Math  /  Algebra

QuestionZack and Trevor got in trouble at football practice and have to run laps as a consequence. Zack, who runs at a rate of 1 lap per minute, had completed 7 laps already when he was joined on the track by Trevor. Trevor's pace is 2 laps per minute. At some point, the two will have run the same distance. How many laps will each boy have run? How long will that take?
Zack and Trevor will each have run \square laps in \square minutes.

Studdy Solution

STEP 1

What is this asking? When will Zack and Trevor have run the same number of laps, given Zack's head start and their different speeds? Watch out! Don't forget Zack's head start!
It's tempting to just use the speeds, but Zack's already run a few laps.

STEP 2

1. Set up the problem
2. Find when they've run the same distance
3. Calculate the total laps and time

STEP 3

Alright, let's **define** what we know!
Zack runs at a rate of 11 lap per minute and has a head start of 77 laps.
Trevor runs at 22 laps per minute.
Let's use zz for Zack's laps and tt for Trevor's laps.
We also need a variable for the time, in minutes, when they've both run the same distance; let's call that mm.

STEP 4

Now, let's **write equations** that represent the number of laps each boy has run based on the time mm.
Zack's total laps will be his head start plus his rate times the time: z=7+1mz = 7 + 1 \cdot m.
Trevor's laps will just be his rate times the time: t=2mt = 2 \cdot m.

STEP 5

We want to find when Zack and Trevor have run the **same number of laps**.
This means their total laps are equal, so we can set their lap equations equal to each other: 7+1m=2m7 + 1 \cdot m = 2 \cdot m.

STEP 6

Now, let's **solve for** mm!
We can subtract 1m1 \cdot m from both sides of the equation: 7+1m1m=2m1m7 + 1 \cdot m - 1 \cdot m = 2 \cdot m - 1 \cdot m.
This simplifies to 7=1m7 = 1 \cdot m, which means m=7m = 7.
So, they'll have run the same distance after **7 minutes**!

STEP 7

We know it takes 77 minutes.
Let's **plug that back** into our equations to find out how many laps each boy has run.
For Zack: z=7+17=7+7=14z = 7 + 1 \cdot 7 = 7 + 7 = 14 laps.
For Trevor: t=27=14t = 2 \cdot 7 = 14 laps.

STEP 8

So, after 77 minutes, both Zack *and* Trevor will have run 1414 laps.
Awesome!

STEP 9

Zack and Trevor will each have run 1414 laps in 77 minutes.

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