`A's` age is seven times `B's` age. Four years hence, the age of A will be six times `B's` age. Find the present ages (in years), of person A and B.

Answer

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

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

### Step 2: Set Up the First Equation
According to the problem, A's age is seven times B's age. This can be expressed as:
\[ A = 7B \]

### Step 3: Set Up the Second Equation
The problem also states that four years from now, A's age will be six times B's age. In four years, A's age will be \( A + 4 \) and B's age will be \( B + 4 \). This gives us the second equation:
\[ A + 4 = 6(B + 4) \]

### Step 4: Simplify the Second Equation
Expanding the second equation:
\[ A + 4 = 6B + 24 \]
Now, rearranging it gives:
\[ A - 6B = 20 \quad \text{(Equation 2)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( A = 7B \) (Equation 1)
2. \( A - 6B = 20 \) (Equation 2)

We can substitute Equation 1 into Equation 2:
\[ 7B - 6B = 20 \]
This simplifies to:
\[ B = 20 \]

### Step 6: Find A's Age
Now that we have the value of \( B \), we can find \( A \) using Equation 1:
\[ A = 7B = 7 \times 20 = 140 \]

### Conclusion
The present ages are:
- A's age = 140 years
- B's age = 20 years