On a certain sum of money, compound interest earned at the end of three years = Rs. 1456. Compound interest at the end of two years is Rs. 880. Compute the principal invested

To solve this problem, we can use the formula for compound interest:

\[
A = P \left( 1 + \frac{r}{100} \right)^n
\]

Where:
- \( A \) is the total amount (principal + interest) after \( n \) years,
- \( P \) is the principal,
- \( r \) is the rate of interest per annum,
- \( n \) is the number of years.

The compound interest (CI) after \( n \) years is given by:

\[
CI = A - P = P \left( 1 + \frac{r}{100} \right)^n - P
\]

From the problem:
- CI after 3 years is Rs. 1456,
- CI after 2 years is Rs. 880.

Let \( P \) be the principal and \( r \) be the rate of interest per annum.

### Step 1: CI after 2 years
After 2 years:
\[
CI_2 = P \left( 1 + \frac{r}{100} \right)^2 - P = 880
\]

\[
P \left( \left( 1 + \frac{r}{100} \right)^2 - 1 \right) = 880
\tag{1}
\]

### Step 2: CI after 3 years
After 3 years:
\[
CI_3 = P \left( 1 + \frac{r}{100} \right)^3 - P = 1456
\]

\[
P \left( \left( 1 + \frac{r}{100} \right)^3 - 1 \right) = 1456
\tag{2}
\]

### Step 3: Solve the system of equations
We have two equations:

\[
P \left( \left( 1 + \frac{r}{100} \right)^2 - 1 \right) = 880
\]
\[
P \left( \left( 1 + \frac{r}{100} \right)^3 - 1 \right) = 1456
\]

Let me solve this system to find the principal \( P \) and interest rate \( r \).The principal invested is Rs. 2000, and the rate of interest is 20% per annum.

Would you like a detailed explanation of how the solution was derived?

Here are 5 questions to expand on this topic:
1. How does compound interest differ from simple interest?
2. What is the formula for compound interest when interest is compounded quarterly?
3. How does the time period affect the compound interest earned on a sum of money?
4. What would the compound interest be after 4 years at the same interest rate?
5. How can we compute the rate of interest if only the principal and final amount are known?

**Tip:** In compound interest, interest is calculated on the accumulated interest as well, which is why the amount grows faster than in simple interest.

On a certain sum of money, compound interest earned at the end of three years = Rs. 1456. Compound interest at the end of two years is Rs. 880. Compute the principal invested.

Learn how to compute the principal invested when the compound interest earned at the end of 3 years is Rs. 1456, and the compound interest at the end of 2 years is Rs. 880. This problem uses the compound interest formula and is suitable for students in Grades 9-12.

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