The average age of A and B, 2 years ago was 26. If the age of A, 5 years hence is 40 yrs, and B is 5 years younger to C, then find the difference between the age of A and C?

Options

A. A. 11 yrs
B. B. 9 yrs
C. C. 7 yrs
D. D. 13 yrs

Answer

To solve the problem step by step, we will denote the ages of A, B, and C as follows:

- Let the current age of A be \( x \).
- Let the current age of B be \( y \).
- Let the current age of C be \( z \).

### Step 1: Establish the average age of A and B two years ago
According to the problem, the average age of A and B two years ago was 26. This can be expressed mathematically as:

\[ \frac{(x - 2) + (y - 2)}{2} = 26 \]

### Step 2: Simplify the equation
Multiplying both sides by 2 gives:

\[ (x - 2) + (y - 2) = 52 \]

This simplifies to:

\[ x + y - 4 = 52 \]

Adding 4 to both sides results in:

\[ x + y = 56 \quad \text{(Equation 1)} \]

### Step 3: Determine the current age of A
The problem states that the age of A five years hence will be 40 years. This can be expressed as:

\[ x + 5 = 40 \]

### Step 4: Solve for x
Subtracting 5 from both sides gives:

\[ x = 40 - 5 = 35 \]

### Step 5: Substitute x back into Equation 1 to find y
Now that we have the age of A, we can substitute \( x = 35 \) into Equation 1:

\[ 35 + y = 56 \]

### Step 6: Solve for y
Subtracting 35 from both sides gives:

\[ y = 56 - 35 = 21 \]

### Step 7: Determine the age of C
The problem states that B is 5 years younger than C, which can be expressed as:

\[ y = z - 5 \]

Substituting \( y = 21 \) into this equation gives:

\[ 21 = z - 5 \]

### Step 8: Solve for z
Adding 5 to both sides results in:

\[ z = 21 + 5 = 26 \]

### Step 9: Find the difference between the ages of A and C
Now we need to find the difference between the ages of A and C:

\[ \text{Difference} = z - x = 26 - 35 = -9 \]

Since we are looking for the absolute difference, we take the positive value:

\[ \text{Difference} = |26 - 35| = 9 \]

### Final Answer
The difference between the age of A and C is \( 9 \) years.

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