#
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.