Translating a Tough Rate Word Problem

by Stacey Koprince Nov 13, 2012

Recently, we discussed various strategies for translating word problems into math. Let’s put that knowledge to the test on a challenging problem from a category that everybody hates: Rates.

Set your timer for 2 minutes and GO! (© ManhattanPrep)

*  A bullet train leaves Kyoto for Tokyo traveling 240 miles per hour at 12 noon. Ten minutes later, a train leaves Tokyo for Kyoto traveling 160 miles per hour. If Tokyo and Kyoto are 300 miles apart, at what time will the trains pass each other?

(A) 12:40pm

(B) 12:49pm

(C) 12:55pm

(D) 1:00pm

(E) 1:05pm

 

One of the strategies we discussed in the translation article was make the situation real. Put yourself into the situation and imagine you’re the one doing whatever the problem is describing. That will help you to set things up cleanly and correctly.

So what’s going on in this particular situation? First, you’re the conductor on the Kyoto train. At noon, you pull out of the station (instantly and magically traveling 240 miles per hour from the very start!). The track is 300 miles long; after one hour, where are you?

After one hour, it’s 1pm and you’ve gone 240 miles, so you’re just 300 “ 240 = 60 miles from Tokyo.

Okay, now switch jobs. You’re the Tokyo train conductor and you leave Tokyo at 12:10pm. After one hour, where are you? You’re going 160 miles an hour, so after 1 hour, it’s 1:10pm and you’re 300 “ 160 = 140 miles from Tokyo.

By 1:10p, have the two trains passed each other? Definitely, because train K (for Kyoto) is even further towards Tokyo at that point. Now, make a guess: do you think that the trains had already passed each other by 1p? Think about it before you read the next paragraph.

By 1p, the two trains have already passed each other. Train K is only 60 miles from Tokyo, and train T has been going for most of an hour already at a rate of 160 miles per hour. Great, we can cross off two answer choices! It can’t be D or E.

Let’s pause to talk about something for a moment. When I start a rate problem with moving objects, I always first try to figure out Where are my objects after 1 ˜unit of time’ (in this case, an hour)? I do this just so that I can understand how the problem works, because often this will give me ideas about how to continue. Sometimes, I can even cross off some answer choices! Win-win.

Now, I’m at a decision point. I can draw up a table and start writing some equations. I can also draw out the scenario and keep trying to figure it out using real world sense, like I did above. In the actual moment, we need to choose just one scenario and go for it, but we’ll talk about both of them here.

First, let’s talk about the more traditional solution method: drawing up a table and writing equations. The textbook method always works, of course, but this is one of the reasons people hate rate problems “ the textbook method can be difficult, confusing, and non-intuitive. People often get lost along the way and end up having to guess.

Rate problems hinge on the RTD formula: Rate ´ Time = Distance. Set up a table:

	Rate	Time	Distance
Train K	240 mph	t	d
Train T	160 mph	t “ 10 MINUTES	300 “ d

 

The rates are easy; the problem tells us these directly. Here’s where things get tricky: what’s the time taken? We need to assign a variable here “ but the two trains take a different length of time. We have a choice here. If we say that train K’s time is t minutes, then train T’s time is 10 minutes less than that (because train T starts 10 minutes later). This is what the table shows. Alternatively, we could say that train T takes t minutes, and then train K takes t plus 10 minutes.

Here’s complication number two: notice that I put minutes in all capital letters in the problem. The rates were given in miles per hour, not minutes, so I have to convert something somewhere. 10 minutes out of 60 minutes is 1/6 of an hour so, really, I need to write t “ 1/6 hours in that blank.

Finally, what’s the distance? Is it 300 miles? Here’s another tricky step. The two trains start out 300 miles apart. The question asks what time it is when they pass each other. Collectively, how far will they have gone when they pass each other?

