Ayush invested Rs.Pat R% p.a. Cl. If compound interest received in second year and third year is Rs.3450 and Rs.3967.5 respectively, then find the value of P.

Options

A. Rs.30000
B. Rs.20000
C. Rs.25000
D. Rs.15000

Answer

To solve the problem step by step, we will use the information given about the compound interest received in the second and third years.

### Step 1: Understand the given information
We know that:
- Compound Interest (CI) in the second year = Rs. 3450
- Compound Interest (CI) in the third year = Rs. 3967.5

### Step 2: Find the difference in compound interest
To find the difference in compound interest between the third year and the second year:
\[ \text{Difference} = \text{CI in third year} - \text{CI in second year} = 3967.5 - 3450 = 517.5 \]

### Step 3: Relate the difference to the principal and rate of interest
The difference in compound interest between two consecutive years can be expressed as:
\[ \text{Difference} = \text{CI in second year} \times \frac{R}{100} \]
Where \( R \) is the rate of interest. Thus, we can write:
\[ 517.5 = 3450 \times \frac{R}{100} \]

### Step 4: Solve for R
Rearranging the equation to find \( R \):
\[ R = \frac{517.5 \times 100}{3450} \]
Calculating this gives:
\[ R = \frac{51750}{3450} = 15 \]
So, the rate of interest \( R \) is 15%.

### Step 5: Find the interest for the first year
The compound interest for the second year is calculated on the amount at the end of the first year. Therefore, the amount at the end of the first year can be represented as:
\[ \text{Amount after 1 year} = P + \text{CI for first year} \]
Let the CI for the first year be \( CI_1 \). We know:
\[ CI_2 = CI_1 \times \frac{R}{100} = 3450 \]
Thus,
\[ CI_1 = \frac{3450 \times 100}{R} = \frac{3450 \times 100}{15} = 23000 \]

### Step 6: Calculate the principal
Using the formula for compound interest:
\[ CI_1 = \frac{P \times R \times T}{100} \]
We can rearrange to find \( P \):
\[ P = \frac{CI_1 \times 100}{R \times T} \]
Substituting \( CI_1 = 3000 \), \( R = 15 \), and \( T = 1 \):
\[ P = \frac{3000 \times 100}{15 \times 1} = 20000 \]

### Conclusion
The value of \( P \) is Rs. 20,000.