# Q. A principal P becomes Q in 1 year when compounded half-yearly with R% annual rate of interest. If the same principal P becomes Q in 1 year when compounded annually with S% annual rate of interest, then which one of the following is correct?

(a) R=S

(b) R>S

(c) R<S

(d) R≤S

Correct Answer: (c) R<S

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Principal P becomes Q in 1 year. Compounded half-yearly at R% and annually at S%.

When interest is compounded half-yearly, it means that the interest is calculated twice in a year. This results in a higher amount at the end of the year compared to when the interest is compounded annually.

To understand this, let’s take an example. Suppose the principal amount is ₹1000, the annual interest rate is 10%, and the interest is compounded half-yearly.

After the first 6 months, the amount will be ₹1000 + ₹1000*(10/2)/100 = ₹1050.

After the next 6 months, the amount will be ₹1050 + ₹1050*(10/2)/100 = ₹1102.5.

So, at the end of the year, the amount is ₹1102.5 when the interest is compounded half-yearly.

Now, if the same principal amount is compounded annually, the amount at the end of the year will be ₹1000 + ₹1000*10/100 = ₹1100.

So, even though the annual interest rate is the same, the amount at the end of the year is higher when the interest is compounded half-yearly.

Therefore, for the same principal P to become Q in 1 year, the annual interest rate R% when compounded half-yearly has to be less than the annual interest rate S% when compounded annually. Hence, R<S.