Find the sum:
$(i)$. $\frac{5}{4}+(-\frac{11}{4})$
$(ii)$. $\frac{5}{3}+\frac{3}{5}$
$(iii)$. $\frac{-9}{10}+\ \frac{22}{15}$
$(iv)$. $\frac{-3}{11}+\frac{5}{9}$
$(v)$. $\frac{-8}{19}+(-\frac{2}{57})$
$(vi)$. $-\frac{2}{3}+0$
$(vii)$. $-2\frac{1}{3}\ +\ 4\frac{3}{5}$

Given: $(i)$. $\frac{5}{4}+(-\frac{11}{4})$

$(ii)$. $\frac{5}{3}+\frac{3}{5}$

$(iii)$. $\frac{-9}{10}+\ \frac{22}{15}$

$(iv)$. $\frac{-3}{-11}+\frac{5}{9}$

$(v)$. $\frac{-8}{19}+(-\frac{2}{57})$ 

$(vi)$. $-\frac{2}{3}+0$

$(vii)$. $-2\frac{1}{3}\ +\ 4\frac{3}{5}$

To do: To find the sum of the given expressions.

Solution: $(i)$. $\frac{5}{4}+(-\frac{11}{4})$

$=\frac{5}{4}-\frac{11}{4}$

$=\frac{5-11}{4}$

$=\frac{-6}{4}$

$(ii)$. $\frac{5}{3}+\frac{3}{5}$

$=\frac{5\times 5}{3\times 5}+\frac{3\times 3}{5\times 3}$                [On taking LCM of 3 and 5, we obtain 15]

$=\frac{25}{15}+\frac{9}{15}$

$=\frac{25+9}{15}$

$=\frac{34}{15}$

$(iii)$. $\frac{-9}{10}+\ \frac{22}{15}$

$=\frac{-9\times 3}{10\times 3}+\frac{22\times 2}{15\times 2}$         [on taking LCM of 10 and 15, we obtain 30]

$=\frac{-27}{30}+\frac{44}{30}$

$=\frac{-27+44}{30}$

$=\frac{13}{30}$

$(iv)$. $\frac{-3}{-11}+\frac{5}{9}$

$=\frac{3\times 9}{11\times 9}+\frac{5\times 11}{9\times 11}$         [On taking LCM of 11 and 9 we obtain 99]

$=\frac{27}{99}+\frac{55}{99}$

$=\frac{27+55}{99}$

$=\frac{82}{99}$

$(v)$. $\frac{-8}{19}+(-\frac{2}{57})$

$=\frac{-8\times 3}{19\times 3}-\frac{2\times 1}{57\times 1}$           [On taking LCM of 19 and 57 we obtain 57]

$=-\frac{24}{57}-\frac{2}{57}$

$=-(\frac{24}{57}+\frac{2}{57}$)

$=-(\frac{24+2}{57})$

$=-\frac{26}{57}$

$(vi)$. $-\frac{2}{3}+0$

$=-\frac{2\times 1}{3\times 1}+\frac{0\times 3}{1\times 3}$          [On taking LCM of 3 and 1 we obtain 3]

$=-\frac{2}{3}+\frac{0}{3}$

$=-\frac{2-0}{3}$

$=-\frac{2}{3}$

$(vii)$. $-2\frac{1}{3}\ + 4\frac{3}{5}$

$=-\frac{2\times 3+1}{3}+\frac{5\times 4+3}{5}$

$=-\frac{7}{3}+\frac{23}{5}$

$=-\frac{7\times 5}{3\times 5}+\frac{23\times 3}{5\times 3}$           [On taking LCM of 3 and 5 we obtain 15]

$=-\frac{35}{15}+\frac{69}{15}$

$=\frac{-35+69}{15}$

$=\frac{34}{15}$