Suppose you are at a river resort and rent a motorboat for 5

Suppose you are at a river resort and rent a motorboat for 5 hours starting at 7am. You are told that the boat will travel at 8 miles per hour upstream and 12 miles per hour returning. you decide that you would like to go as far up the river as you can and still be back at noon. at what time should you turn back, and how far from the resort will you be at that time?

Solution

Distance(d) equals Rate(r) times Time(t) or d=rt; r=d/t and t=d/r
Let t=amount of time used to travel upstream
And (5-t)= amount of time used to travel downstream
Distance travelled upstream=8*t
Distance travelled downstream=12(5-t)
And we know that the above two distances have to be equal, so:
8t=12(5-t) get rid of parens
8t=60-12t add 12t to each side
8t+12t=60-12t+12t collect like terms
20t=60
t=3 hr-----------------time travelled upstream
So, 7am + 3 hr=10am--------------time to turn back
5-t=5-3=2 hr ------------time travelled downstream
distance upstream =8*t=8*3=24 mi from resort
ANOTHER WAY:
let t=time used to travel upstream
Then 5-t=time used to travel downstream
Let d=distance travelled upstream; also distance travelled downstream as well as distance from resort when turning back
7am + t=time to turn back
t=d/8-----------------------------eq1
5-t=d/12---------------------------eq2
substitute t=d/8 from eq1 into eq2
5-d/8=d/12 multiply each term by 24
120-3d=2d add 3d to each side
5d=120
d=24 mi----------------------distance from resort when turning back
substitute d=24 into eq1
t=24/8=3 hrs----------------time used to travel upstream
7am+3=10am-------------time to turn back