Two years ago, the ratio of A's age at that time of B's age at that time was 5:9. A's age three year ago was 13 years less than B's age six years ago. What is B's present age?

Options

A. 38 years
B. 30 years
C. 34 years
D. 32 years

Answer

To solve the problem step-by-step, we will define the ages of A and B and set up equations based on the information provided.

### Step 1: Define the variables
Let:
- A's present age = A years
- B's present age = B years

### Step 2: Set up the first equation based on the ratio of ages two years ago
Two years ago, A's age was (A - 2) and B's age was (B - 2). According to the problem, the ratio of A's age to B's age at that time was 5:9. Thus, we can write the equation:
\[ \frac{A - 2}{B - 2} = \frac{5}{9} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 9(A - 2) = 5(B - 2) \]
Expanding both sides:
\[ 9A - 18 = 5B - 10 \]
Rearranging this gives us:
\[ 9A = 5B + 8 \quad \text{(Equation 1)} \]

### Step 4: Set up the second equation based on the age difference
The problem states that A's age three years ago was 13 years less than B's age six years ago. Thus, we can express this as:
\[ A - 3 = (B - 6) - 13 \]
Simplifying this gives:
\[ A - 3 = B - 19 \]
Rearranging gives us:
\[ A = B - 16 \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
Now we will substitute \(A\) from Equation 2 into Equation 1:
\[ 9(B - 16) = 5B + 8 \]
Expanding this gives:
\[ 9B - 144 = 5B + 8 \]
Rearranging terms gives:
\[ 9B - 5B = 144 + 8 \]
This simplifies to:
\[ 4B = 152 \]

### Step 6: Solve for B
Dividing both sides by 4 gives:
\[ B = \frac{152}{4} = 38 \]

### Conclusion
Thus, B's present age is **38 years**.

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