 # Problem of the Week Problem B and Solution High Noon, Two Trains, and a Bee

## Problem

The Kitchener and London train stations are $$100$$ km apart on a straight section of railroad. A train leaves Kitchener Station at noon, and travels toward London Station. A different train leaves London Station at noon, and travels toward Kitchener Station on a parallel track. 1. The train from Kitchener is travelling at $$60$$ km per hour. What is its speed in km per minutes?

2. The train from London is travelling at $$90$$ km per hour. What is its speed in km per minutes?

3. Find the time that the trains begin to pass each other. To do so, you may find completing the table below helpful.

Time (in minutes after noon) Distance travelled by the train leaving Kitchener Distance travelled by the train leaving London Total distance travelled by the two trains
10
20
30
40
50
60

Extension: Bert the magical bee flies at $$120$$ km per hour back and forth between the two trains as they travel towards one another. What is the total distance he has travelled just as the trains begin to pass one another?
(Note: Bert will not lose any time as he changes direction.)

## Solution

1. The train from Kitchener has a speed of $$60 \text{ km/hour }= \dfrac{60 \text{ km}}{60 \text{ min}} = 1 \text{ km/min}$$.

2. The train from London has a speed of $$\ \dfrac{90 \mbox{ km}}{60 \mbox{ min}} = 1.5\,$$km/min.

3. The train from Kitchener has a speed of $$1$$ km/min. Thus in $$10$$ minutes, it will travel $$10$$ km.
The train from London has a speed of $$1.5$$ km/min. Thus in $$10$$ minutes, it will travel $$15$$ km.
We can now fill in the table. This is done on the previous page.
The two trains will begin to pass each other when the total distance travelled is equal to the distance between the two stations. The table values reveal that this occurs $$40$$ minutes after noon.
Therefore, the trains begin to pass each other at 12:40 pm.

Time (in minutes after noon) Distance travelled by the train leaving Kitchener Distance travelled by the train leaving London Total distance travelled by the two trains
10 10 15 25
20 20 30 50
30 30 45 75
40 40 60 100
50 50 75 125
60 60 90 150

Extension:
Magical Bert flies at $$120$$ km/hr, or $$\dfrac{120 \text{ km}}{60 \text{ min}} = 2 \text{ km/min}$$, or $$20$$ km in $$10$$ minutes.
Since the trains begin to pass one another after $$40$$ minutes, Bert is flying back and forth between the trains for $$40$$ minutes. Thus after $$40$$ minutes, he will have flown $$20 \text{ km} \times 4 = 80 \text{ km}$$ in total.