac{21}{9} \\ =\frac{(-5)+21}{9} \\ =\frac{-5+21}{9} \\ = \frac{16}{9} \\ \left(\mathrm{iii}\right) \\ -3 +\frac{3}{5}=\frac{-3}{1} +\frac{3}{5} \\ \mathrm{The L.C.M. of} \mathrm{denominators} 1 \mathrm{and} 5 \mathrm{is} 5. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-3}{1} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{it} \mathrm{takes} \mathrm{the} \mathrm{denominator} 5. \\ \frac{-3\times 5}{1\times 5}=\frac{-15}{5} \\ \mathrm{So} \\ \frac{-3}{1} +\frac{3}{5}=\frac{-15}{5} +\frac{3}{5} \\ = \frac{-15+3}{5} \\ = \frac{-12}{5} \\ \left(\mathrm{iv}\right) \\ \mathrm{The L.C.M. of} \mathrm{denominators} 27 \mathrm{and} 18 \mathrm{is} 54. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-7}{27} \mathrm{and} \frac{11}{18} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{they} \mathrm{take} \mathrm{the} \mathrm{denominator} 54. \\ \frac{-7\times 2}{27\times 2}=\frac{-14}{54} \\ \frac{11\times 3}{18\times 3}=\frac{33}{54} \\ \frac{-7}{27} + \frac{11}{18}=\frac{-14}{54}+\frac{33}{54} \\ = \frac{-14+33}{54} \\ = \frac{19}{54} \\ \left(\mathrm{v}\right) \\ \mathrm{We} \mathrm{have} \\ \frac{31}{-4} + \frac{-5}{8}=\frac{-31}{4} + \frac{-5}{8} \\ \mathrm{The L.C.M. of} \mathrm{denominators} 4 \mathrm{and} 8 \mathrm{is} 8. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-31}{4} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{it} \mathrm{takes} \mathrm{the} \mathrm{denominator} 8. \\ \frac{-31\times 2}{4\times 2}=\frac{-62}{8} \\ \mathrm{So} \\ \frac{-31}{4} + \frac{-5}{8}=\frac{-62}{8} + \frac{-5}{8} \\ = \frac{(-62)+(-5)}{8} \\ = \frac{-62-5}{8} \\ =\frac{-67}{8} \\ \left(\mathrm{vi}\right) \\ \mathrm{The L.C.M. of} \mathrm{denominators} 36 \mathrm{and} 12 \mathrm{is} 36. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-7}{12} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{whic}h \mathrm{it} \mathrm{takes} \mathrm{the} \mathrm{denominator} 36. \\ \frac{-7\times 3}{12\times 3}=\frac{-21}{36} \\ \mathrm{So} \\ \frac{5}{36} + \frac{-7}{12}=\frac{5}{36} + \frac{-21}{36} \\ = \frac{5+(-21)}{36} \\ =\frac{5-21}{36} \\ =\frac{-16}{36} \\ =\frac{-4}{9} \\ \left(\mathrm{vii}\right) \\ \mathrm{The L.C.M. of} \mathrm{denominators} 16 \mathrm{and} 24 \mathrm{is} 48. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-5}{16} \mathrm{and} \frac{7}{24} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{they} \mathrm{take} \mathrm{the} \mathrm{denominator} 48. \\ \frac{-5\times 3}{16\times 3}=\frac{-15}{48} \\ \frac{7\times 2}{24\times 2}=\frac{14}{48} \\ \mathrm{So} \\ \frac{-5}{16} + \frac{7}{24}=\frac{-15}{48}+\frac{14}{48} \\ = \frac{(-15)+14}{48} \\ = \frac{-15+14}{48} \\ = \frac{-1}{48} \\ \left(\mathrm{viii}\right) \\ \frac{7}{-18} + \frac{8}{27}=\frac{-7}{18} + \frac{8}{27} \\ \mathrm{The L.C.M. of} \mathrm{denominators} 18 \mathrm{and} 27 \mathrm{is} 54. \\ \mathrm{Now}, \mathrm{we} \mathrm{will} \mathrm{express} \frac{-7}{18} \mathrm{and} \frac{8}{27} \mathrm{in} \mathrm{the} \mathrm{form} \mathrm{in} \mathrm{which} \mathrm{they} \mathrm{take} \mathrm{the} \mathrm{denominator} 54. \\ \frac{-7\times 3}{18\times 3}=\frac{-21}{54} \\ \frac{8\times 2}{27\times 2}=\frac{16}{54} \\ \mathrm{So} \\ \frac{-7}{18} + \frac{8}{27}=\frac{-21}{54}+\frac{16}{54} \\ = \frac{-21+16}{54} \\ = \frac{-5}{54} \\ =\frac{-5}{54} \end{gathered}$$

