S borrowed some amount from R and promised to pay him 8% interest. Then S invested the borrowed amount in a scheme, upon which he earned a profit of 5% after paying R, the principal amount with interest. How much percentage would R have gained if he had invested in the scheme directly?

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Let the principal amount borrowed by S from R be 100 units.

S pays R 8% interest, so amount paid to R = 100 + 8 = 108 units.

S invests the borrowed amount (100 units) in a scheme and earns 5% profit on the invested amount after paying R.

This means the total amount S gets from the scheme = amount paid to R + profit earned by S = 108 + 5 = 113 units.

So, the scheme returns 113 units on an investment of 100 units, which means the scheme's rate of return is 13% (since 113 - 100 = 13).

Therefore, if R had invested directly in the scheme, he would have gained 13% profit.

However, the question asks for the percentage gain R would have if he invested directly, considering the 8% interest he received from S.

To find the effective gain for R:

R receives 108 units from S (principal + 8% interest).

If R had invested directly, he would get 113 units (principal + x% profit).

The gain for R over 108 units is 113 - 108 = 5 units.

Percentage gain over 108 units = (5 / 108) * 100 ≈ 4.63%.

But this is the gain over the amount received from S, not over the original principal.

To find the total gain over the original principal (100 units), we calculate:

Total gain = 13% (scheme profit) + 8% (interest) + (8% * 5%) / 100 (compound effect) = 8 + 5 + 0.4 = 13.4%.

Hence, R's effective gain if he invested directly in the scheme is 13.4%.

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