MAT Exam 2025 Paper

Concept:Form linear equations from the given age relationships and solve them to find the required ratio.Explanation:Let Allen's present age be $A$ years and his son's present age be $S$ years.Six years ago, Allen's age was $A - 6$ and his son's age was $S - 6$.Given: $A - 6 = 9(S - 6)$.Simplify: $A - 6 = 9S - 54$, so $A = 9S - 48$. ... (1)Four years ago, Allen's age was $A - 4$ and his son's age was $S - 4$.Given: $A - 4 = 6\frac{1}{3}(S - 4) = \frac{19}{3}(S - 4)$. ... (2)Substitute $A = 9S - 48$ into equation (2):$(9S - 48) - 4 = \frac{19}{3}(S - 4)$$9S - 52 = \frac{19}{3}(S - 4)$Multiply through by 3: $27S - 156 = 19S - 76$So, $27S - 19S = 156 - 76$, giving $8S = 80$.Thus, $S = 10$ years.Using equation (1): $A = 9(10) - 48 = 90 - 48 = 42$ years.After 6 years, Allen's age $= 42 + 6 = 48$ years.After 6 years, son's age $= 10 + 6 = 16$ years.Required ratio $= \frac{48}{16} = 3$ times.Answer:Allen's age would be 3 times his son's age after 6 years, which matches option C.