Question 3:

Simplify:
(i) $\frac{8}{9}+\frac{-11}{6}$
(ii) $3+\frac{5}{-7}$
(iii) $\frac{1}{-12}+\frac{2}{-15}$
(iv) $\frac{-8}{19}+\frac{-4}{57}$
(v) $\frac{7}{9}+\frac{3}{-4}$
(vi) $\frac{5}{26}+\frac{11}{-39}$
(vii) $\frac{-16}{9}+\frac{-5}{12}$
(viii) $\frac{-13}{8}+\frac{5}{36}$
(ix) $0+\frac{-3}{5}$
(x) $1+\frac{-4}{5}$

Answer 3:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \frac{8}{9} + \frac{-11}{6} \\ \text{L.C.M. of the denominators 9 and 6 is 18. } \\ \text{Now, we will express }\frac{8}{9}\text{and }\frac{-11}{6}\text{ in the form in which they take the denominator 18.} \\ \frac{8\times 2}{9\times 2}=\frac{16}{18} \\ \frac{-11\times 3}{6\times 3}=\frac{-33}{18} \\ \frac{8}{9} + \frac{-11}{6}=\frac{16}{18} + \frac{-33}{18} \\ = \frac{16+(-33)}{18} \\ = \frac{16-33}{18} \\ =\frac{-17}{18} \\ \left(\mathrm{ii}\right) \\ 3+\frac{5}{-7}=\frac{3}{1}+\frac{-5}{7} \\ \mathrm{L.C.M. of the}\text{ denominators 1 and 7 is 7. } \\ \mathrm{Now, we will }\text{express }\frac{3}{1}\mathrm{in the form in which} \mathrm{it} \mathrm{takes} \mathrm{the }\text{ denominator 7.} \\ \frac{3\times 7}{1\times 7}=\frac{21}{7} \\ \frac{3}{1}+\frac{-5}{7}=\frac{21}{7}+\frac{-5}{7} \\ =\frac{21+(-5)}{7} \\ =\frac{21-5}{7} \\ =\frac{16}{7} \\ \left(\mathrm{iii}\right) \\ \frac{1}{-12}+\frac{2}{-15}=\frac{-1}{12}+\frac{-2}{15} \\ \mathrm{L.C.M. of the }\text{denominators 12 and 15 is 60. } \\ \mathrm{Now, we will }\text{ express }\frac{-1}{12}\text{and }\frac{-2}{15}\mathrm{in the form in which they take the }\text{denominator 60.} \\ \frac{-1\times 5}{12\times 5}=\frac{-5}{60} \\ \frac{-2\times 4}{15\times 4}=\frac{-8}{60} \\ \frac{-1}{12}+\frac{-2}{15}=\frac{-5}{60}+\frac{-8}{60} \\ =\frac{(-5)+(-8)}{60} \\ =\frac{-5-8}{60} \\ =\frac{-13}{60} \\ \left(\mathrm{iv}\right) \\ \frac{-8}{19} + \frac{-4}{57} \\ \mathrm{L.C.M. of the }\text{denominators 19 and 57 is 57. } \\ \mathrm{Now, we will }\text{express }\frac{-8}{19}\mathrm{in the form in which }it takes the \text{denominator 57.} \\ \frac{-8\times 3}{19\times 3}=\frac{-24}{57} \\ \frac{-8}{19} + \frac{-4}{57}=\frac{-24}{57} + \frac{-4}{57} \\ =\frac{-24-4}{57} \\ =\frac{-28}{57} \\ \left(\mathrm{v}\right) \\ \frac{7}{9} + \frac{3}{-4}=\frac{7}{9} + \frac{-3}{4} \\ \mathrm{L.C.M. of the }\text{denominators 9 and 4 is 36. } \\ \mathrm{Now, we will }\text{express }\frac{7}{9}\text{and }\frac{-3}{4}\mathrm{in the form in which they take the }\text{ denominator 36.