The simple interest on a certain sum of money for 3 yr at 8% pa is half the compound interest on X 8000 for 2 yr at 10% pa. Find the sum on which simple interest is calculated.

Options

A. X 3500
B. X 3800
C. X 4000
D. X 3600

Answer

To solve the problem step by step, we will break it down as follows:

### Step 1: Understand the Problem
We need to find the sum (let's call it \( P \)) on which the simple interest is calculated. The simple interest for 3 years at 8% per annum is half the compound interest on \( 8000 \) for 2 years at 10% per annum.

### Step 2: Calculate the Compound Interest on \( 8000 \)
The formula for compound interest (CI) is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) is the amount after time \( T \)
- \( P \) is the principal amount
- \( R \) is the rate of interest
- \( T \) is the time in years

For our case:
- \( P = 8000 \)
- \( R = 10\% \)
- \( T = 2 \)

Calculating \( A \):
\[ A = 8000 \left(1 + \frac{10}{100}\right)^2 = 8000 \left(1 + 0.1\right)^2 = 8000 \left(1.1\right)^2 \]
Calculating \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
Now substituting back:
\[ A = 8000 \times 1.21 = 9680 \]

### Step 3: Calculate the Compound Interest
Now we can find the compound interest (CI):
\[ CI = A - P = 9680 - 8000 = 1680 \]

### Step 4: Relate Simple Interest to Compound Interest
According to the problem, the simple interest (SI) for \( P \) is half of the compound interest:
\[ SI = \frac{1}{2} \times CI = \frac{1}{2} \times 1680 = 840 \]

### Step 5: Use the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI = 840 \)
- \( R = 8\% \)
- \( T = 3 \)

Substituting the known values:
\[ 840 = \frac{P \times 8 \times 3}{100} \]
\[ 840 = \frac{24P}{100} \]
Multiplying both sides by 100:
\[ 84000 = 24P \]
Now, solving for \( P \):
\[ P = \frac{84000}{24} = 3500 \]

### Conclusion
The sum on which the simple interest is calculated is \( P = 3500 \) rupees.

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