Question

The ages of Ajay and Vijay differ by 22.5 years. If \(6\frac{1}{2}\) years ago, Vijay was \(3\frac{1}{2}\) times as old as Ajay, find their present ages.

• A. \(14\frac{1}{2}\) yrs, 37 yrs
• B. 15 yrs, \(37\frac{1}{2}\) yrs
• C. \(15\frac{1}{2}\) yrs, 38 yrs
• D. 16 yrs, \(38\frac{1}{2}\) yrs

Hint:

When past age ratios and present differences are given, always use variable expressions and set up an equation based on the older statement to solve efficiently.

Answer 1

The Correct Option is C

Let Ajay's present age be \( x \) years.
Then Vijay's present age = \( x + 22.5 \) years. 6.5 years ago:
Ajay's age = \( x - 6.5 \)
Vijay's age = \( x + 22.5 - 6.5 = x + 16 \) Given: \( x + 16 = 3.5(x - 6.5) \) Step 1: Expand the equation \[ x + 16 = 3.5x - 22.75 \] Step 2: Simplify \[ x - 3.5x = -22.75 - 16 \Rightarrow -2.5x = -38.75 \Rightarrow x = \frac{38.75}{2.5} = 15.5 \] Step 3: Vijay's present age \[ x + 22.5 = 15.5 + 22.5 = 38 \] Final Answer:
Ajay = \(15\frac{1}{2}\) years, Vijay = 38 years.