Context: Shobhit invests a sum of money at a fixed annual interest rate, compounded annually. If the interest accrued during the second year and the third year is Rs. 550 and Rs. 605, respectively, what is the interest (in Rs.) accrued during the fourth year?

Answer question

To solve this problem, let's analyze the given information and use the formula for compound interest.

### Given Information:
1. Interest accrued during the second year (I2) = Rs. 550
2. Interest accrued during the third year (I3) = Rs. 605

The interest accrued each year in compound interest is calculated on the principal amount for that year, which includes the principal and all previously accumulated interest.

### Compound Interest Formula:
The interest for a particular year in compound interest can be represented by:
\[
I_n = P \times (1 + r)^{n-1} \times r
\]
where:
- \( I_n \) is the interest for the \( n \)-th year,
- \( P \) is the principal amount,
- \( r \) is the annual interest rate (in decimal form),
- \( n \) is the year number.

Using the provided information for the second and third years:

\[
I_2 = P \times (1 + r) \times r = 550
\]
\[
I_3 = P \times (1 + r)^2 \times r = 605
\]

### Step-by-Step Solution:

1. **Find the Interest Rate \( r \):**

   Divide the interest of the third year by the interest of the second year to find the interest rate \( r \):

\[
\frac{I_3}{I_2} = \frac{P \times (1 + r)^2 \times r}{P \times (1 + r) \times r} = \frac{605}{550}
\]

Simplifying this equation gives:

\[
\frac{(1 + r) \times r}{r} = \frac{605}{550}
\]
\[
1 + r = \frac{605}{550}
\]

Calculate \( \frac{605}{550} \):

\[
1 + r = \frac{605}{550} = 1.1
\]
\[
r = 1.1 - 1 = 0.1
\]

So, the annual interest rate \( r \) is \( 0.1 \) or \( 10\% \).

2. **Find the Interest for the Fourth Year \( I_4 \):**

   Using the value of \( r \), we can calculate the interest for the fourth year:

\[
I_4 = P \times (1 + r)^3 \times r
\]

Since \( P \times (1 + r)^2 \times r = 605 \), multiply by \( 1 + r \) to get the interest for the fourth year:

\[
I_4 = 605 \times (1 + 0.1) = 605 \times 1.1
\]

Calculate \( 605 \times 1.1 \):

\[
I_4 = 665.5
\]

### Conclusion:
The interest accrued during the fourth year is **Rs. 665.5**.

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Would you like more details or have any questions? Here are some related questions to further your understanding:

1. How would the interest change if it was compounded semi-annually instead of annually?
2. What would be the interest rate if the interest accrued in the second year was Rs. 600 instead of Rs. 550?
3. How much total interest would be accrued over 5 years?
4. If the interest accrued in the fifth year was Rs. 731, what would the principal amount be?
5. What is the difference in total accrued interest after 4 years if the principal was Rs. 10,000 instead of the calculated amount?

**Tip:** Remember, in compound interest, the interest for each year is calculated on the updated principal amount, which includes all previously accumulated interest.

Learn how to calculate compound interest with a detailed solution. Given interest accrued in the second and third years, find the interest accrued during the fourth year using the compound interest formula. Suitable for high school students studying financial mathematics.

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