Ramesh invests rs 12800 for three years at the rate of 10% p.a compound interest. Find : (i)  the sum due to Ramesh at the end of the first year. (ii) the interest he earn for the second year . (iii) the total amount due to him at the end of three years

To solve the problem involving compound interest, we can use the formula for compound interest and the formula for calculating the amount due at the end of a specific period.

### Compound Interest Formula:
The formula to calculate the compound interest is:

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

Where:
- \( A \) is the amount due at the end of \( n \) years.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate.
- \( n \) is the number of years.

Given:
- \( P = 12800 \) (initial investment in rupees)
- \( r = 10\% \) per annum
- \( n = 3 \) years

#### (i) The sum due to Ramesh at the end of the first year:
For the first year (\( n = 1 \)),

\[
A_1 = 12800 \left(1 + \frac{10}{100}\right)^1 = 12800 \times 1.1 = 14080
\]

So, the sum due to Ramesh at the end of the first year is **Rs. 14,080**.

#### (ii) The interest he earns for the second year:
To find the interest for the second year, we first calculate the amount at the end of the second year and then subtract the amount at the end of the first year.

For the second year (\( n = 2 \)):

\[
A_2 = 12800 \left(1 + \frac{10}{100}\right)^2 = 12800 \times 1.1^2 = 12800 \times 1.21 = 15488
\]

Interest earned in the second year:

\[
\text{Interest for 2nd year} = A_2 - A_1 = 15488 - 14080 = 1408
\]

So, the interest earned by Ramesh in the second year is **Rs. 1,408**.

#### (iii) The total amount due to him at the end of three years:
For the third year (\( n = 3 \)):

\[
A_3 = 12800 \left(1 + \frac{10}{100}\right)^3 = 12800 \times 1.1^3
\]

Calculating \( 1.1^3 \):

\[
1.1^3 = 1.331
\]

So,

\[
A_3 = 12800 \times 1.331 = 17036.8
\]

The total amount due to Ramesh at the end of three years is **Rs. 17,036.80**.

### Summary of the Results:
1. Sum due at the end of the first year: **Rs. 14,080**
2. Interest earned in the second year: **Rs. 1,408**
3. Total amount due at the end of three years: **Rs. 17,036.80**

Would you like to delve into more details or have any questions?

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### Related Questions:
1. What is the formula for calculating compound interest quarterly?
2. How would the total amount change if the interest rate were 12% instead of 10%?
3. What would be the amount due if Ramesh invested for 5 years instead of 3 years?
4. How does compound interest differ from simple interest?
5. What would be the effect of changing the investment amount to Rs. 15,000?
6. Can compound interest result in greater returns than simple interest over the same period? How?
7. How do you calculate the interest earned in the third year specifically?
8. What is the impact of compounding frequency (monthly, quarterly) on the total amount?

### Tip:
When dealing with compound interest, remember that the frequency of compounding (monthly, quarterly, annually) can significantly affect the total amount due.

Explore how to solve Ramesh's compound interest problem step-by-step. Calculate the amount due at the end of each year and the interest earned annually. Learn the compound interest formula and its application with detailed examples.

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