If the CI and SI of a certain sum of money for 2 years is 400 and 400 respectively, find the rate of compound interest and principal.

Steps to Solve

1. Determine the Simple Interest (SI) for one year

Since the simple interest for 2 years is €400, the simple interest for 1 year is:

$SI_{1} = \frac{400}{2} = 200$

2. Find the Compound Interest (CI) for two years

The Compound Interest for 2 years is €410.

3. Calculate the difference between CI and SI for 2 years

This difference is the interest on the first year's interest.

$Difference = CI - SI = 410 - 400 = 10$

4. Calculate the rate of interest

The rate of interest $r$ is the interest earned on the first year's interest, divided by the first year's interest.

$r = \frac{Difference}{SI_{1}} \times 100$

$r = \frac{10}{200} \times 100 = 5%$

5. Calculate the principal

We know that the simple interest for 1 year is €200, and the rate of interest is 5%. We can use the simple interest formula to find the principal ($P$):

$SI_{1} = \frac{P \times r \times t}{100}$

Where: $SI_{1} = 200$ $r = 5$ $t = 1$

$200 = \frac{P \times 5 \times 1}{100}$

$P = \frac{200 \times 100}{5} = 4000$