Bunuel wrote:

Two cyclists start biking from a trail's start 3 hours apart. The second cyclist travels at 10 miles per hour and starts 3 hours after the first cyclist who is traveling at 6 miles per hour. How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking?

A. 2 hours

B. 4 ½ hours

C. 5 ¾ hours

D. 6 hours

E. 7 ½ hours

I use rate and speed interchangeably in these problems. Speed \(s\) is the same as rate \(r\)

Short version, no explanationCyclist 1, at a speed of 6 mph * 3 hrs, travels 18 miles

Cyclist 2 rides at 10 mph, and will cover that distance at relative speed (=relative rate) of 10 - 6 = 4 mph

18 miles/ 4 mph = 4.5 hours

Second cyclist catches first cyclist in 4.5 hours from the time second cyclist starts.

Answer B

Answer with explanationOne way to think about this question: it's a gap and chase problem.

1) Find the distance between the cyclists, which is the "gap"

The first cyclist creates the distance between the two cyclists. At a rate of 6 mph, he rides for 3 hours before the other cyclist starts.

D = r*t. Cyclist 1, riding alone, creates

\(D: (6 mph * 3 hrs)\) =

18 miles between them At the moment the second cyclist starts from the same place, the first cyclist is exactly 18 miles ahead. That's the "gap's" distance.

2) Find the rate at which the gap is closed in a "chase"

Because they start from the same place, and ride in the same direction, this is a chase* problem. She chases him to close the gap.*

She rides faster, at 10 mph.

The rate* at which she closes the gap is the difference of their speeds, called "relative rate" or "relative speed."

Relative rate: 10 - 6 = 4 mph

3) Find the time it takes for Cyclist 2 to catch up to Cyclist 1

while both travel at the same time*

How much time will pass before the second cyclist catches up with the first, from the time the second cyclist started biking?

Time = D/r

Distance = 18 miles

(Relative) Rate = (10 - 6) = 4 mph

Time:

\(\frac{18miles}{4mph}\) = \(4.5\) hours

Cyclist 2 will catch Cyclist 1 in 4.5 hours after Cyclist 2 begins.

Answer B

*

"Chase" means two travelers travel in the same direction, where one travels faster than the other. One traveler typically starts behind the other, either in time, or in distance. The person behind has to catch the person ahead. She is "chasing" him.

*rate = "relative rate." How this relative rate is calculated depends on travelers' directions, original positions, and speeds. Same direction, same starting point: subtract slower rate from faster rate

*"closing the gap" = "catching up to" the person ahead

*this question has nothing to do with the first cyclist's first three hours, except for its role in creating the distance of the gap between them. After that, the question asks only about the time segment that begins when Cyclist 2 starts to ride.
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