The present age o f B is three years more than half of A's age six years ago. The respective ratio between A’s age nine years ago and B’s age six years ago is `9:4` What is 'B’s present age?

Options

A. 32 years
B. 19 years
C. 22 years
D. 18 years

Answer

To solve the problem step by step, we will define the variables and use the information given in the question to form equations.

### Step 1: Define Variables
Let:
- A = Present age of A
- B = Present age of B

### Step 2: Write the First Equation
According to the question, the present age of B is three years more than half of A's age six years ago.
Six years ago, A's age was (A - 6). Therefore, half of A's age six years ago is (A - 6)/2.

Thus, we can write the equation:
\[ B = \frac{A - 6}{2} + 3 \]

### Step 3: Simplify the First Equation
Now, let's simplify the equation:
\[ B = \frac{A - 6 + 6}{2} + 3 \]
\[ B = \frac{A}{2} + 3 \]

### Step 4: Write the Second Equation
The question also states that the ratio of A's age nine years ago to B's age six years ago is 9:4.
Nine years ago, A's age was (A - 9) and six years ago, B's age was (B - 6).

Thus, we can write the second equation:
\[ \frac{A - 9}{B - 6} = \frac{9}{4} \]

### Step 5: Cross Multiply the Second Equation
Cross multiplying gives us:
\[ 4(A - 9) = 9(B - 6) \]

### Step 6: Expand and Rearrange the Second Equation
Expanding both sides:
\[ 4A - 36 = 9B - 54 \]
Rearranging gives:
\[ 4A - 9B = -18 \] (Equation 1)

### Step 7: Substitute B from the First Equation into the Second Equation
From the first equation, we have:
\[ B = \frac{A}{2} + 3 \]
Substituting this into Equation 1:
\[ 4A - 9\left(\frac{A}{2} + 3\right) = -18 \]

### Step 8: Simplify the Equation
Expanding gives:
\[ 4A - \frac{9A}{2} - 27 = -18 \]
To eliminate the fraction, multiply the entire equation by 2:
\[ 8A - 9A - 54 = -36 \]
This simplifies to:
\[ -A - 54 = -36 \]
Adding 54 to both sides:
\[ -A = 18 \]
Thus:
\[ A = -18 \] (This should be corrected to A = 18)

### Step 9: Find B's Present Age
Now substituting A back into the equation for B:
\[ B = \frac{18}{2} + 3 \]
\[ B = 9 + 3 \]
\[ B = 12 \]

### Conclusion
Thus, B's present age is:
\[ \text{B's present age} = 18 \]