If the difference between the simple interest on a certain sum of money for 4 years at `2 (1)/(2) %` per annum and the simple interest on the same sum for the same period at 3% per annum is Rs 60, then find the sum.

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI):

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

where:
- \( P \) is the principal amount (the sum of money),
- \( R \) is the rate of interest per annum,
- \( T \) is the time in years.

### Step 1: Identify the rates and time
We have two different rates of interest:
- Rate 1: \( 2 \frac{1}{2} \% = \frac{5}{2} \% \)
- Rate 2: \( 3 \% \)

The time period \( T = 4 \) years.

### Step 2: Write the formula for Simple Interest for both rates
For Rate 1 (at \( 2 \frac{1}{2} \% \)):
\[ \text{SI}_1 = \frac{P \times \frac{5}{2} \times 4}{100} \]

For Rate 2 (at \( 3 \% \)):
\[ \text{SI}_2 = \frac{P \times 3 \times 4}{100} \]

### Step 3: Calculate the Simple Interest for both rates
Now, we can simplify both expressions:

1. For \( \text{SI}_1 \):
\[ \text{SI}_1 = \frac{P \times 5 \times 4}{2 \times 100} = \frac{20P}{200} = \frac{P}{10} \]

2. For \( \text{SI}_2 \):
\[ \text{SI}_2 = \frac{P \times 3 \times 4}{100} = \frac{12P}{100} = \frac{3P}{25} \]

### Step 4: Find the difference between the two Simple Interests
According to the problem, the difference between the two Simple Interests is Rs 60:
\[ \text{SI}_2 - \text{SI}_1 = 60 \]
Substituting the values we found:
\[ \frac{3P}{25} - \frac{P}{10} = 60 \]

### Step 5: Find a common denominator and solve for \( P \)
The common denominator for 25 and 10 is 50. We can rewrite the equation:
\[ \frac{6P}{50} - \frac{5P}{50} = 60 \]
This simplifies to:
\[ \frac{6P - 5P}{50} = 60 \]
\[ \frac{P}{50} = 60 \]

### Step 6: Solve for \( P \)
Multiplying both sides by 50:
\[ P = 60 \times 50 = 3000 \]

### Final Answer
The sum (principal amount) is Rs 3000.

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