The sum of present ages of A and B is 63 years. The ratio of their ages 3 years later will be 11 : 12. What is the present age (in years) of A?

Options

A. 30
B. 33
C. 36
D. 27

Answer

To solve the problem step by step, we will follow the information given in the question.

### Step 1: Set up the equations based on the information given.
Let the present age of A be \( a \) years and the present age of B be \( b \) years. According to the problem, we have:
\[ a + b = 63 \quad \text{(1)} \]

### Step 2: Express the ages after 3 years.
Three years later, the ages of A and B will be:
- Age of A: \( a + 3 \)
- Age of B: \( b + 3 \)

### Step 3: Set up the ratio of their ages after 3 years.
According to the problem, the ratio of their ages after 3 years will be \( 11 : 12 \). Therefore, we can write:
\[ \frac{a + 3}{b + 3} = \frac{11}{12} \quad \text{(2)} \]

### Step 4: Cross-multiply to eliminate the fraction.
From equation (2), we can cross-multiply:
\[ 12(a + 3) = 11(b + 3) \]
Expanding both sides gives:
\[ 12a + 36 = 11b + 33 \]

### Step 5: Rearrange the equation to express in terms of \( a \) and \( b \).
Rearranging the equation gives:
\[ 12a - 11b = -3 \quad \text{(3)} \]

### Step 6: Solve the system of equations.
Now we have a system of equations:
1. \( a + b = 63 \) (equation 1)
2. \( 12a - 11b = -3 \) (equation 3)

From equation (1), we can express \( b \) in terms of \( a \):
\[ b = 63 - a \]

### Step 7: Substitute \( b \) in equation (3).
Substituting \( b \) in equation (3):
\[ 12a - 11(63 - a) = -3 \]
Expanding gives:
\[ 12a - 693 + 11a = -3 \]
Combining like terms:
\[ 23a - 693 = -3 \]
Adding 693 to both sides:
\[ 23a = 690 \]
Dividing by 23:
\[ a = 30 \]

### Step 8: Find the present age of A.
Thus, the present age of A is:
\[ \boxed{30} \text{ years} \]