What sum of money at compound interest will amount to ? 4499.04 in 3 yr, if the rate of interest is 3% for the 1st yr, 4% for the 2nd yr and 5% for the 3rd yr ?

Answer

To find the principal amount (the sum of money) that will amount to ₹4499.04 in 3 years with varying interest rates, we can use the formula for compound interest. The formula for the amount \( A \) after \( n \) years is given by:

\[ A = P(1 + r_1)(1 + r_2)(1 + r_3) \]

Where:
- \( A \) is the amount after \( n \) years (₹4499.04 in this case)
- \( P \) is the principal amount (the sum we want to find)
- \( r_1, r_2, r_3 \) are the rates of interest for each year expressed as decimals.

### Step-by-Step Solution:

1. **Identify the interest rates for each year:**
- For the 1st year, the interest rate \( r_1 = 3\% = 0.03 \)
- For the 2nd year, the interest rate \( r_2 = 4\% = 0.04 \)
- For the 3rd year, the interest rate \( r_3 = 5\% = 0.05 \)

2. **Set up the equation using the compound interest formula:**
\[ 4499.04 = P(1 + 0.03)(1 + 0.04)(1 + 0.05) \]

3. **Calculate the factors:**
- \( 1 + 0.03 = 1.03 \)
- \( 1 + 0.04 = 1.04 \)
- \( 1 + 0.05 = 1.05 \)

4. **Multiply the factors together:**
\[ 1.03 \times 1.04 \times 1.05 \]
First, calculate \( 1.03 \times 1.04 \):
\[ 1.03 \times 1.04 = 1.0712 \]
Now multiply this result by \( 1.05 \):
\[ 1.0712 \times 1.05 = 1.12476 \]

5. **Substitute back into the equation:**
\[ 4499.04 = P \times 1.12476 \]

6. **Solve for \( P \):**
\[ P = \frac{4499.04}{1.12476} \]
Now perform the division:
\[ P \approx 3999.00 \]

### Final Answer:
The sum of money (the principal amount) that will amount to ₹4499.04 in 3 years is approximately ₹3999.00.

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