If the compound interest on a certain sum for $2$ years is Rs. $21$. What could be the simple interest? A. Rs. $18$ B. Rs. $24$ C. Rs. $20$ D. Rs. $27$
My approach: interest after first year= pr/100
interest after second year= [p+(pr/100)]* r/100
so pr/100 + [p+(pr/100)]* r/100 =21
but it seems difficult procedding further this way. Kindly looking for help, thanks a lot.
also there is 1 online solution, which i don't think it's correct, link of online solution
Assuming the $21$ was meant to be the percent interest (instead of a cash figure).
Note that the problem does not ask what the simple interest is, it only asks which of the given answers is possible. Thus we can instantly eliminate $B,D$ since they are greater than the compound interest. As we don't know the compounding periods, many (but not all) rates are possible.
Now, $20$ is certainly possible. Indeed, if the annualized rate $r=10$ and we compound annually, we get the two year interest as $(1.1)^2=1.21$ would give us a simple interest of $10+10=20\%$.
To see that $18$ is not possible, regardless of the compounding period, consider continuous compounding. If the annualized continuous rate is $r$ then we have $$\exp(2r)=1.21\implies r\approx .0953$$ which would give a simple interest of $9.53+9.53=19.06>18$. Thus no compounding rate is sufficiently brief to get us to $18$ for the simple interest, and the answer is therefore $C$.
Note: the $20\%$ here is the same as in the "official solution" but the official solution has a different set of possible answers to choose from so the letter of the solution does not match. And, of course, the official solution assumes the compounding period to be annual which is not stated in the problem. If you make that assumption then we can compute the rate exactly (which is what they do) but in that case the phrasing of the problem is odd (why ask what the simple rate "could be" when it is determined uniquely?)