Finding the Present Ages of Father and Son

Video Explanation

Question

The present age of a father is three years more than three times the age of his son. Three years hence, the father’s age will be 10 years more than twice the age of his son. Find their present ages.

Solution

Step 1: Let the Variables

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

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

Step 2: Form the Equations

Present age condition:

\[ x = 3y + 3 \]

\[ x – 3y = 3 \quad (1) \]

Three years hence:

\[ x + 3 = 2(y + 3) + 10 \]

\[ x + 3 = 2y + 6 + 10 \]

\[ x + 3 = 2y + 16 \]

\[ x – 2y = 13 \quad (2) \]

Step 3: Solve by Elimination Method

Subtract equation (1) from equation (2):

\[ (x – 2y) – (x – 3y) = 13 – 3 \]

\[ x – 2y – x + 3y = 10 \]

\[ y = 10 \]

Step 4: Find the Value of x

Substitute \(y = 10\) in equation (1):

\[ x – 3(10) = 3 \]

\[ x – 30 = 3 \]

\[ x = 33 \]

Conclusion

Present age of father:

\[ \boxed{33 \text{ years}} \]

Present age of son:

\[ \boxed{10 \text{ years}} \]