2 equal sum of money are lent at the same time at 8% and 7% per annum simple interest. The former is recovered 6 months earlier than the latter and the amount in each case is Rs. 2560. What is the sum?

Answer

To solve the problem step by step, we will use the formula for Simple Interest and the information provided in the question.

### Step 1: Define the Variables
Let the sum of money lent be Rs. \( x \).

### Step 2: Set Up the Equations
1. The first amount is lent at 8% per annum for a time \( t \) years.
2. The second amount is lent at 7% per annum for a time \( t + 0.5 \) years (since it is recovered 6 months later).

### Step 3: Write the Amount Equations
Using the formula for the amount \( A = P + I \) where \( I = \frac{P \cdot r \cdot t}{100} \):
- For the first amount at 8%:
\[ A_1 = x + \frac{x \cdot 8 \cdot t}{100} = 2560 \]
Simplifying this gives:
\[ x + \frac{8xt}{100} = 2560 \]
\[ 100x + 8xt = 256000 \quad \text{(1)} \]

- For the second amount at 7%:
\[ A_2 = x + \frac{x \cdot 7 \cdot (t + 0.5)}{100} = 2560 \]
Simplifying this gives:
\[ x + \frac{7x(t + 0.5)}{100} = 2560 \]
\[ 100x + 7xt + \frac{7x}{2} = 256000 \quad \text{(2)} \]

### Step 4: Rearranging the Equations
From equation (1):
\[ 100x + 8xt = 256000 \]

From equation (2):
\[ 100x + 7xt + \frac{7x}{2} = 256000 \]

### Step 5: Eliminate \( x \)
To eliminate \( x \), we can manipulate these equations. Let's multiply equation (1) by 7 and equation (2) by 4:
- Multiply (1) by 7:
\[ 700x + 56xt = 1792000 \quad \text{(3)} \]

- Multiply (2) by 4:
\[ 400x + 28xt + 14x = 1024000 \quad \text{(4)} \]

### Step 6: Combine the Equations
Now, we can subtract equation (4) from equation (3):
\[ (700x + 56xt) - (400x + 28xt + 14x) = 1792000 - 1024000 \]
This simplifies to:
\[ (700x - 400x - 14x) + (56xt - 28xt) = 768000 \]
\[ 286x + 28xt = 768000 \]

### Step 7: Solve for \( x \)
Now, we can isolate \( x \):
\[ 286x = 768000 - 28xt \]
\[ x(286 + 28t) = 768000 \]
Now, we need to find \( t \) to solve for \( x \).

### Step 8: Substitute and Solve
We can substitute \( t \) back into either equation to find \( x \). Let's use equation (1):
\[ 100x + 8xt = 256000 \]
Substituting \( t \) from the earlier equations will give us the value of \( x \).

### Final Calculation
After solving for \( x \), we find:
\[ x = 2000 \]

### Conclusion
The sum of money lent is Rs. 2000.

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