Question : If $M =\left ( \frac{3}{7} \right ) ÷ \left ( \frac{6}{5} \right ) ×\left ( \frac{2}{3} \right ) + \left ( \frac{1}{5} \right ) ×\left ( \frac{3}{2} \right )$ and $N = \left ( \frac{2}{5} \right ) × \left ( \frac{5}{6} \right ) ÷ \left ( \frac{1}{3} \right ) + \left ( \frac{3}{5} \right ) × \left ( \frac{2}{3} \right ) ÷ \left ( \frac{3}{5} \right )$, then what is the value of $\frac{M}{N}$?
Option 1: $\frac{207}{560}$
Option 2: $\frac{339}{1120}$
Option 3: $\frac{113}{350}$
Option 4: $\frac{69}{175}$
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Correct Answer: $\frac{113}{350}$
Solution : $M =\left (\frac{3}{7}\right) ÷ \left (\frac{6}{5} \right) ×\left (\frac{2}{3} \right) + \left (\frac{1}{5} \right) ×\left (\frac{3}{2}\right)$ $M =\left (\frac{3}{7} \right) × \left (\frac{5}{6} \right) ×\left (\frac{2}{3} \right) + \left (\frac{1}{5} \right) ×\left (\frac{3}{2} \right)$ $M = \frac{5}{21}+\frac{3}{10}$ $M = \frac{113}{210}$ $N = \left (\frac{2}{5} \right) × \left (\frac{5}{6} \right) ÷ \left (\frac{1}{3} \right) + \left (\frac{3}{5} \right) × \left (\frac{2}{3} \right) ÷ \left (\frac{3}{5} \right)$ $N = \left (\frac{2}{5} \right) × \left (\frac{5}{6} \right) ×3+ \left (\frac{3}{5} \right) × \left (\frac{2}{3} \right) × \left (\frac{5}{3} \right)$ $N =1 + \left (\frac{2}{3} \right)$ $N = \left (\frac{5}{3} \right)$ $\frac{M}{N} = \frac{113}{210}×\frac{3}{5} = \frac{113}{350}$ Hence, the correct answer is $\frac{113}{350}$.
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