Vikram invests some money in three diffrerent schemes for 4 years. 8 years and 12 years at 10%.15% and 20% Simple Interest respectively. At the completion of each scheme. he gets the same interest. The ratio of his investments is

Options

A. `6:2:1`
B. `5:2:1`
C. `5:2:3`
D. `5:2:7`

Explanation

To solve the problem, we need to find the ratio of Vikram's investments in three different schemes based on the simple interest earned from each scheme. Let's denote his investments in the three schemes as \( x_1 \), \( x_2 \), and \( x_3 \).

### Step-by-Step Solution:

1. **Identify the parameters for each investment:**
- For the first investment \( x_1 \):
- Time = 4 years
- Rate = 10%
- Interest = \( I_1 = \frac{x_1 \times 10 \times 4}{100} = \frac{40x_1}{100} = 0.4x_1 \)

- For the second investment \( x_2 \):
- Time = 8 years
- Rate = 15%
- Interest = \( I_2 = \frac{x_2 \times 15 \times 8}{100} = \frac{120x_2}{100} = 1.2x_2 \)

- For the third investment \( x_3 \):
- Time = 12 years
- Rate = 20%
- Interest = \( I_3 = \frac{x_3 \times 20 \times 12}{100} = \frac{240x_3}{100} = 2.4x_3 \)

2. **Set the interest amounts equal:**
Since Vikram receives the same interest from all three investments, we have:
\[ 0.4x_1 = 1.2x_2 = 2.4x_3 \]

3. **Express \( x_1 \), \( x_2 \), and \( x_3 \) in terms of a common variable:**
- From \( 0.4x_1 = 1.2x_2 \):
\[ x_1 = \frac{1.2}{0.4} x_2 = 3x_2 \]
- From \( 1.2x_2 = 2.4x_3 \):
\[ x_2 = \frac{2.4}{1.2} x_3 = 2x_3 \]

4. **Substituting back to find relationships:**
- Substitute \( x_2 = 2x_3 \) into \( x_1 = 3x_2 \):
\[ x_1 = 3(2x_3) = 6x_3 \]

5. **Now we have the relationships:**
- \( x_1 = 6x_3 \)
- \( x_2 = 2x_3 \)

6. **Express the ratio \( x_1 : x_2 : x_3 \):**
- Let \( x_3 = 1 \) (for simplicity),
- Then \( x_1 = 6 \) and \( x_2 = 2 \).
- Therefore, the ratio \( x_1 : x_2 : x_3 = 6 : 2 : 1 \).

### Final Answer:
The ratio of Vikram's investments is \( 6 : 2 : 1 \).