Sum of present ages of A and B is 81. Age of A 2 year hence is equal to age of C, 1 year ago. Ageof A, 4 year hence is equal to age of B, 1 year ago and ratio of present age of A and D is `2:3`. Find the difference of age of B and D.

Options

A. 13 years
B. 15 years
C. 16 years
D. 14 years

Explanation

To solve the problem step by step, we will define the present ages of A, B, C, and D using algebraic expressions based on the information provided in the question.

### Step 1: Define the variables
Let:
- Present age of A = A
- Present age of B = B
- Present age of C = C
- Present age of D = D

### Step 2: Set up the equations based on the information given
1. The sum of the present ages of A and B is 81:
\[ A + B = 81 \quad \text{(Equation 1)} \]

2. Age of A 2 years hence (A + 2) is equal to the age of C 1 year ago (C - 1):
\[ A + 2 = C - 1 \quad \Rightarrow \quad C = A + 3 \quad \text{(Equation 2)} \]

3. Age of A 4 years hence (A + 4) is equal to the age of B 1 year ago (B - 1):
\[ A + 4 = B - 1 \quad \Rightarrow \quad B = A + 5 \quad \text{(Equation 3)} \]

4. The ratio of the present ages of A and D is 2:3:
\[ \frac{A}{D} = \frac{2}{3} \quad \Rightarrow \quad D = \frac{3}{2} A \quad \text{(Equation 4)} \]

### Step 3: Substitute Equation 3 into Equation 1
From Equation 3, we have:
\[ B = A + 5 \]
Substituting this into Equation 1:
\[ A + (A + 5) = 81 \]
\[ 2A + 5 = 81 \]
\[ 2A = 81 - 5 \]
\[ 2A = 76 \]
\[ A = 38 \]

### Step 4: Find the age of B using Equation 3
Now, substituting A back into Equation 3:
\[ B = A + 5 = 38 + 5 = 43 \]

### Step 5: Find the age of C using Equation 2
Now, substituting A back into Equation 2:
\[ C = A + 3 = 38 + 3 = 41 \]

### Step 6: Find the age of D using Equation 4
Now, substituting A back into Equation 4:
\[ D = \frac{3}{2} A = \frac{3}{2} \times 38 = 57 \]

### Step 7: Find the difference between the ages of B and D
Now, we can find the difference between the ages of B and D:
\[ \text{Difference} = D - B = 57 - 43 = 14 \]

### Final Answer
The difference of age between B and D is **14 years**.

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