# Problem of the Week Problem D and Solution Head Start

## Problem

Gabi and Silvio are training for a cycling race. They live on the same street, but Silvio’s house is $$2$$ km east of Gabi’s. On Sunday morning at $$7$$ a.m. they each start biking east from their house. If Gabi bikes at a constant speed of $$24$$ km/h and Silvio bikes at a constant speed of $$18$$ km/h, at what time will Gabi catch up to Silvio?

## Solution

For the first two solutions we will use the formula: $$\text{time}=\frac{\text{distance}}{\text{speed}}$$.

For the third solution we will use the formula: $$\text{distance}=\text{speed} \times \text{time}$$.

Solution 1

Since Gabi bikes at $$24$$ km/h and Silvio bikes at $$18$$ km/h, then Gabi gains $$6$$ km/h on Silvio.

Since Silvio starts $$2$$ km east of of Gabi, then it takes Gabi $$\frac{2}{6}=\frac{1}{3}$$ of an hour or $$\frac{1}{3} \times 60 = 20$$ minutes to catch up to Silvio. Since they started biking at $$7$$ a.m., Gabi will catch up to Silvio at $$7$$:$$20$$ a.m.

Solution 2

Silvio bikes at $$18$$ km/h or $$\frac{18}{60} = \frac{3}{10}$$ km/min. Gabi bikes at $$24$$ km/h or $$\frac{24}{60} = \frac{2}{5}$$ km/min. Therefore Gabi gains $$\frac{2}{5} - \frac{3}{10} = \frac{1}{10}$$ km/min on Silvio.

Since Silvio started $$2$$ km east of Gabi, then it takes Gabi $$2 \div \frac{1}{10} = 20$$ minutes to catch Silvio. Since they started biking at $$7$$ a.m., Gabi will catch up to Silvio at $$7$$:$$20$$ a.m.

Solution 3

Suppose it takes $$t$$ hours for Gabi to catch up to Silvio. Then Silvio has biked $$18 \text{ km/h} \times t\text{ h} = 18t \text{ km}$$, and Gabi has biked $$24 \text{ km/h} \times t\text{ h} = 24t \text{ km}$$.

Since Silvio starts $$2$$ km east of Gabi, then when they meet, Gabi will have travelled $$2$$ km further than Silvio. That is, \begin{aligned} 24t& = 18t + 2\\ 6t &= 2\\ t &= \tfrac{1}{3} \end{aligned} Therefore, it takes Gabi $$\frac{1}{3}$$ of an hour, or $$20$$ minutes to catch up to Silvio. Since they started biking at $$7$$ a.m., Gabi will catch up to Silvio at $$7$$:$$20$$ a.m.