A sum of money becomes Rs 8,928 in two years and Rs 10,224 in `3(1/2)` years at simple interest. Find out the rate of interest.

Options

A. 0.09
B. 0.15
C. 0.12
D. 0.1

Answer

To find the rate of interest, we can use the information given about the amounts after 2 years and 3.5 years. Let's denote the principal amount as \( P \) and the rate of interest as \( R \).

### Step 1: Calculate the Simple Interest for the first 2 years
We know that the amount after 2 years is Rs 8,928. The formula for the amount \( A \) after \( t \) years at simple interest is given by:

\[ A = P + SI \]

Where \( SI \) (Simple Interest) can be calculated as:

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

For 2 years, we can express the amount as:

\[ 8928 = P + \frac{P \times R \times 2}{100} \]

### Step 2: Calculate the Simple Interest for 3.5 years
Next, we know that the amount after 3.5 years is Rs 10,224. We can express this as:

\[ 10224 = P + \frac{P \times R \times 3.5}{100} \]

### Step 3: Set up the equations
Now we have two equations:

1. \( 8928 = P + \frac{2PR}{100} \) (Equation 1)
2. \( 10224 = P + \frac{3.5PR}{100} \) (Equation 2)

### Step 4: Solve the equations
We can rearrange both equations to isolate \( P \):

From Equation 1:

\[ P = 8928 - \frac{2PR}{100} \]

From Equation 2:

\[ P = 10224 - \frac{3.5PR}{100} \]

### Step 5: Set the two expressions for \( P \) equal to each other
Now, we can set the two expressions for \( P \) equal:

\[ 8928 - \frac{2PR}{100} = 10224 - \frac{3.5PR}{100} \]

### Step 6: Simplify the equation
Rearranging gives us:

\[ \frac{3.5PR}{100} - \frac{2PR}{100} = 10224 - 8928 \]

This simplifies to:

\[ \frac{1.5PR}{100} = 1296 \]

### Step 7: Solve for \( PR \)
Multiplying both sides by 100:

\[ 1.5PR = 129600 \]

Now, divide both sides by 1.5:

\[ PR = \frac{129600}{1.5} = 86400 \]

### Step 8: Substitute \( P \) back to find \( R \)
Now we can substitute \( P \) back into one of the equations to find \( R \). Using Equation 1:

\[ 8928 = P + \frac{2 \times 86400}{100} \]

This gives us:

\[ 8928 = P + 1728 \]

So,

\[ P = 8928 - 1728 = 7200 \]

### Step 9: Find the rate of interest \( R \)
Now we can find \( R \) using \( PR = 86400 \):

\[ 7200R = 86400 \]

Dividing both sides by 7200:

\[ R = \frac{86400}{7200} = 12 \]

### Final Answer
The rate of interest is \( \boxed{12\%} \).

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