Question

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 1

$$\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}ittakesthe\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}ittakesthe\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}$$