To solve the problem step by step, we will follow the given information and apply the formula for simple interest.
### Step 1: Define the Variables
Let the principal amount for each sum be \( P \).
### Step 2: Set Up the Interest Formulas
For the first sum lent at 8% for \( t \) years, the total amount received can be expressed as:
\[
A_1 = P + \left( \frac{P \times 8 \times t}{100} \right) = P \left(1 + \frac{8t}{100}\right)
\]
For the second sum lent at 4% for \( t + 2 \) years, the total amount received can be expressed as:
\[
A_2 = P + \left( \frac{P \times 4 \times (t + 2)}{100} \right) = P \left(1 + \frac{4(t + 2)}{100}\right)
\]
### Step 3: Set the Amounts Equal to 14,500
According to the problem, both amounts received are Rs. 14,500:
\[
P \left(1 + \frac{8t}{100}\right) = 14,500 \quad \text{(1)}
\]
\[
P \left(1 + \frac{4(t + 2)}{100}\right) = 14,500 \quad \text{(2)}
\]
### Step 4: Solve Equation (1)
From equation (1):
\[
P \left(1 + \frac{8t}{100}\right) = 14,500
\]
Rearranging gives:
\[
P = \frac{14,500}{1 + \frac{8t}{100}}
\]
### Step 5: Solve Equation (2)
From equation (2):
\[
P \left(1 + \frac{4(t + 2)}{100}\right) = 14,500
\]
Rearranging gives:
\[
P = \frac{14,500}{1 + \frac{4(t + 2)}{100}}
\]
### Step 6: Set the Two Expressions for P Equal
Now we have two expressions for \( P \):
\[
\frac{14,500}{1 + \frac{8t}{100}} = \frac{14,500}{1 + \frac{4(t + 2)}{100}}
\]
We can cancel \( 14,500 \) from both sides (as long as \( P \neq 0 \)):
\[
1 + \frac{8t}{100} = 1 + \frac{4(t + 2)}{100}
\]
### Step 7: Simplify the Equation
Removing the 1 from both sides gives:
\[
\frac{8t}{100} = \frac{4(t + 2)}{100}
\]
Multiplying through by 100 to eliminate the fraction:
\[
8t = 4(t + 2)
\]
Expanding the right side:
\[
8t = 4t + 8
\]
Subtracting \( 4t \) from both sides:
\[
4t = 8
\]
Dividing by 4:
\[
t = 2
\]
### Step 8: Substitute Back to Find P
Now that we have \( t = 2 \), we can substitute it back into either equation for \( P \). Using equation (1):
\[
P = \frac{14,500}{1 + \frac{8 \times 2}{100}} = \frac{14,500}{1 + \frac{16}{100}} = \frac{14,500}{1.16}
\]
Calculating \( P \):
\[
P = \frac{14,500}{1.16} = 12,500
\]
### Final Answer
Each sum lent is Rs. 12,500.
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