Brent@GMATPrepNow wrote: ↑

Wed Nov 16, 2022 7:00 am

Joe regularly drives the 400 miles from A-ville to B-town at the same constant speed of r miles per hour. If Joe drives 30 mph faster than usual, it takes him 3 fewer hours to complete the trip. What is the value of r?

(A) 45
(B) 50
(C) 55
(D) 65
(E) 80

Answer: B
Source: gmatprepnow.com

STRATEGY: As with all GMAT Problem Solving questions, we should immediately ask ourselves, Can I use the answer choices to my advantage?
In this case, we can easily test the answer choices.
In fact, testing the answer choices is most likely the fastest approach because some of the answers choices will definitely not divide nicely into 400 miles (so, we need not check those answers choices)
So, let's start checking answer choices...

Since 50 and 80 are the only two numbers that will divide nicely into 400 miles, let's start by testing those two answer choices, beginning with answer choice B.

(B) 50
If Joe drives 400 miles at a speed of 50 mph, then his travel time = distance/rate = 400/50 = 8 hours.
For the hypothetical situation, Joe is driving 30 miles per hour faster than usual. In other words, Joe is driving 80 mph.
If Joe drives 400 miles at a speed of 80 mph, then his travel time = distance/rate = 400/80 = 5 hours.
Since Joe's hypothetical travel time is 3 hours less than his regular travel time, answer choice B satisfies the given information.

Answer: B



ALTERNATE APPROACH - Algebraic
Let's start with a word equation.
In this question, we're comparing trips when Joe drives at his REGULAR speed with a HYPOTHETICAL trip at a faster speed
The question tells us that the HYPOTHETICAL trip takes 3 fewer hours than the REGULAR trip.
So, we can write: (time to complete REGULAR trip) - (time to complete HYPOTHETICAL trip) = 3 hours

If Joe's REGULAR speed is r miles per hour, then his HYPOTHETICAL speed must be r + 30
time = distance/speed
So we can substitute values into our word equation to get: 400/r - 400/(r + 30) = 3

IMPORTANT: If you haven't already noticed, this equation is going to be a total pain to solve. On the other hand, plugging each answer choice into the above equation is super quick, especially if you start with the answer choices that divide nicely into 400

Multiply both sides of the equation by r to get: 400 - 400r/(r+30) = 3r
Multiply both sides of the equation by r + 30 to get: 400(r + 30) - 400r = 3r(r+30)
Expand both sides: 400r + 12,000 - 400r = 3r² +90r
Simplify the left side: 12,000 = 3r² +90r
Subtract 12,000 from both sides: 0 = 3r² +90r - 12,000
Divide both sides by 3 to get: 0 = r² +30r - 4,000
Factor: 0 = (r + 80)(r - 50)
So, EITHER r = -80 OR r = 50
Since Bob's speed can't be negative, the correct answer is r = 50

Answer: 50