Name: Sarah

Who is asking: Student
Level: Secondary

Question:
Joe Spout left a campsite on a trip down the river in a canoe, traveling at 6 km/h. Four hours later, Joe's father set out after him in a motorboat. The motorboat traveled 30 km/h. How long after Joe`s father started did he overtake the canoe? How far had Joe traveled down the river when his father overtook him?

The first thing I would do to solve this problem is draw a diagram illustrating the information you know.

Joe = 6 km/h , Dad = 30 km/h

   |-----------------------------------------X-------------------|

Joe start                               dad starts           dad catches up
 time = 0                            Joe's time = 4 h         time = ?

Now ask yourself the question "how far did Joe travel before his dad started?"

Using rate x time = distance we can calculate that Joe travelled 6 km/h x 4 h = 24 km before his dad left in the motor boat. This is the distance that his father must "make up" in order to overtake Joe.

Now ask yourself the question, "how much faster is the motorboat going than Joe?"

Joe goes 6 km/h and his dad goes 30 km/h so dad travels 24 km/h faster than Joe. That means that every hour he travels he will "make up" 24 km. From this we see that Joe's dad will overtake him in one hour. At that time, that is, after 5 hours total, Joe will have travelled a total of 6 km/h x 5 h = 30 km.

Here's another way to view it:

Joe and his dad will have both traveled the same distance at the time Joe's canoe is overtaken by his dad's motorboat.

Use our distance formula: rate x time = distance together with our known info:

Joe goes 6 km/h
Dad goes 30 km/h
Dad's time travelling is unknown - call it "t"
Joe travels for 4 hours plus the time his dad travels - 4 + t

Joe travels a distance = 6 km/h x (4 + t)
Dad travels a distance = 30 km/h x t

Now equate the two distances and solve for t:

(for simplicity's sake I will eliminate the units km/h and h, but it is important to understand that they are in fact there in the equation)

which is the same answer as we came up with previously for the time it takes his father to overtake him.

Hope this helps,
Leeanne