Prakash lends a part of Rs 20,000 at 8% simple interste and remaining at `(4)/(3)%` simple interset. His total income after a year was Rs 800. Find the sum lent at 8%.

Options

A. A) Rs 8,000
B. B) Rs 12,000
C. C) Rs 6,000
D. D) Rs 10,000

Answer

To solve the problem step by step, let's define the variables and use the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = amount lent at 8% simple interest
- \( 20000 - x \) = amount lent at \( \frac{4}{3}\% \) simple interest

### Step 2: Calculate Interest from Each Part
The interest earned from the amount lent at 8% for one year is:
\[ \text{Interest at 8%} = \frac{x \times 8 \times 1}{100} = \frac{8x}{100} = 0.08x \]

The interest earned from the amount lent at \( \frac{4}{3}\% \) for one year is:
\[ \text{Interest at } \frac{4}{3}\% = \frac{(20000 - x) \times \frac{4}{3} \times 1}{100} = \frac{(20000 - x) \times 4}{300} = \frac{4(20000 - x)}{300} = \frac{80000 - 4x}{300} \]

### Step 3: Total Interest Earned
According to the problem, the total interest earned after one year is Rs 800. Therefore, we can set up the equation:
\[ 0.08x + \frac{80000 - 4x}{300} = 800 \]

### Step 4: Clear the Fraction
To eliminate the fraction, we can multiply the entire equation by 300:
\[ 300 \times 0.08x + 80000 - 4x = 300 \times 800 \]
This simplifies to:
\[ 24x + 80000 - 4x = 240000 \]

### Step 5: Combine Like Terms
Now, combine the terms involving \( x \):
\[ 20x + 80000 = 240000 \]

### Step 6: Isolate \( x \)
Subtract 80000 from both sides:
\[ 20x = 240000 - 80000 \]
\[ 20x = 160000 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 20:
\[ x = \frac{160000}{20} = 8000 \]

### Conclusion
The amount lent at 8% simple interest is Rs 8000.