What sum of money lent at `6*5%` per annum will produce the same interest in 4 years as Rs. 7,500 produce in 6 years at 5% per annum ?

Answer

To solve the problem step by step, we need to find the sum of money (let's call it P1) that, when lent at an interest rate of 6.5% per annum for 4 years, will yield the same interest as Rs. 7,500 at 5% per annum for 6 years.

### Step 1: Calculate the interest earned on Rs. 7,500 at 5% for 6 years.

The formula for calculating simple interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For this case:
- \( P = 7500 \)
- \( R = 5 \)
- \( T = 6 \)

Substituting the values:
\[ \text{SI} = \frac{7500 \times 5 \times 6}{100} \]

### Step 2: Calculate the interest.

Calculating the above expression:
\[ \text{SI} = \frac{7500 \times 30}{100} = \frac{225000}{100} = 2250 \]

So, the interest earned on Rs. 7,500 at 5% for 6 years is Rs. 2,250.

### Step 3: Set up the equation for the unknown principal (P1).

Now, we need to find the principal amount (P1) that will yield the same interest of Rs. 2,250 at 6.5% for 4 years.

Using the same formula for simple interest:
\[ \text{SI} = \frac{P1 \times R1 \times T1}{100} \]
Where:
- \( P1 = x \) (the unknown principal we want to find)
- \( R1 = 6.5 \)
- \( T1 = 4 \)

We know the interest is the same, so we can set up the equation:
\[ 2250 = \frac{x \times 6.5 \times 4}{100} \]

### Step 4: Solve for P1.

Rearranging the equation to solve for \( x \):
\[ 2250 = \frac{x \times 26}{100} \]
Multiplying both sides by 100:
\[ 225000 = 26x \]
Now, divide both sides by 26:
\[ x = \frac{225000}{26} \approx 8653.846 \]

### Step 5: Round the answer.

Rounding off to two decimal places, we get:
\[ x \approx 8653.85 \]

Thus, the sum of money that needs to be lent at 6.5% per annum to produce the same interest in 4 years as Rs. 7,500 produces in 6 years at 5% per annum is approximately Rs. 8,653.85.

### Summary of the Solution:
The principal amount (P1) is Rs. 8,653.85.

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