By selling mangoes at the rate of 64 for Rs. 2,000, the vendor loses 40%. How many should he sell for Rs.1000 so as to gain 20%?

If 64 mangoes are sold at Rs.2000, each mango will be sold at Rs. $$\frac{2000}{64}$$

Hence Selling price (S.P) of each mango = Rs. 31.25

Given loss percentage of vendor at this S.P = 40%

Loss percentage = $$\frac{C.P - S.P}{C.P}\times 100$$ 

$$\Rightarrow \frac{40}{100} = \frac{C.P - S.P}{C.P}$$

$$\Rightarrow S.P = 0.6\times C.P$$

$$\Rightarrow C.P = \frac{31.25}{0.6} = 52$$

Therefore Cost Price of 1 mango (C.P) = Rs. 52

Let us calculate the S.P of each mango in order to get a 20% gain.

Gain percentage = $$\frac{S.P - C.P}{C.P}\times 100$$

$$\Rightarrow \frac{20}{100} = \frac{S.P - C.P}{C.P}$$

$$\Rightarrow S.P = 1.2\times C.P$$

$$\Rightarrow S.P = 62.5$$

So, to get a gain of 20% we need to sell each mango at Rs. 62.5 

Let say we sold 'x' number of mangoes.

Selling price of these 'x' number of mangoes (S.P) = Rs. 62.5x

But given that this S.P = Rs. 1000

$$\Rightarrow 62.5x = 1000$$

$$\Rightarrow x = \frac{1000}{62.5} = 16$$.

Therefore a total of 16 mangoes are to be sold for Rs. 1000 to get a gain of 20%.