The distance between Town R and Town S is 209 miles GMAT

The distance between Town R and Town S is 209 miles…

Here is the full question text:

The distance between Town R and Town S is 209 miles. At noon yesterday, a motorist started from Town R and drove to Town S at a constant speed of 40 miles per hour, and another motorist started from Town S and drove along the same route to Town R at a constant speed of 55 miles per hour. What was the motorists’ distance, in miles, from Town R when they passed each other?

A. 80

B. 88

C. 95

D. 100

E. 114

From an overall perspective, in this question we know the total distance is 209 miles. 

We also know that the distance traveled by the driver starting at R (Driver R) is:

Likewise, we know that the distance traveled by the driver starting at S (Driver S) is: 

Note that the time t hr will be the same for both because we’re talking about the fixed time that it takes for the two to meet; that is, the time that it takes to cover the full 209 miles. 

Therefore, all we need to do is add the drivers’ respective values and set this equal to the total distance: 

The tricky part of this question is actually in the arithmetic.

You can use the pulling out primes technique because you’re looking for a specific value and it’s not immediately clear what common factor 95 and 209 have–but I assure you there will be one. Otherwise, how would you do this without a calculator?

So prime factorize 95 first (basically because it’s easier): 

Clearly we don’t have a 5 in 209, so let’s assume 19: 

Well, there we go. Now all we have to do is set up the equation like this:

Now that we have the time, the only thing that we need to do is to determine the distance from R, which we do by multiplying the time by Driver R’s speed:

40 \frac{miles}{hour} \times 2.2 hours = 88 miles

The answer is (B). 

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