Question : On a certain sum, the interest is compounded annually. If the compound interest for the second year is INR 400 and the compound interest for the fourth year is 576, then what is the rate of interest per annum?
Option 1: 20%
Option 2: 25%
Option 3: 15%
Option 4: 44%
Correct Answer: 20%
Solution :
When compounded annually, $ A= P(1+\frac{R}{100})^{T}$, Where $A$ is the total amount, $P$ is the principal amount, $R$ is the rate of interest per annum, and $T$ is the time in years.
According to the question,
$P(1+\frac{R}{100})^2-P(1+\frac{R}{100})=400$ -----------------(1)
$P(1+\frac{R}{100})^4-P(1+\frac{R}{100})^3=576$ -----------------(2)
From equation (1) we get,
$P(1+\frac{R}{100})[(1+\frac{R}{100})-1]=400$ ---------------(3)
From equation (2) we get,
$P(1+\frac{R}{100})^3[(1+\frac{R}{100})-1]=576$ ---------------(4)
Dividing equation (4) by equation (3),
$(1+\frac{R}{100})^2=\frac{576}{400}$
⇒ $(1+\frac{R}{100})^2=(\frac{24}{20})^2$
⇒ $(1+\frac{R}{100})=(\frac{24}{20})$
⇒ $\frac{R}{100}=\frac{24}{20}-1$
⇒ $R=20$%
Hence, the correct answer is 20%.
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