A certain amount is invested in a scheme of simple interest. It amounts to Rs 3096 in 6 years and Rs 3744 in 9 years. What is the rate of interest (in percentage)?

Options

A. A)12
B. B)15
C. C)18
D. D)20

Answer

To solve the problem step by step, let's break it down:

**Step 1: Understand the problem**
We are given that an amount invested in a scheme of simple interest amounts to Rs 3096 in 6 years and Rs 3744 in 9 years. We need to find the rate of interest.

**Step 2: Set up the equations**
Let the principal amount be \( P \) and the rate of interest be \( R \% \). The formula for the amount in simple interest is given by:
\[ A = P + I \]
where \( I \) is the interest earned.

The interest \( I \) can be calculated using the formula:
\[ I = P \times \frac{R}{100} \times T \]
where \( T \) is the time in years.

From the problem, we have two amounts:
1. After 6 years:
\[ A_1 = P + I_1 = P + P \times \frac{R}{100} \times 6 = 3096 \]
Simplifying this gives:
\[ P(1 + \frac{6R}{100}) = 3096 \quad \text{(Equation 1)} \]

2. After 9 years:
\[ A_2 = P + I_2 = P + P \times \frac{R}{100} \times 9 = 3744 \]
Simplifying this gives:
\[ P(1 + \frac{9R}{100}) = 3744 \quad \text{(Equation 2)} \]

**Step 3: Find the difference between the two amounts**
Now, we can subtract Equation 1 from Equation 2 to eliminate \( P \):
\[ P(1 + \frac{9R}{100}) - P(1 + \frac{6R}{100}) = 3744 - 3096 \]
This simplifies to:
\[ P \left( \frac{9R}{100} - \frac{6R}{100} \right) = 648 \]
\[ P \left( \frac{3R}{100} \right) = 648 \]

**Step 4: Solve for \( P \) and \( R \)**
From the above equation, we can express \( P \) in terms of \( R \):
\[ P = \frac{648 \times 100}{3R} = \frac{21600}{R} \]

**Step 5: Substitute \( P \) back into one of the equations**
Now, substitute \( P \) back into Equation 1:
\[ \frac{21600}{R} \left( 1 + \frac{6R}{100} \right) = 3096 \]
Multiplying through by \( R \):
\[ 21600 \left( 1 + \frac{6R}{100} \right) = 3096R \]
Expanding this gives:
\[ 21600 + \frac{129600R}{100} = 3096R \]
\[ 21600 + 1296R = 3096R \]
Rearranging gives:
\[ 21600 = 3096R - 1296R \]
\[ 21600 = 1800R \]
So,
\[ R = \frac{21600}{1800} = 12 \]

**Step 6: Conclusion**
The rate of interest \( R \) is \( 12\% \).

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