Question on percentages. 
Cost of packing of mangoes is 40% of cost of fresh mangoes. Price of
  mangoes is increased by 30% but the cost of packing of mangoes is
  decreased by 50%. If cost of packed mangoes is equal to sum of cost of
  fresh mangoes and packing of mangoes then, find what is the percentage
  change in the cost of packed mangoes.

Let M be the cost of mangoes.
Initial cost:
$$=M+\frac{4}{10}M$$
Cost after revision:
$$\frac{13}{10}M+\frac{1}{2}\frac{4}{10}M$$
I get the answer as $\frac{50}{7} percent$, however the anwer is $\frac{15}{7}percent$.

I agree with you. Since the result should be independent of the actual prices involved, we can set original price of fresh mangoes to 100. Then cost of packing in 40 and original cost of packed mangoes is 140.
After price changes, fresh mangoes cost 130, cost of packing is 20, and so packed mangoes now cost 150.
Increase in cost of packed mangoes is $\frac{10}{140} = \frac{1}{14}$. As a percentage this is $\frac{100}{14} \% = \frac{50}{7} \%$.
