A sum amounts to `Rs 3600` at `2%` per annum under simple interest and `Rs 4800` at `4%` per annum under simple interest. The time taken is _____.

Options

A. `2.5 years`
B. `3 years`
C. `30 years`
D. `25 years`

Answer

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

### Given:
1. Amount (A1) = Rs 3600 at 2% per annum
2. Amount (A2) = Rs 4800 at 4% per annum

### Step 1: Write the formula for Simple Interest
The formula for calculating the amount under simple interest is:
\[ A = P + SI \]
Where:
- \( A \) = Amount
- \( P \) = Principal
- \( SI \) = Simple Interest

The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( R \) = Rate of interest
- \( T \) = Time in years

### Step 2: Set up the equations
For the first case (Rs 3600 at 2%):
\[ 3600 = P + \frac{P \times 2 \times T}{100} \]
This can be rearranged to:
\[ 3600 - P = \frac{2PT}{100} \]
Let's call this **Equation 1**.

For the second case (Rs 4800 at 4%):
\[ 4800 = P + \frac{P \times 4 \times T}{100} \]
This can be rearranged to:
\[ 4800 - P = \frac{4PT}{100} \]
Let's call this **Equation 2**.

### Step 3: Divide the two equations
Now, we will divide Equation 1 by Equation 2:
\[ \frac{3600 - P}{4800 - P} = \frac{\frac{2PT}{100}}{\frac{4PT}{100}} \]
This simplifies to:
\[ \frac{3600 - P}{4800 - P} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for P
Cross-multiplying gives:
\[ 2(3600 - P) = 1(4800 - P) \]
Expanding both sides:
\[ 7200 - 2P = 4800 - P \]

### Step 5: Rearranging the equation
Now, rearranging the equation:
\[ 7200 - 4800 = 2P - P \]
\[ 2400 = P \]
So, the principal \( P = Rs 2400 \).

### Step 6: Substitute P back to find T
Now, substitute \( P \) back into either Equation 1 or Equation 2 to find \( T \). Let's use Equation 1:
\[ 3600 - 2400 = \frac{2 \times 2400 \times T}{100} \]
This simplifies to:
\[ 1200 = \frac{4800T}{100} \]
\[ 1200 = 48T \]

### Step 7: Solve for T
Now, divide both sides by 48:
\[ T = \frac{1200}{48} = 25 \]

### Final Answer
The time taken is **25 years**.

---