A person invests money in three different schemes for 6 years, 10 years and 12 years at 10%, 12% and 15% simple interest respectively. At the completion of each scheme, he gets the same interest. The ratio of his investments is :

Options

A. `6: 3:2`
B. `2 : 3:4`
C. `3 : 4 :6`
D. `3 : 4 : 2`

Answer

To solve the problem, we need to find the ratio of investments made in three different schemes with different time periods and interest rates, given that the interest earned from each scheme is the same.

### Step-by-Step Solution:

1. **Define Variables**:
Let the investments in the three schemes be \( x \), \( y \), and \( z \) respectively.

2. **Use the Simple Interest Formula**:
The formula for Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount (investment), \( R \) is the rate of interest, and \( T \) is the time in years.

3. **Calculate Simple Interest for Each Investment**:
- For the first scheme (6 years at 10%):
\[ \text{SI}_1 = \frac{x \times 10 \times 6}{100} = \frac{60x}{100} = 0.6x \]
- For the second scheme (10 years at 12%):
\[ \text{SI}_2 = \frac{y \times 12 \times 10}{100} = \frac{120y}{100} = 1.2y \]
- For the third scheme (12 years at 15%):
\[ \text{SI}_3 = \frac{z \times 15 \times 12}{100} = \frac{180z}{100} = 1.8z \]

4. **Set the Interests Equal**:
Since the interest earned from each scheme is the same, we can set the equations equal to each other:
\[ 0.6x = 1.2y = 1.8z \]

5. **Express Each Investment in Terms of a Common Variable**:
Let \( K \) be the common interest earned from each scheme. Then we can express:
\[ 0.6x = K \implies x = \frac{K}{0.6} = \frac{5K}{3} \]
\[ 1.2y = K \implies y = \frac{K}{1.2} = \frac{K}{\frac{6}{5}} = \frac{5K}{6} \]
\[ 1.8z = K \implies z = \frac{K}{1.8} = \frac{K}{\frac{9}{5}} = \frac{5K}{9} \]

6. **Find the Ratio of Investments**:
Now we can find the ratio \( x : y : z \):
\[ x : y : z = \frac{5K}{3} : \frac{5K}{6} : \frac{5K}{9} \]
To simplify, we can ignore \( 5K \) and focus on the coefficients:
\[ = \frac{1}{3} : \frac{1}{6} : \frac{1}{9} \]
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators (3, 6, 9), which is 18:
\[ = 18 \times \frac{1}{3} : 18 \times \frac{1}{6} : 18 \times \frac{1}{9} = 6 : 3 : 2 \]

7. **Final Ratio**:
Thus, the ratio of the investments \( x : y : z \) is:
\[ \boxed{6 : 3 : 2} \]