Algebra: Speed and Distance Problems

Speed and Distance Problems

Have you ever heard of a word problem like this one? "Train A heads north at an average speed of 95 miles per hour, leaving its station at the precise moment as another train, Train B, departs a different station, heading south at an average speed of 110 miles per hour. If these trains are inadvertently placed on the same track and start exactly 1,300 miles apart, how long until they collide?"

If that problem sounds familiar, it's probably because you watch a lot of television (like me). Whenever TV shows talk about math, it's usually in the context of a main character trying but failing miserably to solve the classic "impossible train problem." I have no idea why that is, but time and time again, this problem is singled out as the reason people hate math so much.

In fact, it's not so hard. This, like any distance and rate of travel problem, only requires one simple formula:

• D = r · t

Distance traveled (D) is equal to your rate of speed (r) multiplied by the time (t) you traveled that speed. What makes most distance and rate problems tricky is that you usually have two things traveling at once, so you need to use the formula twice at the same time. In this problem, you'll use it once for Train A and once for Train B.

To keep things straight in your mind, you should use little descriptive subscripts. For example, use the formula D_A = r_A · t_A for Train A's distance, speed, and time values and use the formula D_B = r_B · t_B for Train B.

Example 4: Train A heads north at an average speed of 95 miles per hour, leaving its station at the precise moment as another train, Train B, departs a different station, heading south at an average speed of 110 miles per hour. If these trains are inadvertently placed on the same track and start exactly 1,300 miles apart, how long until they collide?

Solution: Two trains means two distance formulas: D_A = r_A · t_A and D^B = r_B · t^B. Your first goal is to plug in any values you can determine from the problem. Since Train A travels 95 mph, r_A = 95; similarly, r_B = 110.

Notice that the problem also says that the trains leave at the same time. This means that their travel times match exactly. Therefore, instead of denoting their travel times as t_A and t_B (which suggests they are different), I will write them both as t (which suggests they are equal). At this point, your formulas look like this:

Here's the tricky step. The trains are heading toward one another on a track that's 1,300 miles long. Therefore, they must collide when, together, both trains have traveled a total of 1,300 miles. Of course, Train B is going to travel more of those 1,300 miles than Train A, since it's traveling faster, but that doesn't matter. You don't even have to figure out how far each train will go. All that matters is that when D_A + D_B = 1300, it's curtains. Luckily, you happen to know what D_A and D_B are (95t and 110t, respectively) so plug those into the equation and solve.

• D_A + D_B = 1300
• 95t + 110t = 1300
• 205t = 1300
• t ‰ˆ 6.341 hours

So, the trains will collide in approximately 6.341 hours.

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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