A sum of money was invested for 14 years was in Scheme A which offers simple interest at a rate of 8 % p.a. The amount received from Scheme A after 14 years was then invested for two years in Scheme B which offers com pound interest (compounded annually) at a rate of 10% p.a. If the interest received from Scheme B was Rs. 6,678, what was the sum invested in Scheme A ?

Options

A. Rs. 15,500
B. Rs. 14,500
C. Rs. 16,500
D. Rs. 15,000

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Calculate the amount received from Scheme A
We know that the formula for Simple Interest (SI) is given by:

\[ \text{SI} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (the initial sum invested)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

Let the principal amount invested in Scheme A be \( P \).

Given:
- \( R = 8\% \)
- \( T = 14 \) years

The total amount \( A \) received after 14 years from Scheme A will be:

\[ A = P + \text{SI} = P + \frac{P \times 8 \times 14}{100} \]

This simplifies to:

\[ A = P + \frac{112P}{100} = P \left(1 + \frac{112}{100}\right) = P \left(\frac{212}{100}\right) = \frac{212P}{100} \]

### Step 2: Invest the amount from Scheme A in Scheme B
The amount received from Scheme A is then invested in Scheme B for 2 years at a compound interest rate of 10% per annum.

The formula for the amount \( A \) in compound interest is:

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

Where:
- \( R = 10\% \)
- \( T = 2 \) years

Substituting the values, we have:

\[ A = \frac{212P}{100} \left(1 + \frac{10}{100}\right)^2 = \frac{212P}{100} \left(1.1\right)^2 = \frac{212P}{100} \times 1.21 = \frac{256.52P}{100} \]

### Step 3: Calculate the compound interest from Scheme B
The compound interest (CI) earned from Scheme B is given as:

\[ \text{CI} = A - P \]

Substituting the values we have:

\[ \text{CI} = \frac{256.52P}{100} - \frac{212P}{100} = \frac{256.52P - 212P}{100} = \frac{44.52P}{100} \]

### Step 4: Set up the equation with the given interest from Scheme B
We know that the interest received from Scheme B is Rs. 6,678. Therefore, we can set up the equation:

\[ \frac{44.52P}{100} = 6678 \]

### Step 5: Solve for \( P \)
To find \( P \), we rearrange the equation:

\[ P = \frac{6678 \times 100}{44.52} \]

Calculating this gives:

\[ P = \frac{667800}{44.52} \approx 15000 \]

Thus, the sum invested in Scheme A is Rs. 15,000.

### Final Answer
The sum invested in Scheme A is **Rs. 15,000**.

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