A and B borrowed Rs 2,000 and Rs 3,000 respectively at the same rate of interest for `2.5` years. If B paid Rs 125 more interest than A, find the rate of interest.

Options

A. `8%`
B. `7%`
C. `5%`
D. `9%`

Answer

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

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

Where:
- \( P \) = Principal amount
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 1: Identify the principal amounts and time
- A borrowed Rs 2000, so \( P_A = 2000 \)
- B borrowed Rs 3000, so \( P_B = 3000 \)
- The time for both A and B is \( t = 2.5 \) years.

### Step 2: Write the formula for Simple Interest for both A and B
- For A:
\[ \text{SI}_A = \frac{P_A \times r \times t}{100} = \frac{2000 \times r \times 2.5}{100} \]

- For B:
\[ \text{SI}_B = \frac{P_B \times r \times t}{100} = \frac{3000 \times r \times 2.5}{100} \]

### Step 3: Set up the equation based on the information given
According to the problem, B paid Rs 125 more interest than A:
\[ \text{SI}_B - \text{SI}_A = 125 \]

Substituting the expressions for SI:
\[ \frac{3000 \times r \times 2.5}{100} - \frac{2000 \times r \times 2.5}{100} = 125 \]

### Step 4: Simplify the equation
Factor out the common terms:
\[ \frac{2.5r}{100} (3000 - 2000) = 125 \]

This simplifies to:
\[ \frac{2.5r}{100} \times 1000 = 125 \]

### Step 5: Further simplify
\[ 2.5r = 125 \]

### Step 6: Solve for \( r \)
Dividing both sides by 2.5:
\[ r = \frac{125}{2.5} \]

Calculating the right side:
\[ r = 50 \]

### Step 7: Convert to percentage
Since \( r \) is in terms of per annum, we express it as a percentage:
\[ r = 50\% \]

### Final Answer
The rate of interest is **5%**.

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