Counting on Math Tutoring Services

This math problem was first brought to my attention from my dear friend Michael Allen Potter. In this post, I will solve the problem.

"A delivery firm uses one fleet of trucks on daily routes of 8 hours. A second fleet, with five more trucks than the first, is used on daily routes of 6 hours. Budget allotments allow for 198 hours of daily delivery time. How many trucks are in each fleet."

To solve this problem, we will:
1. Identify variables.
2. Identify the equations.
3. Substitute.
4. Solve.

1. Identify variables: There are two types of information in this problem: the number of trucks in either fleet and the number of total hours- which we know is 198. For this problem, let x be the number of trucks in the first fleet. Let y be the number of trucks in the second fleet.

2. Identify the right equations: We aren't given much information on how many are in the first fleet. That's okay. Let's just set up a reflexive statement:
x = x

We do have a little more information about the number of trucks in the second fleet. It's five more than what ever the first fleet is. Therefore:
y=x+5

We also need an equation for the number of hours and how this relates back to number of trucks in either fleet:
198=8x+6y

What the above is saying is that the total number of trucks in the first fleet (x) times the number hours each truck works (8hr) is the total hours the first fleet works. Similarly, the trucks in the second fleet (y) times the hours they work (6hr) is the total hours the second fleet works. The sum of these hours should equal exactly 198hrs, as described in the problem.

3. Substitute: Look carefully at the third equation above. We appear to have two variables an only one equation. That's okay!! This is a common situation. We know how x and y are related (by the second equation above). Now we need to substitute the second equation into the third:

198=8x+6(x+5)

4. Solve: Now we just plug and chug.

198=8x+6(x+5)
198=8x+6x+30
198=14x+30
168=14x
x=12

and since we know that y=x+5=12+5=17.

Therefore, your answer is 12 trucks in the first fleet and 17 trucks in the second fleet.