} \\ \frac{7\times 4}{9\times 4}=\frac{28}{36} \\ \frac{-3\times 9}{4\times 9}=\frac{-27}{36} \\ \text{So} \\ \frac{7}{9} + \frac{-3}{4}=\frac{28}{36}+\frac{-27}{36} \\ =\frac{28+(-27)}{36} \\ =\frac{28-27}{36} \\ = \frac{1}{36} \\ \left(\mathrm{vi}\right) \\ \frac{5}{26} + \frac{11}{-39}=\frac{5}{26} + \frac{-11}{39} \\ \mathrm{L.C.M. of the }\text{denominators 26 and 39 is 78. } \\ \mathrm{Now, we will }\text{ express }\frac{5}{26}\text{and }\frac{-11}{39}\mathrm{in the form in which they take the }\text{ denominator 78.} \\ \frac{5\times 3}{26\times 3}=\frac{15}{78} \\ \frac{-11\times 2}{39\times 2}=\frac{-22}{78} \\ \text{So} \\ \frac{5}{26} + \frac{-11}{39}=\frac{15}{78}+\frac{-22}{78} \\ =\frac{15-22)}{78} \\ = \frac{-7}{78} \\ \left(\mathrm{vii}\right) \\ \frac{-16}{9} + \frac{-5}{12} \\ \mathrm{L.C.M. of the }\text{denominators 9 and 12 is 36. } \\ \mathrm{Now, we will }\text{express }\frac{-16}{9}\text{and }\frac{-5}{12}\mathrm{in the form in which they take the }\text{ denominator 36.} \\ \frac{-16\times 4}{9\times 4}=\frac{-64}{36} \\ \frac{-5\times 3}{12\times 3}=\frac{-15}{36} \\ \frac{-16}{9} + \frac{-5}{12}=\frac{-64}{36}+\frac{-15}{36} \\ = \frac{(-64)+(-15)}{36} \\ = \frac{-64-15}{36} \\ =\frac{-79}{36} \\ \left(\mathrm{viii}\right) \\ \frac{-13}{8}+ \frac{5}{36} \\ \mathrm{L.C.M. of the }\text{denominators 8 and 36 is 72. } \\ \mathrm{Now, we will }\text{express }\frac{-13}{8}\text{and }\frac{5}{36}\mathrm{in the form in which they take the }\text{ denominator 72.} \\ \frac{-13\times 9}{8\times 9}=\frac{-117}{72} \\ \frac{5\times 2}{36\times 2}=\frac{10}{72} \\ \frac{-13}{8}+ \frac{5}{36}=\frac{-117}{72}+\frac{10}{72} \\ = \frac{-117+10}{72} \\ = \frac{-107}{72} \\ \left(\mathrm{ix}\right) 0 + \frac{-3}{5} \\ \mathrm{Taking} L.C.M. \mathrm{of} \mathrm{the} \mathrm{denominator}: \\ = \frac{0\times 5-3}{5} \\ = \frac{-3}{5} \\ \left(\mathrm{x}\right) \\ 1 + \frac{-4}{5}= \frac{1}{1} + \frac{-4}{5} \\ \mathrm{L.C.M. of the }\text{denominators 1 and 5 is 5. } \\ \text{Now, we will express }\frac{1}{1}\mathrm{in the form in which }it takes the \text{ denominator 5.} \\ \frac{1\times 5}{1\times 5}=\frac{5}{5} \\ \frac{1}{1} + \frac{-4}{5}=\frac{5}{5} + \frac{-4}{5} \\ = \frac{5-4}{5} \\ = \frac{1}{5} \end{gathered}$$

