Sonali invests money in three different schemes for 6 years, 10 years and 12 years at 10% p.a., 12% p.a. and 15% p.a. at simple interest respectively. At the completion of each scheme, she gets the same interest. What is the ratio of her investments ?

Options

A. `6:3:2`
B. `3:2:1`
C. `3:4:5`
D. `2:3:6`

Answer

To solve the problem step by step, we will denote Sonali's investments in the three schemes as \( x \), \( y \), and \( z \) respectively.

### Step 1: Write the formula for Simple Interest
The formula for Simple Interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount (investment), \( R \) is the rate of interest per annum, and \( T \) is the time in years.

### Step 2: Set up the equations for each investment
1. For the first scheme (6 years at 10%):
\[ SI_1 = \frac{x \times 10 \times 6}{100} = \frac{60x}{100} = 0.6x \]

2. For the second scheme (10 years at 12%):
\[ SI_2 = \frac{y \times 12 \times 10}{100} = \frac{120y}{100} = 1.2y \]

3. For the third scheme (12 years at 15%):
\[ SI_3 = \frac{z \times 15 \times 12}{100} = \frac{180z}{100} = 1.8z \]

### Step 3: Set the interests equal to each other
Since the interest from all three schemes is the same, we can set the equations equal:
\[ 0.6x = 1.2y = 1.8z \]

### Step 4: Express \( y \) and \( z \) in terms of \( x \)
From \( 0.6x = 1.2y \):
\[ y = \frac{0.6x}{1.2} = \frac{x}{2} \]

From \( 0.6x = 1.8z \):
\[ z = \frac{0.6x}{1.8} = \frac{x}{3} \]

### Step 5: Write the ratio of investments
Now we have:
- \( x \) (for the first scheme)
- \( y = \frac{x}{2} \) (for the second scheme)
- \( z = \frac{x}{3} \) (for the third scheme)

The ratio of investments \( x : y : z \) can be expressed as:
\[ x : \frac{x}{2} : \frac{x}{3} \]

### Step 6: Simplify the ratio
To simplify this ratio, we can multiply through by the least common multiple (LCM) of the denominators (which is 6):
\[ x : \frac{x}{2} : \frac{x}{3} = 6x : 3x : 2x \]

Thus, the ratio simplifies to:
\[ 6 : 3 : 2 \]

### Final Result
The ratio of Sonali's investments in the three schemes is:
\[ \boxed{6 : 3 : 2} \]