A is 16 years older than B. But half the age of B is equal to one third of A's age. Find their present age.

Options

A. (a) A = 40, B = 32
B. (b) A = 48, B = 32
C. (c) A = 46, B = 30
D. (d) A = 32, B = 16

Explanation

To solve the problem, we need to set up equations based on the information given about the ages of A and B.

1. **Define the variables:**
Let B's age be \( b \).
Then A's age can be expressed as \( a = b + 16 \) (since A is 16 years older than B).

2. **Set up the second equation:**
According to the problem, half the age of B is equal to one third of A's age. This can be written as:
\[ \frac{1}{2}b = \frac{1}{3}a \]

3. **Substitute A's age in the second equation:**
We already have \( a = b + 16 \). Substitute this into the second equation:
\[ \frac{1}{2}b = \frac{1}{3}(b + 16) \]

4. **Eliminate the fractions:**
To eliminate the fractions, we can multiply both sides of the equation by 6 (the least common multiple of 2 and 3):
\[ 6 \cdot \frac{1}{2}b = 6 \cdot \frac{1}{3}(b + 16) \]
This simplifies to:
\[ 3b = 2(b + 16) \]

5. **Distribute on the right side:**
Now distribute the 2 on the right side:
\[ 3b = 2b + 32 \]

6. **Isolate b:**
Subtract \( 2b \) from both sides:
\[ 3b - 2b = 32 \]
This simplifies to:
\[ b = 32 \]

7. **Find A's age:**
Now that we have B's age, we can find A's age using the equation \( a = b + 16 \):
\[ a = 32 + 16 = 48 \]

8. **Conclusion:**
Therefore, the present ages are:
- A's age = 48 years
- B's age = 32 years

### Summary of the Solution:
- A is 48 years old.
- B is 32 years old.