The ratio between the present ages of A and B is 5 : 3 respectively. The ratio between A's age 4 years ago and B's age 4 years hence is 1 : 1. What is the ratio between A's age 4 years hence and B's age 4 years ago?

### Concept & Strategy When one person's past age equals another person's future age (a $1 : 1$ ratio), it means their present ages are separated by the sum of those two time shifts. We can model this linearly using their present ratio. $$ \frac{A - t_1}{B + t_2} = \frac{1}{1} \implies A - t_1 = B + t_2 $$ ### Step-by-Step Solution * **Given:** Present ratio $A : B = 5 : 3$. Condition: (A's age $- 4$) : (B's age $+ 4$) = $1 : 1$. * **Calculation:** Let A's present age be $5x$ and B's present age be $3x$. * A's age 4 years ago = $5x - 4$. * B's age 4 years hence = $3x + 4$. * Equate them according to the $1 : 1$ ratio: $5x - 4 = 3x + 4$ * Solve for $x$: $2x = 8 \implies x = 4$. * Calculate their exact present ages: $A = 5(4) = 20$ years. $B = 3(4) = 12$ years. * We need the ratio of A's age 4 years hence to B's age 4 years ago. * A's age in 4 years = $20 + 4 = 24$. * B's age 4 years ago = $12 - 4 = 8$. * Find the final ratio: $24 : 8 = 3 : 1$. ### Exam Strategy & Shortcut **Logical Gap Analysis:** The condition $5x - 4 = 3x + 4$ shows that the difference between their current ages ($2x$) is exactly equal to the total time shift ($8$ years). This immediately gives $x=4$. Since you want A $+ 4$ and B $- 4$, you are essentially pushing A one $x$-unit forward and pulling B one $x$-unit back. The new ratio is $(5+1) : (3-1) = 6 : 2 = 3 : 1$. ### Common Pitfall Misreading the final question and finding the ratio of their *present* ages or accidentally calculating B's future age against A's past age again. Always double-check the final prompt direction. ### Final Answer Therefore, the correct answer is **3 : 1**.