Present ages from future sum – “X” and her grandfather differ by 50 years. Six years from now, their ages will sum to 152. What are their present ages?

Introduction / Context:Two linear relations are provided: a present-time difference and a future-time sum. Solving them together cleanly yields the present ages without trial-and-error.

Given Data / Assumptions:

• Let x = X’s present age, g = grandfather’s present age.
• g − x = 50.
• (x + 6) + (g + 6) = 152 ⇒ x + g = 140.

Concept / Approach:Solve the simple system: x + g = 140 and g − x = 50. Adding gives 2g = 190 ⇒ g = 95, then x = 45.

Step-by-Step Solution:x + g = 140.g − x = 50.Add: 2g = 190 ⇒ g = 95.Then x = 140 − 95 = 45.

Verification / Alternative check:Six years later: 51 + 101 = 152, as required; difference now: 95 − 45 = 50.

Why Other Options Are Wrong:The other pairs do not simultaneously satisfy both equations.

Common Pitfalls:Subtracting the equations in the wrong order or misreading “after 6 years” as “in 6 years total.”

Final Answer:45, 95