John drove on a highway at a constant speed of r miles per hour in 13:

When Tom started driving, the distance the John had already traveled is: 2r miles
Each hour the distance between Tom and John shortened by (4r/3-r) = r/3 miles
Then it will take Tom \(\frac{2r}{(r/3)}=6\) hours to catch up John, which means till Tom caught up with John, John had been driving for 8 hours => The distance John drove is 8r
Answer C.

for 2 hrs, J travelled 2r miles.. now tom starts with 4r/3 speed.. suppose they meet after time T.. distance travelled by tom = 4rT/3

also John will travell rT miles in this time..
now.. 2r+rT = 4rT/3
2r=rT/3
T=6

Therefore John travels = 2r +6r = 8r miles
C

The key thing to understand here is that when Tom catches up with John, they have covered the same distance. What the question really asks for is this distance, which is expressed as \(T * R\) in the answer choices. So our goal is to express the distance covered by John and Tom with rate and time.

Distance covered by John :
\(2 * r\)(distance covered during the first 2 hours at r miles)
We define a new variable, \(T\) = the time it took John to travel the second portion of the trip. The distance he covers is \(r * T\).
Total distance covered by John: \(2r + rT\).

Distance covered by Tom :
Tom travels at a speed of \(\frac{4}{3}r\).
The total travel time of Tom is equal to the time it takes John to travel the second part of his trip --> \(T\).
Total distance covered by Tom: \(\frac{4}{3}rT\)

Since John and Tom have covered the same distance when they meet up, we can solve the following equation for T :
\(2r + rT = \frac{4}{3}rT\)
\(2r = \frac{1}{3}rT\)
\(T = 6\).
So it took Tom 6 hours to catch up with John.

And the distance covered is :
\(\frac{4}{3}r * 6 = 8r\)

Answer C.

MATH REVOLUTION OFFICIAL SOLUTION:

It is important to note that in speed problems, time passes at the same time.



John traveled 2r miles during the first 2 hours and he traveled rt miles during the next t hours. In case of Tom, he traveled 4rt/3 miles during t hours. The distances they traveled are rt and 4rt/3, respectively,instead of rt1 and 4rt2/3,respectively because both traveled during the same time period. Then, \(2r+rt=\frac{4rt}{3}\) → \(2+t=\frac{4t}{3}\) → t=6. This means that Tom caught up with John after John traveled 6 hours. So we can solve 2r+6r=8r, and the correct answer is C.

[quote="Bunuel"]

Math Revolution and GMAT Club Contest Starts!

QUESTION #12:

John drove on a highway at a constant speed of r miles per hour in 13:00. Then, 2 hours later, Tom drove on the same highway at a constant speed of 4r/3 miles per hour in 15:00. If both drivers maintained their speed, how many did John drive on a highway, in miles, when Tom caught up with John?

A. 3r
B. 5r
C. 8r
D. 9r
E. 10r


Let t be the time john drove before meeting with tom
distance drove by john = rt
distance drove by tom = 4/3r(t-2)

so rt=4/3r(t-2)
solving for t, t=8
therefore john drove 8r miles before meeting with tom

By the way, the redaction of this question is very poor

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