Question

An automobile financier claims to be lending money at simple interest, but includes the interest every six months for calculating the principal. If he is charging an interest of \[10%\], find the effective rate of interest.
(a) $10%$
(b) $10.25%$
(c) $10.5%$
(d) $15%$

Answer 1

Hint: Assume that the principal is $Rs.100$. Calculate simple interest on this principal for the first 6 months at the rate of $10%$ using the formula $SI=\dfrac{P\times R\times T}{100}$. Calculate the amount at the end of the first six months by adding simple interest to the principal. Calculate the simple interest taking this amount as principal for the next 6 months at the rate of $10%$. Calculate the amount again by adding simple interest to the principal. Find the difference between the final amount and principal to calculate the effective rate of interest.

Complete step-by-step answer:
We have data regarding the rate of interest and the time after which an automobile financier adds interest. We have to calculate the effective rate of interest.

Let’s assume that the principal amount on which interest is added is $Rs.100$.
We will calculate the interest on this principal for first 6 months at the rate of $10%$, using the formula $SI=\dfrac{P\times R\times T}{100}$, where P represents the amount which on which interest is added, R represents the rate of interest, T represents the time for which interest is added and SI represents the simple interest.

Substituting $P=Rs.100,R=10%,T=6months=\dfrac{1}{2}yr$ in the above equation, we have $SI=\dfrac{100\times 10\times \dfrac{1}{2}}{100}=Rs.5$ .
Now, we will calculate the amount to be paid after 6 months. We can calculate the amount by adding principal and simple interest.
Thus, the amount after 6 months $=Rs.100+Rs.5=Rs.105$.
We will now calculate simple interest using this amount as principal for the next 6 months.
Substituting $P=Rs.105,R=10%,T=6months=\dfrac{1}{2}yr$ in the formula $SI=\dfrac{P\times R\times T}{100}$, we have $SI=\dfrac{105\times 10\times \dfrac{1}{2}}{100}=Rs.5.25$.
We will now calculate the amount to be paid for the next 6 months. To do so, we will add the principal and the simple interest.
Thus, the amount paid after 6 months $=Rs.105+Rs.5.25=Rs.110.25$.

We will now calculate the effective rate of interest per annum. To do so, we will subtract the initial principal from the final amount. We know that the initial principal is $Rs.100$ and the final amount is $Rs.110.25$.
Thus, the effective rate of interest per annum \[=110.25-100=10.25%\].
Hence, the effective rate of interest is \[10.25%\], which is option (b).

Note: We can also solve this question by taking the initial principal to be Rs.x and then calculating the final amount on this principal. However, while calculating the rate of interest, we must divide the difference between the final amount and principal by x.