What sum of money at compound interest will amount to Rs. 2249.52 in 3 years, if the rate of interest is 3% for the first year, 4 % for the second year and 5% for the third year ?

Answer

To solve the problem step by step, we need to find the principal amount (P) that will amount to Rs. 2249.52 after 3 years with variable rates of interest for each year.

### Step 1: Understand the Problem
We need to find the principal amount (P) that will grow to Rs. 2249.52 in 3 years with the following interest rates:
- 1st year: 3%
- 2nd year: 4%
- 3rd year: 5%

### Step 2: Set Up the Compound Interest Formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r_1}{100}\right) \left(1 + \frac{r_2}{100}\right) \left(1 + \frac{r_3}{100}\right) \]
where:
- \( A \) is the amount after time \( t \),
- \( P \) is the principal amount,
- \( r_1, r_2, r_3 \) are the rates of interest for each year.

### Step 3: Substitute the Values
In our case:
- \( A = 2249.52 \)
- \( r_1 = 3 \), \( r_2 = 4 \), \( r_3 = 5 \)

Substituting these values into the formula gives:
\[ 2249.52 = P \left(1 + \frac{3}{100}\right) \left(1 + \frac{4}{100}\right) \left(1 + \frac{5}{100}\right) \]

### Step 4: Simplify the Equation
Calculating the terms:
- \( 1 + \frac{3}{100} = 1.03 \)
- \( 1 + \frac{4}{100} = 1.04 \)
- \( 1 + \frac{5}{100} = 1.05 \)

So the equation becomes:
\[ 2249.52 = P \times 1.03 \times 1.04 \times 1.05 \]

### Step 5: Calculate the Product of the Interest Factors
Now we calculate:
\[ 1.03 \times 1.04 \times 1.05 \]
Calculating step by step:
- \( 1.03 \times 1.04 = 1.0712 \)
- \( 1.0712 \times 1.05 = 1.12476 \)

Thus, we have:
\[ 2249.52 = P \times 1.12476 \]

### Step 6: Solve for P
Now, to find \( P \):
\[ P = \frac{2249.52}{1.12476} \]

### Step 7: Perform the Division
Calculating the division:
\[ P \approx 2000 \]

### Conclusion
The principal amount (P) is Rs. 2000.

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