Class 8 Rational Numbers Exercise 1.1-2

\begin{array}{l} \text {Q3. Simplify: } \\ (i) \quad \frac{8}{9}+\frac{-11}{6} \\ (ii) \quad 3+\frac{5}{-7} \\ (iii) \quad \frac{1}{-12}+\frac{2}{-15} \\ (iv) \quad \frac{-8}{19}+\frac{-4}{57} \\ (v) \quad \frac{7}{9}+\frac{3}{-4} \\ (vi) \quad \frac{5}{26}+\frac{11}{-39} \\ (vii) \quad \frac{-16}{9}+\frac{-5}{12} \\ (viii) \quad \frac{-13}{8}+\frac{5}{36} \\ (ix) \quad 0+\frac{-3}{5} \\ (x) \quad 1+\frac{-4}{5} \\ \\\text {Sol. } (i) \quad \frac{8}{9}+\frac{-11}{6} \\ \text {The denominators of the given rational numbers are 9 and 6 respectively.} \\ \text {LCM of the denominators 9 and 6 is 18.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 18, we get } \\\frac{8}{9} = \frac{8 \times 2}{9 \times 2} =\frac{16}{18} \\\frac{-11}{6} = \frac{-11 \times 3}{6 \times 3} =\frac{-33}{18} \\\Rightarrow \quad \frac{8}{9}+\frac{-11}{6} = \frac{16}{18} + \frac{-33}{18} \\ =\frac{16-33}{18} = \frac{-17}{18} \\ \\(ii) \quad 3+\frac{5}{-7} \\\text {LCM of the denominators 1 and 7 is 7.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 7, we get } \\3 = \frac{3 \times 7}{7} =\frac{21}{7} \\\Rightarrow \quad 3+\frac{5}{-7} = \frac{21}{7} + \frac{5}{-7} \\ =\frac{21-5}{7} = \frac{16}{7} \\ \\(iii) \quad \frac{1}{-12}+\frac{2}{-15} \\\text {LCM of the denominators 12 and 15 is 60.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 60, we get } \\\frac{1}{-12} = \frac{1 \times 5}{-12 \times 5} =\frac{5}{-60} \\\frac{2}{-15} = \frac{2 \times 4}{-15 \times 4} =\frac{8}{-60} \\\Rightarrow \quad \frac{1}{-12}+\frac{2}{-15} = \frac{5}{-60} + \frac{8}{-60} \\ =\frac{-5-8}{60} = \frac{-13}{60} \\ \\(iv) \quad \frac{-8}{19}+\frac{-4}{57} \\\text {LCM of the denominators 19 and 57 is 57.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 57, we get } \\\frac{-8}{19} = \frac{-8 \times 3}{19 \times 3} =\frac{-24}{57} \\\Rightarrow \quad \frac{-8}{19}+\frac{-4}{57} = \frac{-24}{57}+ \frac{-4}{57} \\ =\frac{-24-4}{57} = \frac{-28}{57} \\ \\(v) \quad \frac{7}{9}+\frac{3}{-4} \\\text {LCM of the denominators 9 and 4 is 36.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 36, we get } \\\frac{7}{9} = \frac{7 \times 4}{9 \times 4} =\frac{28}{36} \\\frac{3}{-4} = \frac{3 \times 9}{-4 \times 9} =\frac{-27}{36} \\\Rightarrow \quad \frac{7}{9}+\frac{3}{-4} = \frac{28}{36}+ \frac{-27}{36} \\ =\frac{28-27}{36} = \frac{1}{36} \\ \\(vi) \quad \frac{5}{26}+\frac{11}{-39} \\\text {LCM of the denominators 26 and 39 is 78.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 78, we get } \\\frac{5}{26} = \frac{5 \times 3}{26 \times 3} =\frac{15}{78} \\\frac{11}{-39}= \frac{11 \times 2}{-39 \times 2} =\frac{22}{-78} \\\Rightarrow \quad \frac{5}{26}+\frac{11}{-39} = \frac{15}{78} + \frac{22}{-78} \\ =\frac{15-22}{78} = \frac{-7}{78} \\ \\(vii) \quad \frac{-16}{9}+\frac{-5}{12} \\\text {LCM of the denominators 9 and 12 is 36.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 36, we get } \\\frac{-16}{9} = \frac{-16 \times 4}{9 \times 4} =\frac{-64}{36} \\\frac{-5}{12}= \frac{-5 \times 3}{12 \times 3} =\frac{-15}{36} \\\Rightarrow \quad \frac{-16}{9}+\frac{-5}{12} = \frac{-64}{36} + \frac{-15}{36} \\ =\frac{-64-15}{36} = \frac{-79}{36} \\ \\(viii) \quad \frac{-13}{8}+\frac{5}{36} \\\text {LCM of the denominators 8 and 36 is 72.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 72, we get } \\\frac{-13}{8} = \frac{-13 \times 9}{8 \times 9} =\frac{-117}{72} \\\frac{5}{36}= \frac{5 \times 2}{36 \times 2} =\frac{10}{72} \\\Rightarrow \quad \frac{-13}{8}+\frac{5}{36} = \frac{-117}{72} + \frac{10}{72} \\ =\frac{-117 + 10}{72} = \frac{-107}{36} \\ \\(ix) \quad 0+\frac{-3}{5} = \frac{-3}{5} \\ \\(x) \quad 1+\frac{-4}{5} \\\text {LCM of the denominators 1 and 5 is 5.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 5, we get } \\1 = \frac{1 \times 5}{5} =\frac{5}{5} \\\Rightarrow \quad 1+\frac{-4}{5} = \frac{5}{5} +\frac{-4}{5} \\ =\frac{5-4}{5} = \frac{1}{5} \\ \\\text {Q4. Add and express the sum as a mixed fraction:} \\ (i) \quad \frac{-12}{5} \text { and } \frac{43}{10} \\ (ii) \quad \frac{24}{7} \text { and } \frac{-11}{4} \\ (iii) \quad \frac{-31}{6} \text { and } \frac{-27}{8} \\ (iv) \quad \frac{101}{6} \text { and } \frac{7}{8} \\ \\\text {Sol. (i) } \quad \frac{-12}{5} \text { and } \frac{43}{10} \\ \text {The denominators of the given rational numbers are 5 and 10 respectively.} \\ \text {LCM of the denominators 5 and 10 is 10.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 10, we get } \\\frac{-12}{5} = \frac{-12 \times 2}{5 \times 2} =\frac{-24}{10} \\\Rightarrow \quad \frac{-12}{5} + \frac{43}{10} = \frac{-24}{10} + \frac{43}{10} \\ =\frac{-24+43}{10} = \frac{19}{10} = 1 \frac{9}{10} \\ \\(ii) \quad \frac{24}{7} \text { and } \frac{-11}{4} \\\text {LCM of the denominators 7 and 4 is 28.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 28, we get } \\\frac{24}{7} = \frac{24 \times 4}{7 \times 4} =\frac{96}{28} \\ \frac{-11}{4} =\frac{-11 \times 7}{4 \times 7} =\frac{-77}{28} \\\Rightarrow \quad \frac{24}{7} + \frac{-11}{4} = \frac{96}{28} + \frac{-77}{28} \\ =\frac{96-77}{28} = \frac{19}{28} \\ \\(iii) \quad \frac{-31}{6} \text { and } \frac{-27}{8} \\\text {LCM of the denominators 6 and 8 is 24.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 24, we get } \\\frac{-31}{6} = \frac{-31 \times 4 }{6 \times 4} = =\frac{-124}{24} \\ \frac{-27}{8} =\frac{-27 \times 3}{8 \times 3} =\frac{-81}{24} \\\Rightarrow \quad \frac{-31}{6} + \frac{-27}{8} = \frac{-124}{24} + \frac{-81}{24} \\ =\frac{-124-81}{28} = \frac{-205}{24} = -8 \frac{13}{24}\\ \\(iv) \quad \frac{101}{6} \text { and } \frac{7}{8} \\\text {LCM of the denominators 6 and 8 is 24.} \\ \text {Now, re-writing both rational numbers such that they have same denominators as 24, we get } \\\frac{101}{6} = \frac{101 \times 4 }{6 \times 4} = =\frac{404}{24} \\\frac{7}{8} =\frac{7 \times 3}{8 \times 3} =\frac{21}{24} \\\Rightarrow \quad \frac{101}{6} +\frac{7}{8} = \frac{404}{24} + \frac{21}{24} \\ =\frac{404+21}{28} = \frac{425}{24} = 17 \frac{17}{24}\\ \\\end{array}