Add the following rational numbers:
(i) $\frac{3}{4}$ and $\frac{-5}{8}$
(ii) $\frac{5}{-9}$ and $\frac{7}{3}$
(iii) $-3$ and $\frac{3}{5}$
(iv) $\frac{-7}{27}$ and $\frac{11}{18}$
(v) $\frac{31}{-4}$ and $\frac{-5}{8}$
(vi) $\frac{5}{36}$ and $\frac{-7}{12}$
(vii) $\frac{-5}{16}$ and $\frac{7}{24}$
(viii) $\frac{7}{-18}$ and $\frac{8}{27}$
(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
$\frac{-7}{18}+\frac{8}{27}=\frac{-21}{54}+\frac{16}{54}$
$\frac{-7}{18}+\frac{8}{27}=\frac{-21}{54}+\frac{16}{54}$
$=\frac{-21+16}{54}$
$=\frac{-5}{54}$
$=\frac{-5}{54}$
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