Finding the Present Ages of A and B

Video Explanation

Question

Ten years later, A will be twice as old as B. Five years ago, A was three times as old as B. Find their present ages.

Solution

Step 1: Let the Variables

Let present age of A = \(x\) years

Let present age of B = \(y\) years

Step 2: Form the Equations

Ten years later:

\[ x + 10 = 2(y + 10) \]

\[ x + 10 = 2y + 20 \]

\[ x – 2y = 10 \quad (1) \]

Five years ago:

\[ x – 5 = 3(y – 5) \]

\[ x – 5 = 3y – 15 \]

\[ x – 3y = -10 \quad (2) \]

Step 3: Solve by Elimination Method

Subtract equation (2) from equation (1):

\[ (x – 2y) – (x – 3y) = 10 – (-10) \]

\[ x – 2y – x + 3y = 20 \]

\[ y = 20 \]

Step 4: Find the Value of x

Substitute \(y = 20\) in equation (1):

\[ x – 2(20) = 10 \]

\[ x – 40 = 10 \]

\[ x = 50 \]

Conclusion

Present age of A:

\[ \boxed{50 \text{ years}} \]

Present age of B:

\[ \boxed{20 \text{ years}} \]