Hi,
I was hoping someone could help clarify the following problem for me, it's an original problem, to which I don't know how to apply the general RT=D technique taught in the MGMAT guide:
Jane and Karen were in a race, Jane gave Karen a 5 m head start in the 100m race, and Jane was beaten by 0.25m. In how many meters more would Jane have overtaken Karen?
I appreciate any help!
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- boiarski
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- JonathanSchneider
- ManhattanGMAT Staff
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Re: Difficult Rate Problem
Unless I'm overlooking something, you need some more information before this can be solved (ideally some measurement of time).
First, we can construct some sort of a drawing. I can't draw too easily here, but you need to have a good drawing on your page to think this one through.
Show that K has a 5m head start, and J and K are both going in the same direction. K has only 95m to go to the finish line, and she arrives there first, so she goes that full distance. J has 100m to get to the finish line, but she only gets to the 99.75 mark when the race ends. Thus, we know the relative distances run by each:
Distance(J) = 99.75
Distance(K) = 95
Since we know the time was the same for each, we can call their rates:
Rate(J) = 99.75 / t
Rate(K) = 95 / t
where "t" represents the time that they spent running.
Now, for J to "overtake" K, we need to see that they will both continue running forward at their respective rates, until they are at the same place. This means that J must run a distance of "d + 0.25m," and K must run a distance of "d," where "d" represents some unknown distance. Now, we know the individual rates, and so at this point the problem becomes a "chase" scenario; simply subtract the rates to find the *relative* rate between the runners:
(99.75 / t) - (95 / t) = 4.75 / t
Thus,w e can say that J is catching up to K at a rate of 4.75/t. The *relative* distance between them is 0.25m. Thus, we can set up a *relative motion* RTD chart:
R * T = D
(4.75/t) * (T) = 0.25m
Notice that the capitalized "T" is different from the lowercase "t." (On your page you might use sub-1 and sub-2.) This new "T" represents the time it takes for J to overtake K. Because we have two variables here, we cannot determine either time. Thus, we cannot determine the distance they will run before J overtakes K.
This can all be seen in a more intuitive manner, of course. We know that they run a footrace, and we know that one of them wins by a certain distance. However, we don't know their times or exact speeds. Were they fast or slow? If they were fast, then J will not take long to overtake K. But if they were slow, then it may take J a very long time to overtake K.
First, we can construct some sort of a drawing. I can't draw too easily here, but you need to have a good drawing on your page to think this one through.
Show that K has a 5m head start, and J and K are both going in the same direction. K has only 95m to go to the finish line, and she arrives there first, so she goes that full distance. J has 100m to get to the finish line, but she only gets to the 99.75 mark when the race ends. Thus, we know the relative distances run by each:
Distance(J) = 99.75
Distance(K) = 95
Since we know the time was the same for each, we can call their rates:
Rate(J) = 99.75 / t
Rate(K) = 95 / t
where "t" represents the time that they spent running.
Now, for J to "overtake" K, we need to see that they will both continue running forward at their respective rates, until they are at the same place. This means that J must run a distance of "d + 0.25m," and K must run a distance of "d," where "d" represents some unknown distance. Now, we know the individual rates, and so at this point the problem becomes a "chase" scenario; simply subtract the rates to find the *relative* rate between the runners:
(99.75 / t) - (95 / t) = 4.75 / t
Thus,w e can say that J is catching up to K at a rate of 4.75/t. The *relative* distance between them is 0.25m. Thus, we can set up a *relative motion* RTD chart:
R * T = D
(4.75/t) * (T) = 0.25m
Notice that the capitalized "T" is different from the lowercase "t." (On your page you might use sub-1 and sub-2.) This new "T" represents the time it takes for J to overtake K. Because we have two variables here, we cannot determine either time. Thus, we cannot determine the distance they will run before J overtakes K.
This can all be seen in a more intuitive manner, of course. We know that they run a footrace, and we know that one of them wins by a certain distance. However, we don't know their times or exact speeds. Were they fast or slow? If they were fast, then J will not take long to overtake K. But if they were slow, then it may take J a very long time to overtake K.
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