Mr. Bajaj invested 1/7of his total investment at `4%` and 1/2at `5%` and rest at `6%` for the one year and received total interest of ₹ 730. What is the total sum invested?

Options

A. ₹ 70000
B. ₹ 14000
C. ₹ 24000
D. ₹ 38000

Answer

To solve the problem step by step, we will break down the investments made by Mr. Bajaj and calculate the total sum invested based on the interest received.

### Step 1: Define the total investment
Let the total investment be denoted as \( P \).

### Step 2: Calculate the amounts invested at different rates
1. Mr. Bajaj invested \( \frac{1}{7} \) of his total investment at 4%.
\[ \text{Investment at 4%} = \frac{1}{7}P \]

2. He invested \( \frac{1}{2} \) of his total investment at 5%.
\[ \text{Investment at 5%} = \frac{1}{2}P \]

3. The remaining investment is at 6%. To find the remaining amount, we first calculate the total fraction invested:
\[ \text{Total fraction invested} = \frac{1}{7} + \frac{1}{2} \]
To add these fractions, we need a common denominator. The least common multiple of 7 and 2 is 14.
\[ \frac{1}{7} = \frac{2}{14}, \quad \frac{1}{2} = \frac{7}{14} \]
Thus,
\[ \text{Total fraction} = \frac{2}{14} + \frac{7}{14} = \frac{9}{14} \]
Therefore, the fraction remaining for the 6% investment is:
\[ \text{Remaining fraction} = 1 - \frac{9}{14} = \frac{5}{14} \]
Hence, the investment at 6% is:
\[ \text{Investment at 6%} = \frac{5}{14}P \]

### Step 3: Calculate the interest from each investment
1. Interest from the investment at 4%:
\[ \text{Interest at 4%} = \frac{1}{7}P \times \frac{4}{100} = \frac{4P}{700} \]

2. Interest from the investment at 5%:
\[ \text{Interest at 5%} = \frac{1}{2}P \times \frac{5}{100} = \frac{5P}{200} \]

3. Interest from the investment at 6%:
\[ \text{Interest at 6%} = \frac{5}{14}P \times \frac{6}{100} = \frac{30P}{1400} = \frac{3P}{140} \]

### Step 4: Total interest received
According to the problem, the total interest received is ₹730. Therefore, we can set up the equation:
\[ \frac{4P}{700} + \frac{5P}{200} + \frac{3P}{140} = 730 \]

### Step 5: Find a common denominator and simplify
The least common multiple of 700, 200, and 140 is 1400. We will convert each term to have this common denominator:
1. Convert \( \frac{4P}{700} \):
\[ \frac{4P}{700} = \frac{8P}{1400} \]
2. Convert \( \frac{5P}{200} \):
\[ \frac{5P}{200} = \frac{35P}{1400} \]
3. \( \frac{3P}{140} \) is already in terms of 1400:
\[ \frac{3P}{140} = \frac{30P}{1400} \]

Now we can combine the terms:
\[ \frac{8P + 35P + 30P}{1400} = 730 \]
This simplifies to:
\[ \frac{73P}{1400} = 730 \]

### Step 6: Solve for \( P \)
To solve for \( P \), multiply both sides by 1400:
\[ 73P = 730 \times 1400 \]
Calculating the right side:
\[ 73P = 1022000 \]
Now, divide by 73:
\[ P = \frac{1022000}{73} = 14000 \]

### Conclusion
The total sum invested by Mr. Bajaj is ₹14,000.