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 $\frac{7}{8}$
(i) We have $\frac{-12}{2}+\frac{43}{10}$.
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 .
$\frac{-12 \times 2}{5 \times 2}=\frac{-24}{10}$
$\therefore \frac{-12}{5}+\frac{43}{10}=\frac{-24}{10}+\frac{43}{10}$
$=\frac{-24+43}{10}$
$=\frac{19}{10}$
$=1 \frac{9}{10}$
(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 .
$\frac{24 \times 4}{7 \times 4}=\frac{96}{28}$
$\frac{-11 \times 7}{4 \times 7}=\frac{-77}{28}$
$\therefore \frac{24}{7}+\frac{-11}{4}=\frac{96}{28}+\frac{-77}{28}$
$=\frac{96-77}{28}$
$=\frac{19}{28}$
(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 .
$\frac{-31 \times 4}{6 \times 4}=\frac{-124}{24}$
$\frac{-27 \times 3}{8 \times 3}=\frac{-81}{24}$
$\therefore \frac{-31}{6}+\frac{-27}{8}=\frac{-124}{24}+\frac{-81}{24}$
$=\frac{-124-81}{24}$
$=\frac{-205}{24}$
$=-8 \frac{13}{24}$
(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 .
$\frac{101 \times 4}{6 \times 4}=\frac{404}{24}$
$\frac{7 \times 3}{8 \times 3}=\frac{21}{24}$
$\therefore \frac{101}{6}+\frac{7}{8}=\frac{404}{24}+\frac{21}{24}$
$=\frac{404+21}{24}$
$=\frac{425}{24}$
$=17 \frac{17}{24}$
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