Bunuel wrote:

Train A leaves New York at 7:00 am traveling to Boston at 80mph. Train B leaves Boston at 7:45 am traveling to New York at 70 mph on a parallel track. If the distance between New York and Boston is 210 miles, at what time will the two trains pass each other?

A) 8:15 am

B) 8:45 am

C) 9:00 am

D) 9:30 am

E) Cannot be determined from the information given

Kudos for a correct solution.

OFFICIAL SOLUTION:METHOD 1: Setting Up a Multi-Part Journey TableThe “mph” tips off that this is a rates question, and the fact that the two trains are at one point traveling (or “working”) together means we are also dealing with combined rates. However, the fact that Train A travels alone for the first 45 minutes (from 7:00 am – 7:45 am) also gives this problem the characteristic of a multi-part journey question.

Now that we know we are being tested on a multi-part journey with combined rates in the second leg of the journey, we can approach strategically with a multi-part journey chart:

We were given that Train A’s rate is 80mph, and we were given that Train A traveled alone for 45 minutes (0.75 of an hour). Any time you are given two parts of a three-part equation, you can solve for the third part: Train A traveled 60 miles between 7:00 am and 7:45 am.

We were also given that the total distance between the two trains is 210 miles. If we know Train A traveled alone for 60 miles, then the distance left when the trains are traveling together is 150 miles.

We are given that Train B’s rate is 80 mph. Whenever two trains are moving toward each other (or “working together”) you combine the rates: 70+80 = 150 mph. If the trains were traveling at 150mph it would have taken them 1 hour to cover the 150 miles left between them.

Starting from 7:00 am and adding on the 0.75 hours Train A traveled alone with the 1 hour the two trains traveled together, they will meet at 8:45 am. The correct answer is B.

METHOD 2: BacksolvingPerhaps you weren’t familiar or comfortable with the multi-party journey table, and you attempted to solve the problem through backsolving. Backsolving is a popular method when there are numbers in the answer choices, because the method consists of simply plugging each answer choice into the question stem and “backsolving” until you find the right one.

However, because you have to do the math for each of the answer choices, this method can often be lengthy and end up using more time than you should be on one problem – but in a bind it could have led you to the correct answer. With this method you would solve for how far each of the trains would have traveled by the given time in the answer choice, and see if their separate distances traveled, when combined, equaled a total of 210 miles.

METHOD 3: AlgebraYou could have also used an algebraic set-up to solve:

Let T = the time Train A takes to get to the meeting point

210-80T = 70(T-0.75)

T = 1.75 hours

Meeting time = 7:00am + 1.75 hours = 8:45 am

As you can see, there are a multitude of approaches that will all lead you to the correct answer. If you use your practice problems as a way to explore these various methods (as opposed to simply solving with one method and moving on), you will become more and more comfortable with the question types in which each can be applied, and ultimately more successful in knowing when to use them.

Methods 1 & 2 would have been the most efficient for a question type like this, but both are equally quick and effective. Individuals’ minds all work differently, so the only way to know what works best for you is to try them all out and get as much practice as you can.

Next time you do a practice problem, go the extra mile and reflect on what choices you were faced with along the way, what decisions you made, and what methods are available that you didn’t use the first time around.

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