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:

$(i) \frac{8}{9} + \frac{- 11}{6} L.C.M. of the denominators 9 and 6 is 18. Now, we will express \frac{8}{9} and \frac{- 11}{6} 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} (ii) 3 + \frac{5}{- 7} = \frac{3}{1} + \frac{- 5}{7} L.C.M. of the denominators 1 and 7 is 7. Now, we will express \frac{3}{1} in the form in which it takes the 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} (iii) \frac{1}{- 12} + \frac{2}{- 15} = \frac{- 1}{12} + \frac{- 2}{15} L.C.M. of the denominators 12 and 15 is 60. Now, we will express \frac{- 1}{12} and \frac{- 2}{15} in the form in which they take the 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} (iv) \frac{- 8}{19} + \frac{- 4}{57} L.C.M. of the denominators 19 and 57 is 57. Now, we will express \frac{- 8}{19} in the form in which i t t a k e s the 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} (v) \frac{7}{9} + \frac{3}{- 4} = \frac{7}{9} + \frac{- 3}{4} L.C.M. of the denominators 9 and 4 is 36. Now, we will express \frac{7}{9} and \frac{- 3}{4} in the form in which they take the denominator 36. \frac{7 \times 4}{9 \times 4} = \frac{28}{36} \frac{- 3 \times 9}{4 \times 9} = \frac{- 27}{36} So \frac{7}{9} + \frac{- 3}{4} = \frac{28}{36} + \frac{- 27}{36} = \frac{28 + (- 27)}{36} = \frac{28 - 27}{36} = \frac{1}{36} (vi) \frac{5}{26} + \frac{11}{- 39} = \frac{5}{26} + \frac{- 11}{39} L.C.M. of the denominators 26 and 39 is 78. Now, we will express \frac{5}{26} and \frac{- 11}{39} in the form in which they take the denominator 78. \frac{5 \times 3}{26 \times 3} = \frac{15}{78} \frac{- 11 \times 2}{39 \times 2} = \frac{- 22}{78} So \frac{5}{26} + \frac{- 11}{39} = \frac{15}{78} + \frac{- 22}{78} = \frac{15 - 22)}{78} = \frac{- 7}{78} (vii) \frac{- 16}{9} + \frac{- 5}{12} L.C.M. of the denominators 9 and 12 is 36. Now, we will express \frac{- 16}{9} and \frac{- 5}{12} in the form in which they take the 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} (viii) \frac{- 13}{8} + \frac{5}{36} L.C.M. of the denominators 8 and 36 is 72. Now, we will express \frac{- 13}{8} and \frac{5}{36} in the form in which they take the 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} (ix) 0 + \frac{- 3}{5} Taking L . C . M . of the denominator : = \frac{0 \times 5 - 3}{5} = \frac{- 3}{5} (x) 1 + \frac{- 4}{5} = \frac{1}{1} + \frac{- 4}{5} L.C.M. of the denominators 1 and 5 is 5. Now, we will express \frac{1}{1} in the form in which i t t a k e s the 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}$

Page No 1.6:

Question 4:

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

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

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

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

Answer:

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

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