15 years hence, A will be twice as old as B but five years ago A was 4 times as old as B. Find the difference of their present ages.

Options

A. 15 years
B. 45 years
C. 30 years
D. 25 years

Answer

To solve the problem step by step, let's denote the present ages of A and B as follows:

- Let A's present age be \( x \) years.
- Let B's present age be \( y \) years.

### Step 1: Set up the first equation based on the future condition
According to the problem, "15 years hence, A will be twice as old as B." This can be expressed mathematically as:

\[ x + 15 = 2(y + 15) \]

### Step 2: Simplify the first equation
Expanding the equation gives:

\[ x + 15 = 2y + 30 \]

Rearranging this, we get:

\[ x - 2y = 30 - 15 \]
\[ x - 2y = 15 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation based on the past condition
The problem also states, "5 years ago, A was 4 times as old as B." This can be expressed as:

\[ x - 5 = 4(y - 5) \]

### Step 4: Simplify the second equation
Expanding this equation gives:

\[ x - 5 = 4y - 20 \]

Rearranging this, we get:

\[ x - 4y = -20 + 5 \]
\[ x - 4y = -15 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( x - 2y = 15 \)
2. \( x - 4y = -15 \)

We can solve these equations simultaneously. Let's subtract Equation 2 from Equation 1:

\[ (x - 2y) - (x - 4y) = 15 - (-15) \]

This simplifies to:

\[ 2y = 30 \]

Dividing both sides by 2 gives:

\[ y = 15 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):

\[ x - 2(15) = 15 \]
\[ x - 30 = 15 \]
\[ x = 45 \]

### Step 7: Find the difference of their present ages
Now we have the present ages:
- A's age \( x = 45 \)
- B's age \( y = 15 \)

The difference in their present ages is:

\[ x - y = 45 - 15 = 30 \]

Thus, the difference of their present ages is **30 years**.