Calculate the following:

(i) $\dfrac{2}{3} + \dfrac{3}{4}$

(ii) $\dfrac{5}{7} - \dfrac{4}{9}$

(iii) $\dfrac{1}{2} + \dfrac{3}{5}$

(iv) $1\dfrac{4}{9} + 3\dfrac{5}{12}$

(v) $2\dfrac{1}{4} - 1\dfrac{7}{10}$

(vi) $3\dfrac{5}{6} - 2\dfrac{7}{15}$

(i) $\dfrac{2}{3} + \dfrac{3}{4}$

LCM of 3 and 4 = 12.

$\Rightarrow \dfrac{2 \times 4}{3 \times 4} + \dfrac{3 \times 3}{4 \times 3}\\[1em] \Rightarrow \dfrac{8}{12} + \dfrac{9}{12}\\[1em] \Rightarrow \dfrac{8 + 9}{12}\\[1em] \Rightarrow \dfrac{17}{12}\\[1em] \Rightarrow 1\dfrac{5}{12}$

Hence, $\dfrac{2}{3} + \dfrac{3}{4} = 1\dfrac{5}{12}$.

(ii) $\dfrac{5}{7} - \dfrac{4}{9}$

LCM of 7 and 9 = 63.

$\Rightarrow \dfrac{5 \times 9}{7 \times 9} - \dfrac{4 \times 7}{9 \times 7}\\[1em] \Rightarrow \dfrac{45}{63} - \dfrac{28}{63}\\[1em] \Rightarrow \dfrac{45 - 28}{63}\\[1em] \Rightarrow \dfrac{17}{63}$

Hence, $\dfrac{5}{7} - \dfrac{4}{9} = \dfrac{17}{63}$.

(iii) $\dfrac{1}{2} + \dfrac{3}{5}$

LCM of 2 and 5 = 10.

$\Rightarrow \dfrac{1 \times 5}{2 \times 5} + \dfrac{3 \times 2}{5 \times 2}\\[1em] \Rightarrow \dfrac{5}{10} + \dfrac{6}{10}\\[1em] \Rightarrow \dfrac{5 + 6}{10}\\[1em] \Rightarrow \dfrac{11}{10}\\[1em] \Rightarrow 1\dfrac{1}{10}$

Hence, $\dfrac{1}{2} + \dfrac{3}{5} = 1\dfrac{1}{10}$.

(iv) $1\dfrac{4}{9} + 3\dfrac{5}{12}$

LCM of 9 and 12 = 36.

$\Rightarrow \dfrac{13}{9} + \dfrac{41}{12}\\[1em] \Rightarrow \dfrac{13 \times 4}{9 \times 4} + \dfrac{41 \times 3}{12 \times 3}\\[1em] \Rightarrow \dfrac{52}{36} + \dfrac{123}{36}\\[1em] \Rightarrow \dfrac{52 + 123}{36}\\[1em] \Rightarrow \dfrac{175}{36}\\[1em] \Rightarrow 4\dfrac{31}{36}$

Hence, $1\dfrac{4}{9} + 3\dfrac{5}{12} = 4\dfrac{31}{36}$.

(v) $2\dfrac{1}{4} - 1\dfrac{7}{10}$

LCM of 4 and 10 = 20.

$\Rightarrow \dfrac{9}{4} - \dfrac{17}{10}\\[1em] \Rightarrow \dfrac{9 \times 5}{4 \times 5} - \dfrac{17 \times 2}{10 \times 2}\\[1em] \Rightarrow \dfrac{45}{20} - \dfrac{34}{20}\\[1em] \Rightarrow \dfrac{45 - 34}{20}\\[1em] \Rightarrow \dfrac{11}{20}$

Hence, $2\dfrac{1}{4} - 1\dfrac{7}{10} = \dfrac{11}{20}$.

(vi) $3\dfrac{5}{6} - 2\dfrac{7}{15}$

LCM of 6 and 15 = 30.

$\Rightarrow \dfrac{23}{6} - \dfrac{37}{15}\\[1em] \Rightarrow \dfrac{23 \times 5}{6 \times 5} - \dfrac{37 \times 2}{15 \times 2}\\[1em] \Rightarrow \dfrac{115}{30} - \dfrac{74}{30}\\[1em] \Rightarrow \dfrac{115 - 74}{30}\\[1em] \Rightarrow \dfrac{41}{30}\\[1em] \Rightarrow 1\dfrac{11}{30}$

Hence, $3\dfrac{5}{6} - 2\dfrac{7}{15} = 1\dfrac{11}{30}$.