If the CI & SI of a certain sum of money for 2 years is 410 & 400 respectively. Find the rate of compound interest & principle.

Steps to Solve

1. Find the Simple Interest for One Year

Since the simple interest (SI) for 2 years is $400, the simple interest for 1 year is: $SI_{1} = \frac{400}{2} = 200$

2. Find the Difference Between CI and SI for 2 Years

The difference between compound interest (CI) and simple interest (SI) for 2 years is: $CI - SI = 410 - 400 = 10$

3. Realize That the Difference Is the Interest on the First Year's Interest

The difference between CI and SI for 2 years is the interest earned on the first year's simple interest. Therefore, $10 is the interest on $200 for one year.

4. Calculate the Rate of Interest

Let the rate of interest be $R$. We have: $200 \times \frac{R}{100} = 10$

Solving for $R$: $R = \frac{10 \times 100}{200} = \frac{1000}{200} = 5$

Therefore, the rate of interest is $5%$.

5. Calculate the Principal Amount

We know the simple interest for 1 year is $200, and the rate of interest is $5%$. Let the principal amount be $P$. We have:

$P \times \frac{5}{100} = 200$

Solving for $P$: $P = \frac{200 \times 100}{5} = \frac{20000}{5} = 4000$

Therefore, the principal amount is $4000.