Three years ago, ratio of ages of Sohit and Mohit was `2:3`. Average of present ages of Sohit, Mohit and Rohit is 22 years. Rohit is 7 years younger than Mohit. What is the present age of Mohit?

Options

A. 20 years
B. 28 years
C. 22 years
D. 27 years

Answer

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

### Step 1: Define the variables based on the information given.
Let the present age of Sohit be \( S \), the present age of Mohit be \( M \), and the present age of Rohit be \( R \).

### Step 2: Set up the equations based on the ratios and relationships.
From the problem, we know:
- Three years ago, the ratio of ages of Sohit and Mohit was \( 2:3 \).
- Therefore, we can express their ages three years ago as:
- Sohit's age three years ago: \( S - 3 \)
- Mohit's age three years ago: \( M - 3 \)

This gives us the equation:
\[ \frac{S - 3}{M - 3} = \frac{2}{3} \]

### Step 3: Cross-multiply to eliminate the fraction.
Cross-multiplying gives us:
\[ 3(S - 3) = 2(M - 3) \]
Expanding this, we get:
\[ 3S - 9 = 2M - 6 \]
Rearranging gives us:
\[ 3S - 2M = 3 \quad \text{(Equation 1)} \]

### Step 4: Use the average age information.
The average of the present ages of Sohit, Mohit, and Rohit is 22 years. Thus:
\[ \frac{S + M + R}{3} = 22 \]
Multiplying through by 3 gives:
\[ S + M + R = 66 \quad \text{(Equation 2)} \]

### Step 5: Use the relationship between Rohit and Mohit.
We know that Rohit is 7 years younger than Mohit:
\[ R = M - 7 \quad \text{(Equation 3)} \]

### Step 6: Substitute Equation 3 into Equation 2.
Substituting \( R \) from Equation 3 into Equation 2 gives:
\[ S + M + (M - 7) = 66 \]
This simplifies to:
\[ S + 2M - 7 = 66 \]
Rearranging gives:
\[ S + 2M = 73 \quad \text{(Equation 4)} \]

### Step 7: Solve the system of equations.
Now we have two equations:
1. \( 3S - 2M = 3 \) (Equation 1)
2. \( S + 2M = 73 \) (Equation 4)

From Equation 4, we can express \( S \) in terms of \( M \):
\[ S = 73 - 2M \]

### Step 8: Substitute \( S \) back into Equation 1.
Substituting \( S \) into Equation 1:
\[ 3(73 - 2M) - 2M = 3 \]
Expanding gives:
\[ 219 - 6M - 2M = 3 \]
Combining like terms:
\[ 219 - 8M = 3 \]
Rearranging gives:
\[ 8M = 216 \]
Thus:
\[ M = \frac{216}{8} = 27 \]

### Step 9: Conclusion.
The present age of Mohit is \( 27 \) years.