Five years ago, A's age was four times the age of B. Five years hence , A's age will be twice the age of B. Find their present ages.

Answer

To solve the problem of finding the present ages of A and B, we can set up a system of equations based on the information given. Let's go through the solution step by step.

### Step 1: Define the Variables
Let:
- \( x \) = present age of A
- \( y \) = present age of B

### Step 2: Set Up the First Equation
According to the problem, five years ago, A's age was four times the age of B. This can be expressed mathematically as:
\[ x - 5 = 4(y - 5) \]
Expanding this equation:
\[ x - 5 = 4y - 20 \]
Rearranging gives us:
\[ x = 4y - 15 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that five years hence, A's age will be twice the age of B. This can be expressed as:
\[ x + 5 = 2(y + 5) \]
Expanding this equation:
\[ x + 5 = 2y + 10 \]
Rearranging gives us:
\[ x = 2y + 5 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
Now we can substitute Equation 2 into Equation 1:
\[ 2y + 5 = 4y - 15 \]
Rearranging this equation:
\[ 5 + 15 = 4y - 2y \]
This simplifies to:
\[ 20 = 2y \]
Dividing both sides by 2 gives:
\[ y = 10 \]

### Step 5: Find A's Age
Now that we have \( y \), we can find \( x \) using Equation 2:
\[ x = 2y + 5 \]
Substituting \( y = 10 \):
\[ x = 2(10) + 5 = 20 + 5 = 25 \]

### Conclusion
Thus, the present ages are:
- A's age \( x = 25 \)
- B's age \( y = 10 \)

### Final Answer
- Present age of A: 25 years
- Present age of B: 10 years