Equal sum of money are lent to X and Y at 7.5% per annum for a period of 4 years and 5 years respectivelty. If the difference in interst paid by them was Rs 150, the sum lent to each was

Options

A. Rs 500
B. Rs 1000
C. Rs 2000
D. Rs 3000

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 lent),
- \( R \) is the rate of interest,
- \( T \) is the time in years.

### Step 1: Define the variables
Let the principal amount lent to both X and Y be \( P \).

### Step 2: Calculate the interest for X
X is lent money for 4 years at a rate of 7.5%. Therefore, the interest paid by X is:

\[ \text{SI}_X = \frac{P \times 7.5 \times 4}{100} \]

### Step 3: Calculate the interest for Y
Y is lent money for 5 years at the same rate of 7.5%. Therefore, the interest paid by Y is:

\[ \text{SI}_Y = \frac{P \times 7.5 \times 5}{100} \]

### Step 4: Set up the equation for the difference in interest
According to the problem, the difference in interest paid by X and Y is Rs 150. Thus, we can write:

\[ \text{SI}_Y - \text{SI}_X = 150 \]

Substituting the expressions for SI:

\[ \frac{P \times 7.5 \times 5}{100} - \frac{P \times 7.5 \times 4}{100} = 150 \]

### Step 5: Simplify the equation
Factoring out \( \frac{P \times 7.5}{100} \):

\[ \frac{P \times 7.5}{100} (5 - 4) = 150 \]

This simplifies to:

\[ \frac{P \times 7.5}{100} = 150 \]

### Step 6: Solve for P
Now, we can solve for \( P \):

\[ P \times 7.5 = 150 \times 100 \]

\[ P \times 7.5 = 15000 \]

\[ P = \frac{15000}{7.5} \]

Calculating \( P \):

\[ P = 2000 \]

### Final Answer
The sum lent to each X and Y is Rs 2000.

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