Six years hence, a man's age will be three times the age of his son and three years ago he was nine times as old as his son. The present age of the man is __________

Options

A. 28 years
B. 30 years
C. 32 years
D. 34 years

Answer

To solve the problem, let's denote the present age of the man as \( M \) and the present age of his son as \( S \).

### Step 1: Set up the equations based on the problem statement.

1. **From the first condition**: Six years hence, the man's age will be three times the age of his son.
\[ M + 6 = 3(S + 6) \]
Simplifying this equation:
\[ M + 6 = 3S + 18 \]
Rearranging gives us:
\[ M - 3S = 12 \quad \text{(Equation 1)} \]

2. **From the second condition**: Three years ago, the man was nine times as old as his son.
\[ M - 3 = 9(S - 3) \]
Simplifying this equation:
\[ M - 3 = 9S - 27 \]
Rearranging gives us:
\[ M - 9S = -24 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

Now we have two equations:
1. \( M - 3S = 12 \)
2. \( M - 9S = -24 \)

We can solve these equations simultaneously. Let's subtract Equation 2 from Equation 1:

\[ (M - 3S) - (M - 9S) = 12 - (-24) \]
This simplifies to:
\[ 6S = 36 \]
Dividing both sides by 6 gives:
\[ S = 6 \]

### Step 3: Substitute \( S \) back to find \( M \).

Now that we have \( S \), we can substitute it back into Equation 1 to find \( M \):
\[ M - 3(6) = 12 \]
This simplifies to:
\[ M - 18 = 12 \]
Adding 18 to both sides gives:
\[ M = 30 \]

### Conclusion

The present age of the man is \( \boxed{30} \).