Question 4:

Add and express the sum as a mixed fraction:
(i) $\frac{-12}{5} \mathrm{and} \frac{43}{10}$

(ii) $\frac{24}{7} \mathrm{and} \frac{-11}{4}$

(iii) $\frac{-31}{6} \mathrm{and} \frac{-27}{8}$

(iv) $\frac{101}{6} \mathrm{and} \frac{7}{8}$

Answer 4:

$$\begin{gathered} \left(\mathrm{i}\right) \\ \text{We have }\frac{-12}{5} + \frac{43}{10}. \\ \text{ L.C.M. of the denominators 5 and 10 is 10. } \\ \text{Now, we will express }\frac{-12}{5}\text{ in the form in which it takes the denominator 10.} \\ \frac{-12\times 2}{5\times 2}=\frac{-24}{10} \\ \therefore \frac{-12}{5} + \frac{43}{10}=\frac{-24}{10}+\frac{43}{10} \\ = \frac{-24+43}{10} \\ = \frac{19}{10} \\ = 1\frac{9}{10} \\ \left(\mathrm{ii}\right) \\ \text{We have }\frac{24}{7} + \frac{-11}{4}. \\ \mathrm{L.C.M. of the }\text{denominators 7 and 4 is 28. } \\ \mathrm{Now, we will express}\frac{24}{7}\text{and }\frac{-11}{4} in the form in which \text{they take the denominator 28.} \\ \frac{24\times 4}{7\times 4}=\frac{96}{28} \\ \frac{-11\times 7}{4\times 7}=\frac{-77}{28} \\ \therefore \frac{24}{7} + \frac{-11}{4}=\frac{96}{28}+\frac{-77}{28} \\ = \frac{96-77}{28} \\ = \frac{19}{28} \\ \left(\mathrm{iii}\right) \\ \text{We have }\frac{-31}{6}+ \frac{-27}{8}. \\ \mathrm{L.C.M. of the }\text{denominators 6 and 8 is 24. } \\ \mathrm{Now, we will express}\frac{-31}{6}\text{and }\frac{-27}{8} in the form in which\mathrm{they take the denominator} \text{24.} \\ \frac{-31\times 4}{6\times 4}=\frac{-124}{24} \\ \frac{-27\times 3}{8\times 3}=\frac{-81}{24} \\ \therefore \frac{-31}{6}+ \frac{-27}{8}=\frac{-124}{24}+\frac{-81}{24} \\ = \frac{-124-81}{24} \\ = \frac{-205}{24} \\ = -8\frac{13}{24} \\ \left(\mathrm{iv}\right) \\ \text{We have }\frac{101}{6} + \frac{7}{8}. \\ \mathrm{L.C.M. of the }\text{denominators 6 and 8 is 24. } \\ \mathrm{Now, we will express}\frac{101}{6}\text{and }\frac{7}{8} in the form in which\mathrm{they take the denominator} \text{ 24.} \\ \frac{101\times 4}{6\times 4}=\frac{404}{24} \\ \frac{7\times 3}{8\times 3}=\frac{21}{24} \\ \therefore \frac{101}{6} + \frac{7}{8}=\frac{404}{24}+\frac{21}{24} \\ = \frac{404+21}{24} \\ = \frac{425}{24} \\ = 17\frac{17}{24} \end{gathered}$$