A hiker walking at a constant rate of 4 miles per hour

[nwaneri.michael]

a hiker walking at a constant rate of 4 miles per hour is passed by a cyclist travelling in the same direction along the same path at a constant rate of 20 miles per hour. The cyclist stops to wait for the hiker 5 minutes after passing her while the hiker continues to walk at her constant rate. How many minutes must the cyclist wait until the hiker catches up?

6 2/3
15
20
25
26 2/3

What is the Manhattan Gmat way to solving and approaching this?

Thank you in advance.

[RonPurewal]

probably the most efficient way to solve this one is with the concept of relative rates.

start at the time when the cyclist passes the hiker.
since they're both traveling in the same direction, their relative rate - i.e., the rate at which the distance between them is changing - is found by subtraction. so, as long as they're both moving the cyclist is gaining on the hiker at a rate of 20 - 4 = 16 mph.
after the cyclist stops, the hiker is making up that distance at a rate of 4 mph. that's only 1/4 as fast as the relative rate during the first 5 minutes, so it will take the hiker 4 times as long to catch back up. so, 4 x 5 = 20 minutes.

if you wanted to, you could calculate the actual distance between the cyclist and the hiker when the cyclist stops: (16 mi/h)(1/12 h) = d, where "1/12 h" is five minutes. then you could find the time taken by the hiker alone to travel the same distance "d".
you don't have to do this, though, if you realize the 1:4 ratio referenced above.

[chintanjpatel]

i dont get ur explanation. the flaw is tht when u say tht the ratio is 1:4 and hence wht took cycler 5 min will take hiker 20 min is tht during the 5 min tht the cycler overtook and got away, hiker was travelling at 4kmph. hence, the ratio doesn't exlain it.

explanation: distance to be covered by hiker = distance by cycler - d by hiker
i.e. for cycler @ 20kmph for 5 min = 5/3 and for cycler @4kmph for 5 min = 1/3. now, the leftover distance to be covered by cycler = 5/3- 1/3 = 4/3.

time to cover 4/3km for cycler = 4/3 (distance) / 4 (speed) i.e. 1/3 hours i.e. 20 min.

guess this should help to understand the concept better.

[RonPurewal]

i dont get ur explanation. the flaw is tht when u say tht the ratio is 1:4 and hence wht took cycler 5 min will take hiker 20 min is tht during the 5 min tht the cycler overtook and got away, hiker was travelling at 4kmph. hence, the ratio doesn't exlain it.

no flaw.

check out the explanation again; the 1:4 ratio is the ratio of the hiker's catch-up speed to the original RELATIVE speed. all of the motion occurring during the original travel is subsumed under the relative speed, so we're good.

--

your explanation works too, of course.

[GMATDESTROYER]

Can anyone explain this question again?
I am not getting the explanation.

[krishnan.anju1987]

Hi,

There are two ways to do this. Though the second one is quicker once you grab the concept, I will start with the first one.

The person who is walking say X is walking at a speed of 4 miles/hr. The cyclist, riding in the same direction as X, is riding the bike at a speed of 20 miles/hr. Now, the cyclist passes X and then stops after 5 minutes. We need to find out the time he will need to wait so that X can catch up.

Now let's consider the point when the cyclist and X met. At that point, they were at the same starting point. The distance that the cyclist covered in those 5 minutes is found by cross multiplication

20 miles ------> 60 minutes
? ------> 5 minutes

?= 5*20/60= 5/3

At the same time, X continued walking. He did not stop. Hence the distance he covered in 5 minutes is al;so got by cross multiplication.

4 miles ------> 60 minutes
? ------> 5 minutes

?=4*5/60 =1/3

Now the cyclist stopped and waited for X to catch up. Remaining distance that X needs to cover = (Distance covered by cyclist in 5 minutes)- (distance covered by X in those 5 minutes)

4/3 miles

Now, we can find out the time taken by X to travel that distance

4 miles -----> 60 minutes
4/3 miles ------> ?

?= ((4/3)*60)/4= 20

Second way...
There is a shorter way to do this as Ron explained. The difference between the cyclist's and X's speed is 16 miles/hr.

X walks at a speedof 4 miles/hr. Hence after the cyclist stops riding his cycle, the difference between them that would have been built up in those 5 minutes would be at a difference of 16 miles/hour. X walks at one fourth of this speed and thus, will take 4 times the time taken by the cyclist to cover the remaining distance.

Hence, time taken by X to cover the distance= 20 minutes

Hope this helped

[GMATDESTROYER]

Hi

yes i got it now But what confused me was :


"The cyclist stops to wait for the hiker 5 minutes after passing her while the hiker continues to walk at her constant rate. "

For how long Cyclist travelled , anyway its clear now.

thanks a lot!

Hi,

There are two ways to do this. Though the second one is quicker once you grab the concept, I will start with the first one.

The person who is walking say X is walking at a speed of 4 miles/hr. The cyclist, riding in the same direction as X, is riding the bike at a speed of 20 miles/hr. Now, the cyclist passes X and then stops after 5 minutes. We need to find out the time he will need to wait so that X can catch up.

Now let's consider the point when the cyclist and X met. At that point, they were at the same starting point. The distance that the cyclist covered in those 5 minutes is found by cross multiplication

20 miles ------> 60 minutes
? ------> 5 minutes

?= 5*20/60= 5/3

At the same time, X continued walking. He did not stop. Hence the distance he covered in 5 minutes is al;so got by cross multiplication.

4 miles ------> 60 minutes
? ------> 5 minutes

?=4*5/60 =1/3

Now the cyclist stopped and waited for X to catch up. Remaining distance that X needs to cover = (Distance covered by cyclist in 5 minutes)- (distance covered by X in those 5 minutes)

4/3 miles

Now, we can find out the time taken by X to travel that distance

4 miles -----> 60 minutes
4/3 miles ------> ?

?= ((4/3)*60)/4= 20

Second way...
There is a shorter way to do this as Ron explained. The difference between the cyclist's and X's speed is 16 miles/hr.

X walks at a speedof 4 miles/hr. Hence after the cyclist stops riding his cycle, the difference between them that would have been built up in those 5 minutes would be at a difference of 16 miles/hour. X walks at one fourth of this speed and thus, will take 4 times the time taken by the cyclist to cover the remaining distance.

Hence, time taken by X to cover the distance= 20 minutes

Hope this helped

[RonPurewal]

glad it's clear.