Modelling Linear Systems with Distance, Speed and Time

Question

Kareem took 5 hours to drive 470km from Sudbury to Brantford. For part of the trip he drove at 100km/h. For the rest of the trip he drove at 90km/h. How many kilometres were driven at each speed?

Answer 1

The basis of this solution is Distance = Rate x Time. We would usually use "distance" as our variable (unknown), because that is what we are trying to find, but for this problem, "time" will be easier to use. Let's use "t" as our variable, representing the amount of time Kareem drove at 100 km/h. Since the total time of the trip was 5 hours, the remaining time (when he drove at 90 km/h) must be  5-t. Be sure you understand this before you go on.

 

Now, if he drove t hours at 100 km/h, the distance he drove was the product of those two  (t times 100, or   100t)

If he drove   5-t  hours at 90 km/h, the distance he drove was  (5-t) times 90, or  90(5-t).   Be sure you are clear on this before moving on.

 

So, the two parts of the trip were  100t, then  90(5-t), so the total distance he drove was   100t + 90(5-t).

 

Use the distributive property to remove the parentheses:  100t + 450 - 90t.

Combine like terms to get  10t + 450. This is the total distance he drove.

 

From the problem, we know that the total distance is  470 km, so we have the equation   10t + 450 = 470. Solve this equation (which I think you can do without my help) to get   t =2 hours. The remainder of the time is 3 hours.

 

Now, we just need to find the distances at these two speeds, which are found by speed x time:

 

100 km/h x 2 h = 200 km

90 km/h x 3 h = 270 km

This answers the question. You should always check a solution in the words of the original problem. Do these two distances correctly add to the 470 km of the trip? Yes, they do, so we know we have it right.