4 years ago, the ratio of 1/2 of A's age at that time and four times of B's age at the time was 5 : 12. Eight years hence, 1/2 of A's age at that time will be less than B's age at that time by 2 years. What is B's present age?

### Concept & Strategy This requires setting up simultaneous equations across different timelines (past, present, future). The strategy is to clearly define the variables at a single point in time, build the fractional equations, and solve through substitution. ### Step-by-Step Solution * **Given:** Let the ages of A and B **4 years ago** be $a$ and $b$ respectively. * **Calculation (Part 1: The Ratio 4 years ago):** The ratio of $1/2$ of A's age to $4$ times B's age is $5:12$. $$\frac{a/2}{4b} = \frac{5}{12}$$ $$\frac{a}{8b} = \frac{5}{12}$$ Cross-multiply to find a relationship: $$12a = 40b \implies 3a = 10b \implies a = \frac{10b}{3}$$ * **Calculation (Part 2: Eight years hence):** Since $a$ and $b$ are ages 4 years ago, their ages **8 years hence** (which is 12 years after the "4 years ago" timeline) will be: A's future age = $a + 12$ B's future age = $b + 12$ The problem states $1/2$ of A's future age is less than B's future age by 2 years: $$\frac{a + 12}{2} = (b + 12) - 2$$ $$\frac{a + 12}{2} = b + 10$$ $$a + 12 = 2b + 20$$ $$a - 2b = 8$$ * **Calculation (Part 3: Solving for B):** Substitute $a = \frac{10b}{3}$ into the new equation: $$\frac{10b}{3} - 2b = 8$$ $$\frac{10b - 6b}{3} = 8$$ $$\frac{4b}{3} = 8 \implies 4b = 24 \implies b = 6$$ So, B's age 4 years ago was 6. B's present age = $6 + 4 = 10$ years. ### Exam Strategy & Shortcut Working with fractions can be slow. Instead, use the options to work backwards. We are looking for B's present age. Test Option (a) 10: If B is currently 10, then 4 years ago, B was 6. From the first ratio: $\frac{a}{8(6)} = \frac{5}{12} \implies \frac{a}{48} = \frac{5}{12} \implies a = 20$. So 4 years ago A was 20. A's present age is 24. Let's check the second condition (8 years hence): A will be 32, B will be 18. Is half of A's age (16) less than B's age (18) by 2? Yes! $18 - 16 = 2$. Option (a) is verified instantly without solving heavy simultaneous equations. ### Common Pitfall A major trap is mixing up the timelines. When dealing with "4 years ago" and "8 years hence", students often add 8 to the present age correctly, but fail to realize that relative to the "4 years ago" variables ($a$ and $b$), the time jump is actually $4 + 8 = 12$ years. ### Final Answer **Therefore, the correct answer is 10 years.**