On certain sum of money, the compound interest for 2 years is Rs.304.5 and the simple interest for the same period is Rs.290. Find the rate of interest per annum :

A9%

B8%

C11%

D10%

Answer:

Understanding the Difference: Simple Interest (SI) is calculated only on the principal amount. Compound Interest (CI) is calculated on the principal amount plus the accumulated interest from previous periods. This means CI grows faster than SI over time.

Simple Interest (SI) = (P * R * T) / 100, where P is the Principal, R is the Rate of Interest (per annum), and T is the Time (in years).
Compound Interest (CI) = A - P, where A is the Amount, and A = P(1 + R/100)^T.

Let the Principal be P, Rate be R%, and Time be 2 years.
SI for 2 years = (P * R * 2) / 100
CI for 2 years = P[(1 + R/100)^2 - 1]
A key relationship derived from these formulas for 2 years is: CI - SI = P * (R/100)^2. This difference is essentially the interest earned on the first year's simple interest.

From the SI formula, P * R * 2 / 100 = 290, which simplifies to PR/50 = 290 or PR = 14500.
Using the difference formula: CI - SI = P * (R/100)^2.
Substitute the values: 14.5 = P * (R^2 / 10000).
Rearranging, P * R^2 = 14.5 * 10000 = 145000.
Now we have two equations:
1) PR = 14500
2) PR^2 = 145000
Divide equation (2) by equation (1):
(PR^2) / (PR) = 145000 / 14500
R = 10
Therefore, the rate of interest (R) is 10% per annum.
To verify, we can find the Principal: P * 10 = 14500 => P = 1450.
SI = (1450 * 10 * 2) / 100 = 290 (Correct).
CI = 1450 * (1 + 10/100)^2 - 1450 = 1450 * (1.1)^2 - 1450 = 1450 * 1.21 - 1450 = 1754.5 - 1450 = 304.5 (Correct).

The difference between CI and SI for 2 years is the interest on the SI of the first year.
SI for 1 year = SI for 2 years / 2 = 290 / 2 = Rs. 145.
The interest on this Rs. 145 for 1 year at rate R% is the difference between CI and SI, which is Rs. 14.5.
So, (145 * R * 1) / 100 = 14.5
145 * R = 14.5 * 100
145 * R = 1450
R = 1450 / 145
R = 10%

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