Puneet lent out Rs 30000 in two parts, one at 5% and the other at 9% interest. The yearly average interest comes out to be 6%. What are the amounts (in Rs) that are lent at 5% and 9% respectively ?

Options

A. 20000, 10000
B. 22500, 7500
C. 18000, 12000
D. 21000, 9000

Answer

To solve the problem, we can use the concept of allegation to find the amounts lent at different interest rates. Here’s a step-by-step solution:

### Step 1: Define the variables
Let:
- \( x \) = amount lent at 5%
- \( y \) = amount lent at 9%

Given that the total amount lent is Rs 30,000, we can write the equation:
\[ x + y = 30000 \]

### Step 2: Set up the average interest equation
The average interest rate can be calculated using the formula:
\[ \text{Average Interest} = \frac{\text{Total Interest}}{\text{Total Principal}} \]

The total interest earned from both parts can be expressed as:
\[ \text{Interest from } x = \frac{5}{100} \cdot x \]
\[ \text{Interest from } y = \frac{9}{100} \cdot y \]

Thus, the total interest earned is:
\[ \frac{5}{100}x + \frac{9}{100}y \]

Given that the average interest rate is 6%, we can set up the equation:
\[ \frac{\frac{5}{100}x + \frac{9}{100}y}{30000} = \frac{6}{100} \]

### Step 3: Simplify the average interest equation
Multiplying both sides by 30000 gives:
\[ \frac{5}{100}x + \frac{9}{100}y = \frac{6}{100} \cdot 30000 \]
\[ \frac{5}{100}x + \frac{9}{100}y = 1800 \]

### Step 4: Eliminate the fractions
To eliminate the fractions, multiply the entire equation by 100:
\[ 5x + 9y = 180000 \]

### Step 5: Solve the system of equations
Now, we have a system of two equations:
1. \( x + y = 30000 \)
2. \( 5x + 9y = 180000 \)

From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 30000 - x \]

Substituting \( y \) in the second equation:
\[ 5x + 9(30000 - x) = 180000 \]
\[ 5x + 270000 - 9x = 180000 \]
\[ -4x + 270000 = 180000 \]
\[ -4x = 180000 - 270000 \]
\[ -4x = -90000 \]
\[ x = \frac{90000}{4} = 22500 \]

### Step 6: Find \( y \)
Now substituting \( x \) back into the equation for \( y \):
\[ y = 30000 - 22500 = 7500 \]

### Final amounts
Thus, the amounts lent at 5% and 9% are:
- Amount lent at 5%: Rs 22500
- Amount lent at 9%: Rs 7500

### Summary of the solution:
- Amount lent at 5% = Rs 22500
- Amount lent at 9% = Rs 7500