Question

Add the following rational numbers:

(i) $\frac{-2}{5}\mathrm{and}\frac{3}{4}$
(ii) $\frac{-5}{9}\mathrm{and}\frac{2}{3}$
(iii) $-4\mathrm{and}\frac{1}{2}$
(iv) $\frac{-7}{27}\mathrm{and}\frac{5}{18}$
(v) $\frac{-5}{36}\mathrm{and}\frac{-7}{12}$
(vi) $\frac{1}{-9}\mathrm{and}\frac{4}{-27}$
(vii) $\frac{-9}{24}\mathrm{and}\frac{-1}{8}$
(viii) $\frac{27}{-4}\mathrm{and}\frac{-15}{8}$

Answer 1

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

$\begin{array}{l}\text{The denominators of the given rational numbers are 5 and 4}.\\ L.C.M.of\text{5 and 4 is}\text{2}0.\\ \frac{-2}{5}=\frac{(-2)\times 4}{5\times 4}=\frac{-8}{20}\\ \frac{3}{4}=\frac{3\times 5}{4\times 5}=\frac{15}{20}\\ \\ \mathrm{Now},\frac{(-8)}{20}+\frac{15}{20}=\frac{-8+15}{20}=\frac{7}{20}\end{array}$

(ii)$\frac{-5}{9}+\frac{2}{3}$

$\begin{array}{l}\text{The denominators of the given rational numbers are 9 and 3.}\end{array}$



$\begin{array}{l}\text{L.C.M. of 9 and 3 is}\text{9.}\\ \frac{-5}{9}=\frac{(-5)\times 1}{9\times 1}=\frac{-5}{9}\\ \\ \frac{2}{3}=\frac{2\times 3}{3\times 3}=\frac{6}{9}\\ \\ \mathrm{Now},\frac{(-5)}{9}+\frac{6}{9}\\ =\frac{-5+6}{9}\\ =\frac{1}{9}\end{array}$
(iii)$-4+\frac{1}{2}$


$\begin{array}{l}\text{The denominators of the given rational numbers are 1 and 2.}\\ \text{L.C.M. of 1 and 2 is 2}.\\ \frac{-4}{1}=\frac{(-4)\times 2}{1\times 2}=\frac{-8}{2}\\ \\ \frac{1}{2}=\frac{1\times 1}{2\times 1}=\frac{1}{2}\\ \\ \mathrm{Now},\frac{(-8)}{2}+\frac{1}{2}\\ =\frac{-8+1}{2}\\ =\frac{-7}{2}\end{array}$

(iv)
$\frac{-7}{27}+\frac{5}{18}$


The denominators of the given rational numbers are 27 and 18.

L.C.M. of 27 and 18 is 54.

$\begin{array}{l}\\ \frac{-7}{27}=\frac{(-7)\times 2}{27\times 2}=\frac{-14}{54}\\ \frac{5}{18}=\frac{5\times 3}{18\times 3}=\frac{15}{54}\\ \\ \mathrm{Now},\frac{(-14)}{54}+\frac{15}{54}=\frac{-14+15}{54}\\ =\frac{1}{54}\\ \end{array}$

(v)$\frac{-5}{36}+\left(\frac{-7}{12}\right)$


The denominators of the given rational numbers are 36 and 12.
L.C.M. of 36 and 12 is 36.

$\begin{array}{l}\frac{-5}{36}=\frac{(-5)\times 1}{36\times 1}=\frac{-5}{36}\\ \frac{-7}{12}=\frac{-7\times 3}{12\times 3}=\frac{-21}{36}\\ \\ \mathrm{Now},\frac{(-5)}{36}+\frac{(-21)}{36}=\frac{-5-21}{36}\\ \frac{-26}{36}=\frac{-13}{18}(\text{26 and 36 are divided by 2 .)}\\ \end{array}$

(vi)
$\frac{1}{-9}+\left(\frac{4}{-27}\right)$

$\begin{array}{l}\text{We need a positive denominator}.\\ \frac{1}{-9}\times \frac{-1}{-1}=\frac{-1}{9}\text{and}\frac{4}{-27}\times \frac{-1}{-1}=\frac{-4}{27}\\ \\ \text{The denominators of the given rational numbers are 9}\text{and}\text{27}\end{array}$

$\begin{array}{l}\text{L.C.M. of 9 and 27 is}\text{27}\\ \frac{-1}{9}=\frac{(-1)\times 3}{9\times 3}=\frac{-3}{27}\\ \frac{-4}{27}=\frac{-4\times 1}{27\times 1}=\frac{-4}{27}\\ \\ \frac{(-3)}{27}+\frac{(-4)}{27}=\frac{-3-4}{27}\\ \\ =\frac{-7}{27}\end{array}$


(vii)
$$\begin{gathered} \frac{-9}{24}+\left(\frac{-1}{18}\right) \\ \mathrm{The}\mathrm{denominators}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{numbers}\mathrm{are}24\mathrm{and}18 \end{gathered}$$

$$\begin{gathered} \mathrm{L}.\mathrm{C}.\mathrm{M}.\mathrm{of}24\mathrm{and}18\mathrm{is}72. \\ \therefore \frac{-9}{24}=\frac{-9\times 3}{24\times 3}=\frac{-27}{72} \\ \frac{-1}{18}=\frac{-1\times 4}{18\times 4}=\frac{-4}{72} \\ \mathrm{Now},\frac{-27}{72}+\left(\frac{-4}{72}\right) \\ =\frac{-27+(-4)}{72} \\ =\frac{-27-4}{72} \\ =\frac{-31}{72} \end{gathered}$$


(viii)$\frac{27}{-4}+\left(\frac{-15}{8}\right)$

$\begin{array}{l}\text{We need a positive denominator.}\\ \frac{27}{-4}\times \frac{-1}{-1}=\frac{-27}{4}\\ \text{The denominators of the given rational numbers are 4 and 8.}\end{array}$

$\begin{array}{l}\text{L.C.M. of 4 and 8 is 8.}\\ \frac{-27}{4}=\frac{-27\times 2}{4\times 2}=\frac{-54}{8}\\ \frac{(-15)}{8}=\frac{(-15)\times 1}{8\times 1}=\frac{-15}{8}\\ \mathrm{Now},\frac{-54}{8}+\frac{(-15)}{8}\\ =\frac{-54-15}{8}\\ =\frac{-69}{8}\end{array}$