Sudeep invested 1/8 of a certain sum at 5% p. a. for two years and 3/5 of the sum at 6% p. a. for two years and the remaining at 10% p.a. for two years. If the total interest received is Rs 1,674, then the total sum invested is:

Options

A. Rs 12500
B. Rs 12,000
C. Rs 10,500
D. Rs, 13,000

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the total sum
Let the total sum invested be \( S \). We can express it as:
\[ S = 800x \]

### Step 2: Calculate the amounts invested in different rates
1. **Amount invested at 5%**:
\[ \text{Amount at 5%} = \frac{1}{8} S = \frac{1}{8} \times 800x = 100x \]

2. **Amount invested at 6%**:
\[ \text{Amount at 6%} = \frac{3}{5} S = \frac{3}{5} \times 800x = 480x \]

3. **Remaining amount invested at 10%**:
\[ \text{Remaining amount} = S - \left(100x + 480x\right) = 800x - 580x = 220x \]

### Step 3: Calculate the interest for each investment
Using the formula for simple interest \( I = \frac{P \times R \times T}{100} \):

1. **Interest from 100x at 5% for 2 years**:
\[ I_1 = \frac{100x \times 5 \times 2}{100} = 10x \]

2. **Interest from 480x at 6% for 2 years**:
\[ I_2 = \frac{480x \times 6 \times 2}{100} = 57.6x \]

3. **Interest from 220x at 10% for 2 years**:
\[ I_3 = \frac{220x \times 10 \times 2}{100} = 44x \]

### Step 4: Total interest received
The total interest received is given as Rs. 1674:
\[ I_1 + I_2 + I_3 = 1674 \]
Substituting the values:
\[ 10x + 57.6x + 44x = 1674 \]
Combining like terms:
\[ 111.6x = 1674 \]

### Step 5: Solve for x
To find \( x \):
\[ x = \frac{1674}{111.6} = 15 \]

### Step 6: Calculate the total sum invested
Now substituting \( x \) back into the expression for \( S \):
\[ S = 800x = 800 \times 15 = 12000 \]

### Final Answer
The total sum invested is Rs. 12,000.

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