Question: Two friends, Tanaya and Stephen were standing together. Tanaya begins to walk in a straight line away from Stephen at a constant rate of 3 miles per hour. One hour later, Stephen begins to run in a straight line in the exact opposite direction at a constant rate of 9 miles per hour. If both Tanaya and Stephen continue to travel, what is the positive difference between the amount of time it takes Stephen to cover the exact distance that Tanaya has covered and the amount of time it takes Stephen to cover twice the distance that Tanaya has covered?

(A) 60 mins
(B) 72 mins
(C) 90 mins
(D) 100 mins
(E) 120 mins

“Two friends, Tanaya and Stephen were standing together” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solutions and Explanation

Approach Solution : 1

Consider Tanya's journey to take t hours.
Stephen therefore travels for (t-1) hours.

Given that distance is equal
So,
3t=9(t-1)
t=1.5 hours

As a result, Stephen travels the same distance as Tanya in 0.5 hours.
Stephen is now expected to travel twice as far.
Say Tanya travels for t hours once more. Then, Stephen travels for (t-1) hours.
Stephen travels twice as far as Tanya, so it will take three hours

because 9(t-1) = 2*3t
t = 3 hours

Stephen therefore travels twice as far as Tanya in two hours.
Difference therefore equals 2 - 0.5 = 1.5 hours = 90 minutes.

Correct Answer: (C)

Approach Solution : 2

Tanya covered 3 miles after an hour of travel while Stephen traveled 0 miles.
After two hours, Tanya traveled 6 miles, while Stephen traveled 9 miles,
After two hours, Tanya traveled 6 miles, while Stephen traveled 9 miles,
Okay, so Stephen needs two hours to travel twice as far as Tanya.

One hour results in Tanya 3 and Stephen as 0.
We are aware of their rates.
In 20 minutes, Tanya covers 1 miles and Stephen covers 3 miles
Tanya walked 4 miles in 1 hour and 20 minutes and Stephen walked 3 miles in 20 minutes

Therefore, if it takes Tanya 20 minutes to walk 1 mile and Stephen walks 3 miles in that time

So, within 10 minutes time, Tanya covers 0.5 miles and Stephen 1.5 miles
As we can see, Stephen only needs 30 minutes to travel the same distance as Tanya
So, 30-120=-90

Since the positive difference is required, -90*-1 = 90 minutes

Correct Answer: (C)

Approach Solution : 3

Tanaya will be 3 miles from Stephen in an hour.
Let's say Stephen has traveled the same distance as Tanya after she has walked x miles.
Stephen would have traveled three times as far in the same amount of time because his speed is three times that of Tanaya's.

Then, 3x = x + 3
x = 3/2 miles

Tanaya's travel time will be (3/2)/3 = 1/2 hour + 30 minutes for the distance x.
Let's say that after Tanaya has traveled a certain distance (y), Stephen has done so twice as far. Additionally, Stephen would have traveled a 3y distance because his speed is three times that of Tanaya's.

Consequently,
3y = 2(y+3)
y = 6 miles
Tanaya will need 6/3 of an hour, or two hours, to travel the distance which can be said as 120 minutes.

Therefore, the time difference between the two scenarios will be 120 – 30 = 90 minutes.

Correct Answer: (C)

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