SOLUTION: (Question) Two cyclists start biking from a trailhead at different speeds and times. The second cyclist travels at 10 miles per hour and starts 2 hours after the first cyclist, who

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Question 887930: (Question) Two cyclists start biking from a trailhead at different speeds and times. The second cyclist travels at 10 miles per hour and starts 2 hours after the first cyclist, who is traveling at 6 miles per hour. Once the second cyclist starts biking, how much time will pass before he catches up with the first cyclist?
(Answer)
(a)2 hours
(b)4 1/2 hours
(c)5 3/4 hours
(d)6 hours
(e)7 1/2 hours
My attempt:
What is known from the problem-
Distance formula=
r*t=d
1st cyclist: 3 hours and 18 miles ahead 2nd cyclist (0hr, 0mi; 1hr, 6mi; 2hr, 12mi; 3hr, 18mi)
2nd cyclist: 3 hours and 18 miles behind 1st cyclist (0hr, 0mi; 1hr, 0mi; 2hr, 0mi; 3hr, 0mi)
Cyclist 2 starts to ride after Cyclist 1 has traveled for 3 hours or 18 miles.
(1st hour) for 2nd cyclist = 4th hour for 1st cyclist
1st cyclist after 4hrs = 24 miles
2nd cyclist after 1hrs = 10 miles
(2nd hour)
1st cyclist after 5hrs = 30 miles
2nd cyclist after 2hrs = 20 miles
(3rd hour)
1st cyclist after 6hrs = 36 miles
2nd cyclist after 3hrs = 30 miles
(4th hour)
1st cyclist after 7hrs = 42 miles
2nd cyclist after 4hrs = 40 miles
(5th hour)
1st cyclist after 8hrs = 48 miles
2nd cyclist after 5hrs = 50 miles
Choices (c), (d), and (e) are both too long.
Choice (a) is too short.
There for choice (b) is the correct answer.
Checking the answer:
(4 1/2th hour)
1st Cyclist after 7 1/2hrs = 45 miles
2nd cyclist after 4 1/2hrs = 45 miles
That's fine. But what is the algebraic formula for this problem? I solved it using the process of elimination from the given answer choices. There must be a faster way?

Found 2 solutions by josgarithmetic, Alan3354:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!

__________________speed_______time_______distance
FIRST______________6__________t+2
SECOND____________10__________t

Be really certain you understand that data table before continuing. It is a transcription directly from the description. Also notice how the FIRST and slower bike travels for more time than the bike which left later and moves faster.


__________________speed_______time_______distance
FIRST______________6__________t+2________d=6(t+2)
SECOND____________10__________t__________d=10t

Do you find how to handle this now? FIRST and SECOND bikes are expected to reach the same distance when SECOND catches up with FIRST.

Hopefully you can see and understand the TWO equal expressions for the distance d.
highlight_green%286%28t%2B2%29=10t%29
6t%2B12=10t
12=4t
highlight%28t=3%29
The second bike needs three hours travel time to match first bike.

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Two cyclists start biking from a trailhead at different speeds and times. The second cyclist travels at 10 miles per hour and starts 2 hours after the first cyclist, who is traveling at 6 miles per hour. Once the second cyclist starts biking, how much time will pass before he catches up with the first cyclist?
---------------
In 2 hours, the 1st cyclist is 12 miles away (2*6).
The 2nd cyclist "gains on" the 1st are 4 mi/hr (10-6).
12/4 = 3 hours.