Five years ago, A was thrice as old as B and 10 years later, A shall be twice as old as B. What are the present ages of A and B?

Answer

To solve the problem of finding the present ages of A and B, we can follow these steps:

### Step 1: Define the Variables
Let the present age of A be \( X \) years and the present age of B be \( Y \) years.

### Step 2: Set Up the First Equation
According to the problem, five years ago, A was thrice as old as B. This can be expressed mathematically as:
\[ X - 5 = 3(Y - 5) \]
Expanding this equation gives:
\[ X - 5 = 3Y - 15 \]
Rearranging it, we get:
\[ X = 3Y - 10 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
Next, the problem states that ten years later, A will be twice as old as B. This can be expressed as:
\[ X + 10 = 2(Y + 10) \]
Expanding this equation gives:
\[ X + 10 = 2Y + 20 \]
Rearranging it, we get:
\[ X = 2Y + 10 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( X = 3Y - 10 \)
2. \( X = 2Y + 10 \)

Since both expressions are equal to \( X \), we can set them equal to each other:
\[ 3Y - 10 = 2Y + 10 \]
Subtracting \( 2Y \) from both sides gives:
\[ Y - 10 = 10 \]
Adding 10 to both sides results in:
\[ Y = 20 \]

### Step 5: Find the Age of A
Now that we have the value of \( Y \), we can substitute it back into Equation 1 to find \( X \):
\[ X = 3(20) - 10 \]
Calculating this gives:
\[ X = 60 - 10 = 50 \]

### Step 6: Conclusion
Thus, the present ages of A and B are:
- Present age of A: \( 50 \) years
- Present age of B: \( 20 \) years

### Summary
The present age of A is 50 years and the present age of B is 20 years.

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