A walks at a uniform rate of $4$ km an hour; and $4$ hours after his start, B bicycles after him at the uniform rate of $10$ km an hour. How far from the starting point will B catch A?

### Concept & Logic This is a standard "catch-up" problem based on relative speed. When two entities travel in the same direction from the same starting line, the relative speed used to close the gap between them is the difference between their individual speeds. ### Step-by-Step Solution * **Given:** A's speed = $4$ km/hr. B's speed = $10$ km/hr. A has a $4$-hour head start. * **Calculation:** Distance covered by A in his $4$-hour head start = $4 \text{ km/hr} \times 4 \text{ hrs} = 16 \text{ km}$. * Relative speed of B with respect to A = $10 - 4 = 6$ km/hr. * Time taken by B to close this $16$ km gap = $\frac{16}{6} = \frac{8}{3}$ hours. * The distance from the starting point when B catches A is calculated using B's travel data: $\text{Distance} = \text{B's speed} \times \text{Time} = 10 \times \left(\frac{8}{3}\right) = \frac{80}{3}$ km. * Converting the fraction to a decimal gives $\frac{80}{3} \approx 26.66$ km, which rounds to $26.7$ km. ### Exam Strategy & Shortcut You can directly equate the distances traveled by setting up a simple equation: $4 \times (t + 4) = 10 \times t \implies 4t + 16 = 10t \implies 6t = 16 \implies t = \frac{8}{3}$ hours. Total distance = $10 \times \frac{8}{3} \approx 26.7$ km. This linear equation approach is foolproof and extremely fast. ### Common Pitfall Encountering rounding errors early in the calculation process. Always keep fractions like $\frac{8}{3}$ intact until the final step before converting to the requested decimal format. ### Final Answer Therefore, the correct answer is **$26.7$ km**.