A person invested a total sum of Rs. 7900 in three different schemes of simple interset at 3% 5% and 8% per annum. At the end of one year he got same interset 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 step by step, we will denote the amounts invested in the three different schemes as follows:

Let:
- \( P_1 \) = amount invested at 3% per annum
- \( P_2 \) = amount invested at 5% per annum
- \( P_3 \) = amount invested at 8% per annum

According to the problem, the total investment is Rs. 7900:
\[ P_1 + P_2 + P_3 = 7900 \quad \text{(1)} \]

We also know that the interest earned from all three investments is the same at the end of one year. The formula for simple interest is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time in years.

Since the time \( T \) is 1 year for all investments, we can express the interest for each scheme as follows:
- Interest from the 3% scheme:
\[ I_1 = \frac{P_1 \times 3 \times 1}{100} = \frac{3P_1}{100} \]
- Interest from the 5% scheme:
\[ I_2 = \frac{P_2 \times 5 \times 1}{100} = \frac{5P_2}{100} \]
- Interest from the 8% scheme:
\[ I_3 = \frac{P_3 \times 8 \times 1}{100} = \frac{8P_3}{100} \]

Since the interest from all three schemes is the same, we can set them equal to each other:
\[ \frac{3P_1}{100} = \frac{5P_2}{100} = \frac{8P_3}{100} \]

To eliminate the fraction, we can multiply through by 100:
\[ 3P_1 = 5P_2 \quad \text{(2)} \]
\[ 3P_1 = 8P_3 \quad \text{(3)} \]

From equation (2), we can express \( P_2 \) in terms of \( P_1 \):
\[ P_2 = \frac{3P_1}{5} \quad \text{(4)} \]

From equation (3), we can express \( P_3 \) in terms of \( P_1 \):
\[ P_3 = \frac{3P_1}{8} \quad \text{(5)} \]

Now we can substitute equations (4) and (5) into equation (1):
\[ P_1 + \frac{3P_1}{5} + \frac{3P_1}{8} = 7900 \]

To combine these terms, we need a common denominator. The least common multiple of 5 and 8 is 40. Thus, we rewrite the fractions:
\[ P_1 + \frac{24P_1}{40} + \frac{15P_1}{40} = 7900 \]
\[ P_1 + \frac{39P_1}{40} = 7900 \]

Now, we can express \( P_1 \) as:
\[ \frac{40P_1 + 39P_1}{40} = 7900 \]
\[ \frac{79P_1}{40} = 7900 \]

Multiplying both sides by 40:
\[ 79P_1 = 316000 \]

Now, divide by 79:
\[ P_1 = \frac{316000}{79} = 4000 \]

Thus, the amount invested at 3% is:
\[ P_1 = 4000 \]

### Final Answer:
The money invested at 3% is Rs. 4000.