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Alan and Soren are roommates. They both leave their jobs at the same time to go back to their apartment.
Alan works 6 miles away from the apartment and rollerblades from his workplace to the apartment at a constant rate of 7 miles per hour.
Soren works 8 miles away from the apartment and bikes from his workplace to the apartment at a constant rate of 12 miles per hour.

Part A

How much time, in hours, do Alan and Soren have to travel to be the same distance from their apartment? Show all the steps you took to find your answer.

Part B

Who will get back to their apartment first? How much less time, in hours, will it take this person to get home than the other person? Show or explain how you determined your answer.

Part A:

Let be the time (in hours) that has passed since Alan/Soren has started to head for their apartment.

Alan works 6 miles away from the apartment and moves at a rate of 7 miles per hour towards his apartment

The distance remaining to Alan's apartment after hours is:

Soren works 8 miles away and moves at a rate of 12 miles per hour. The distance remaining to Soren's apartment after hours is:

Set the equations equal to each other to find how long it takes for them to be at the same distance from their apartment:

Add both sides by 12t

Subtract both sides by 6

Divide both sides by 5

They will be at the same distance from their apartment after 0.4 hours, or 2/5 hours.

Part B:

Set d equal to 0 for each equation

Alan's equation:

Add both sides by 7t

Divide both sides by 7

Soren's equation:

Add both sides by 12t

Divide both sides by 12 and simplify

Since , it takes less time for Soren to get to his apartment, which means Soren gets back first.

The difference in the amount of time (how much less time it took) between Soren and Alan is:

It took Soren 4/21 hours less than Alan to get back to his apartment.

Let me know if you need any clarifications, thanks!

~ Padoru

A) 0.4 hours

B) Soren, 4/21 hours

Step-by-step explanation:

A) 6 - 7t = 8 - 12t

5t = 2

t = 0.4 hours

B) Alan: 6/7 hours

Soren: 8/12 = 2/3 hours

Soren will reach first

6/7 - 2/3

LCM: 21

[(6×3) - (2×7)]/21

(18-14)/21

4/21 hours

power property.

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