A person invested a total sum of Rs. 7900 in three different schemes of simple interest at 3%, 5% and 8% per annum. At the end of one year he got same interest in all three schemes. What is the money (in Rs.) invested at 3%?

Options

A. 2900
B. 3500
C. 4000
D. 5600

Answer

To solve the problem, we need to find out how much money was invested at the 3% interest rate, given that the total investment is Rs. 7900 and the interest earned from each scheme is the same at the end of one year.

Let's denote:
- The amount invested at 3% as \( x \)
- The amount invested at 5% as \( y \)
- The amount invested at 8% as \( z \)

According to the problem, we have the following equations:

1. The total investment equation:
\[ x + y + z = 7900 \]

2. The interest earned from each investment after one year is the same. The interest for each scheme can be calculated using the formula for simple interest:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
Since the time is 1 year, we can express the interest for each scheme as:
- Interest from 3% scheme: \( \frac{x \times 3}{100} \)
- Interest from 5% scheme: \( \frac{y \times 5}{100} \)
- Interest from 8% scheme: \( \frac{z \times 8}{100} \)

Setting these interests equal to each other gives us the following equations:
\[ \frac{x \times 3}{100} = \frac{y \times 5}{100} = \frac{z \times 8}{100} \]

We can simplify this by removing the common factor of \( \frac{1}{100} \):
\[ 3x = 5y \quad \text{(1)} \]
\[ 3x = 8z \quad \text{(2)} \]

Now, we can express \( y \) and \( z \) in terms of \( x \):

From equation (1):
\[ y = \frac{3x}{5} \]

From equation (2):
\[ z = \frac{3x}{8} \]

Next, we substitute \( y \) and \( z \) into the total investment equation:
\[ x + \frac{3x}{5} + \frac{3x}{8} = 7900 \]

To solve this equation, we need a common denominator for the fractions. The least common multiple of 5 and 8 is 40. We can rewrite the equation as:
\[ x + \frac{24x}{40} + \frac{15x}{40} = 7900 \]

Combining the fractions:
\[ x + \frac{39x}{40} = 7900 \]

Now, converting \( x \) to have a common denominator:
\[ \frac{40x}{40} + \frac{39x}{40} = 7900 \]
\[ \frac{79x}{40} = 7900 \]

Now, multiply both sides by 40:
\[ 79x = 316000 \]

Now, divide both sides by 79:
\[ x = \frac{316000}{79} = 4000 \]

Thus, the amount invested at 3% is Rs. 4000.

### Summary of the Solution Steps:
1. Define variables for each investment.
2. Set up the total investment equation.
3. Set up equations for equal interest earned.
4. Express \( y \) and \( z \) in terms of \( x \).
5. Substitute into the total investment equation.
6. Solve for \( x \).