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

Options

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

Answer

To find the total sum invested by Mr. Bajaj, we can follow these steps:

### Step 1: Define the total investment
Let the total investment be Rs. \( X \).

### Step 2: Calculate the amounts invested at different rates
- Mr. Bajaj invested \( \frac{1}{7} \) of his total investment at 4%. Therefore, the amount invested at 4% is:
\[ \text{Amount at 4%} = \frac{1}{7}X \]

- He invested \( \frac{1}{2} \) of his total investment at 5%. Therefore, the amount invested at 5% is:
\[ \text{Amount at 5%} = \frac{1}{2}X \]

- The remaining amount is invested at 6%. To find the remaining amount, we calculate:
\[ \text{Remaining amount} = X - \left(\frac{1}{7}X + \frac{1}{2}X\right) \]
To simplify this, we find a common denominator:
\[ \frac{1}{7}X + \frac{1}{2}X = \frac{2}{14}X + \frac{7}{14}X = \frac{9}{14}X \]
Thus, the remaining amount is:
\[ \text{Remaining amount} = X - \frac{9}{14}X = \frac{5}{14}X \]

### Step 3: Calculate the interest from each investment
Using the formula for simple interest \( \text{SI} = \frac{P \times R \times T}{100} \):
- Interest from the amount at 4%:
\[ \text{Interest at 4%} = \frac{\frac{1}{7}X \times 4 \times 1}{100} = \frac{4X}{700} \]

- Interest from the amount at 5%:
\[ \text{Interest at 5%} = \frac{\frac{1}{2}X \times 5 \times 1}{100} = \frac{5X}{200} \]

- Interest from the amount at 6%:
\[ \text{Interest at 6%} = \frac{\frac{5}{14}X \times 6 \times 1}{100} = \frac{30X}{1400} \]

### Step 4: Sum the total interest
The total interest received is Rs. 730. Therefore, we can set up the equation:
\[ \frac{4X}{700} + \frac{5X}{200} + \frac{30X}{1400} = 730 \]

### Step 5: Find a common denominator and solve for \( X \)
The common denominator for 700, 200, and 1400 is 1400. Rewriting each term:
\[ \frac{4X \times 2}{1400} + \frac{5X \times 7}{1400} + \frac{30X}{1400} = 730 \]
This simplifies to:
\[ \frac{8X + 35X + 30X}{1400} = 730 \]
Combining the terms in the numerator:
\[ \frac{73X}{1400} = 730 \]

### Step 6: Solve for \( X \)
Cross-multiplying gives:
\[ 73X = 730 \times 1400 \]
Calculating the right side:
\[ 73X = 1022000 \]
Now, divide both sides by 73:
\[ X = \frac{1022000}{73} = 14000 \]

### Conclusion
Thus, the total sum invested by Mr. Bajaj is Rs. 14,000.

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