If 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. 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.

Step-by-step explanation:

Let x = distance from house after some time has passed

distance = rate * time

Let time = t

Alan is 6-x miles away when he meets Soren

6-x = 7t

t = (6-x)/7

Soren is 8-x miles away when he meets Alan

8-x = 12t

t = (8-x)/12

The two will have travelled the same amount of time when they meet since we are assuming they left at the same time.

(8-x)/12 = (6-x)/7

7(8-x) = 12(6-x)

56 -7x = 72 - 12x

5x = 16

x = 16/5 = 3.2 miles traveled when they meet

You can plug in (16/5) for x in one of the earlier equations to get the time.

t = (6-(16/5))/7

= ((30/5)-(16/5))/7

= (14/5)/7

= (14/5) × (1/7)

= 14/35

= 2/5 = 0.4 hours

x=30

step-by-step explanation: