Add the following rational numbers:

Solution:

(i) Clearly, denominators of the given numbers are positive.

The L.C.M. of denominators 4 and 8 is 8 .

Now, we will express $\frac{3}{4}$ in the form in which it takes the denominator is 8 .

$\frac{3 \times 2}{4 \times 2}=\frac{6}{8}$

$\frac{3}{4}+\frac{-5}{8}=\frac{6}{8}+\frac{-5}{8}$

$=\frac{6+(-5)}{8}$

$=\frac{6-5}{8}$

$=\frac{1}{8}$

(ii) $\frac{5}{-9}+\frac{7}{3}=\frac{-5}{9}+\frac{7}{3}$

The L.C.M. of denominators 9 and 3 is 9 .

Now, we will express $\frac{7}{3}$ in the form in which it takes the denominator is 9 .

$\frac{7 \times 3}{3 \times 3}=\frac{21}{9}$

$\frac{-5}{9}+\frac{7}{3}=\frac{-5}{9}+\frac{21}{9}$

$=\frac{(-5)+21}{9}$

$=\frac{-5+21}{9}$

$=\frac{16}{9}$

(iii) $-3+\frac{3}{5}=\frac{-3}{1}+\frac{3}{5}$

The L.C.M. of denominators 1 and 5 is $5 .$

Now, we will express $\frac{-3}{1}$ in the form in which it takes the denominator 5 .

So,

$\frac{-3 \times 5}{1 \times 5}=\frac{-15}{5}$

$\frac{-3}{1}+\frac{3}{5}=\frac{-15}{5}+\frac{3}{5}$

$=\frac{-15+3}{5}$

$=\frac{-12}{5}$

(iv) The L.C.M. of denominators 27 and 18 is $54 .$

Now, we will express $\frac{-7}{27}$ and $\frac{11}{18}$ in the form in which they take the denominator 54 .

$\frac{-7 \times 2}{27 \times 2}=\frac{-14}{54}$

$\frac{11 \times 3}{18 \times 3}=\frac{33}{54}$

$\frac{-7}{27}+\frac{11}{18}=\frac{-14}{54}+\frac{33}{54}$

$=\frac{-14+33}{54}$

$=\frac{19}{54}$

(v) We have

$\frac{31}{-4}+\frac{-5}{8}=\frac{-31}{4}+\frac{-5}{8}$

The L.C.M. of denominators 4 and 8 is 8 .

Now, we will express $\frac{-31}{4}$ in the form in which it takes the denominator 8 .

$\frac{-31 \times 2}{4 \times 2}=\frac{-62}{8}$

So,

$\frac{-31}{4}+\frac{-5}{8}=\frac{-62}{8}+\frac{-5}{8}$

$=\frac{(-62)+(-5)}{8}$

$=\frac{-62-5}{8}$

$=\frac{-67}{8}$

(vi) The L.C.M. of denominators 36 and 12 is 36 .

Now, we will express $\frac{-7}{12}$ in the form in whic $h$ it takes the denominator 36 .

$\frac{-7 \times 3}{12 \times 3}=\frac{-21}{36}$

So,

$\frac{5}{36}+\frac{-7}{12}=\frac{5}{36}+\frac{-21}{36}$

$=\frac{5+(-21)}{36}$

$=\frac{5-21}{36}$

$=\frac{-16}{36}$

$=\frac{-4}{9}$

(vii) The L.C.M. of denominators 16 and 24 is 48 .

Now, we will express $\frac{-5}{16}$ and $\frac{7}{24}$ in the form in which they take the denominator 48 .

$\frac{-5 \times 3}{16 \times 3}=\frac{-15}{48}$

$\frac{7 \times 2}{24 \times 2}=\frac{14}{48}$

So,

$\frac{-5}{16}+\frac{7}{24}=\frac{-15}{48}+\frac{14}{48}$

$=\frac{(-15)+14}{48}$

$=\frac{-15+14}{48}$

$=\frac{-1}{48}$

(viii) $\frac{7}{-18}+\frac{8}{27}=\frac{-7}{18}+\frac{8}{27}$

The L.C.M. of denominators 18 and 27 is $54 .$

Now, we will express $\frac{-7}{18}$ and $\frac{8}{27}$ in the form in which they take the denominator 54 .

So

$\frac{-7}{18}+\frac{8}{27}=\frac{-21}{54}+\frac{16}{54}$

$\frac{-7}{18}+\frac{8}{27}=\frac{-21}{54}+\frac{16}{54}$

$=\frac{-21+16}{54}$

$=\frac{-5}{54}$

$=\frac{-5}{54}$