Directions: These questions are based on the following information: $P$ and $Q$ are $120$ km apart. $A$ starts from $P$ towards $Q$ at 6 a.m. $B$ starts from $Q$ towards $P$ at 11 a.m. on the same day. $A$ is $50$ % faster than $B$. They cross each other at 8 p.m. $A$'s speed (in km/hr) is

### Concept & Speed Ratio Calculation When given the ratio of speeds and individual travel times to a common meeting point, you can express the total distance as the sum of algebraic distance terms to cleanly solve for the exact base speeds. ### Step-by-Step Solution * **Given:** Total distance = $120$ km. Meeting point time = 8 p.m. $A$ starts at 6 a.m. ($14$ hours travel). $B$ starts at 11 a.m. ($9$ hours travel). $A$ is $50\%$ faster than $B$. * **Calculation:** Let $B$'s speed = $2x$. Then $A$'s speed = $3x$. * Distance traveled by $A$ = $14 \times 3x = 42x$. * Distance traveled by $B$ = $9 \times 2x = 18x$. * Total distance covered together = $42x + 18x = 60x$. * We know the total gap is $120$ km, so $60x = 120 \implies x = 2$. * The question asks for $A$'s speed, which is represented by $3x$. * $A$'s speed = $3(2) = 6$ km/hr. ### Exam Strategy & Shortcut The key to speed here is setting up a linear equation using a single variable for the baseline speed efficiently. $14 \text{ hrs} \times (1.5v) + 9 \text{ hrs} \times (v) = 120 \implies 21v + 9v = 120 \implies 30v = 120 \implies v = 4$. This is $B$'s speed. $A$'s speed is $1.5 \times 4 = 6$ km/hr. ### Common Pitfall A common arithmetic error is confusing the base of the percentage. "A is $50\%$ faster than B" means $A = B + 0.5B = 1.5B$. Students sometimes mistakenly write the relationship backwards as $B = 0.5A$. ### Final Answer Therefore, the correct answer is **$6$**.