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

Explanation

To solve the problem step by step, we will use the formula for simple interest and the information given in the question.

### Step 1: Understanding the Problem
We have two scenarios:
1. A sum amounts to Rs. 3600 at 2% per annum.
2. The same sum amounts to Rs. 4800 at 4% per annum.

We need to find the time taken (T) for both cases.

### Step 2: Set Up the Equations
For the first case:
- Amount (A1) = Rs. 3600
- Rate (R1) = 2%
- Principal (P) = Unknown
- Time (T) = Unknown

Using the formula for simple interest:
\[ A = P + \text{SI} \]
Where \(\text{SI} = \frac{P \times R \times T}{100}\)

So, we can write:
\[ 3600 = P + \frac{P \times 2 \times T}{100} \]
This simplifies to:
\[ 3600 = P + \frac{2PT}{100} \]
(Equation 1)

For the second case:
- Amount (A2) = Rs. 4800
- Rate (R2) = 4%
- Principal (P) = Unknown
- Time (T) = Unknown

Similarly, we can write:
\[ 4800 = P + \frac{P \times 4 \times T}{100} \]
This simplifies to:
\[ 4800 = P + \frac{4PT}{100} \]
(Equation 2)

### Step 3: Solve the Equations
Now we have two equations:
1. \( 3600 = P + \frac{2PT}{100} \)
2. \( 4800 = P + \frac{4PT}{100} \)

We can subtract Equation 1 from Equation 2:
\[ 4800 - 3600 = \left(P + \frac{4PT}{100}\right) - \left(P + \frac{2PT}{100}\right) \]
This simplifies to:
\[ 1200 = \frac{4PT}{100} - \frac{2PT}{100} \]
\[ 1200 = \frac{2PT}{100} \]

### Step 4: Isolate T
Now, multiply both sides by 100 to eliminate the fraction:
\[ 120000 = 2PT \]

Now, divide both sides by \(2P\):
\[ T = \frac{120000}{2P} \]
\[ T = \frac{60000}{P} \]

### Step 5: Substitute P
Now we need to find the value of \(P\). We can use either Equation 1 or Equation 2. Let's use Equation 1:
\[ 3600 = P + \frac{2PT}{100} \]

Substituting \(T\):
\[ 3600 = P + \frac{2P \times (60000/P)}{100} \]
\[ 3600 = P + \frac{120000}{100} \]
\[ 3600 = P + 1200 \]

Now, isolate \(P\):
\[ P = 3600 - 1200 \]
\[ P = 2400 \]

### Step 6: Find the Time T
Now substitute \(P\) back into the equation for \(T\):
\[ T = \frac{60000}{2400} \]
\[ T = 25 \]

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

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