Draw it out. Together, they will have covered the 300 miles between them, so their two separate distances add to 300. If train K went 200 miles, then train T must have gone 100 miles. If train K went 180 miles, then train T must have gone 120 miles. In that case, we can call one of the distances d, and then the other must be 300 “ d.

Sigh. Okay, now what? Now, we’ve got two equations with two variables and we can solve:

Equation 1: 240t = d

Equation 2: 160(t “ 1/6) = 300 “ d

Substitute and solve. Remember that the problem asks us to find the time, so make sure to solve for t, not d.

160t “ 160/6 = 300 “ 240t

240t + 160t = 300 + 160/6

400t = 980/3

t =

t = 49/60

That was lucky “ we ended up with 60 on the denominator! (Why is that lucky?)

We just solved for t in hours, but the answer choices present the time in hours and minutes. Train K started out at noon, and we need to add 49/60 hours to that time, which corresponds directly to 49 minutes out of 60 minutes, or 12:49pm.

The correct answer is B.

Okay, now, isn’t there a better way? Why, yes there is “ I’m so glad you asked. First, there is a shortcut that we can use to make the above algebra a little bit easier, but I’m not going to explain it here because we already explain it completely in our Word Problems Strategy Guide (chapter 3, problem #8).

I still do, though, want to talk about the other path we can go down, the one where we continue to use real world sense to lay out the problem step by step.

Recall that, at our decision point, we had figured out this:

By 1p, the two trains have already passed each other. Train K is only 60 miles from Tokyo, and train T has been going for most of an hour already at a rate of 160 miles per hour. Great, we can cross off two answer choices! It can’t be D or E.

Now what? Let’s try a smaller time increment, but one that still works well with one hour / 60 minutes. 30 minutes (1/2 of an hour) is generally the easiest one to try next, but I could also try 45 minutes (3/4 of an hour). Before deciding, glance at the answers. I already know that 30 minutes isn’t enough time, because the remaining three answers are between 12:40 and 12:55, so I should try 45 minutes, or ¾ of an hour.

Okay, after ¾ of an hour, how far as train K gone? It’s traveling at 240 mph, so ¾ of that is 180 miles. Train K has gone a bit more than halfway; if they’ve already passed each other, then train T will have to have gone more than 120 miles at this point.

What about train T? First, remember that train T has only gone 35 minutes, not 45; we’ll have to adjust accordingly. Train T is traveling 160 mph. In 30 minutes, it goes 80 miles, so the train will go a bit more than 80 miles in 35 minutes. We need it to have traveled 120 miles in order to have passed train K. Is that going to happen in 5 minutes? Nope, definitely not. If it takes train K 30 minutes to go 80 miles, then the train will need another 15 minutes to go 40 more miles. Okay, so train K has to run for more than 45 minutes, so cross off answer A.

Now we’re down to the last two answers: 12:49p or 12:55p. (And I want to mention one thing: on many problems, the answers would be spread farther apart, and we’d already be able to select one answer by this point. But this is an especially hard problem. : ) )

Now, look at the work we’ve done so far. At 12:45p, train K has gone 180 miles and train T has gone a little bit more than 80 miles. Together, the two trains have gone a bit more than 260 miles.

At 1p, train K has gone 240 miles and train T has gone close to (but not quite) 160 miles. Together, they’ve traveled something less than 400 miles.

Which one is closer to our desired distance, 300? Should the answer be closer to 12:45p or closer to 1p?

Right. A bit more than 260 is a lot closer to 300 than a bit less than 400, so the correct answer should be closer to 12:45. And, once again, the correct answer is B.

 

Key Takeaways for Rate Word Problems:

(1) For wordy story problems in general, we want to insert ourselves into the story and make it real. Understand the sequence of events and actually figure out each step that needs to take place, one by one.

(2) Test out the scenario using the first unit of time “ one hour, one minute, one whatever. What has happened after that first hour? Don’t forget to adjust if the two trains or cars or people didn’t start at the exact same time.

(3) Make a choice: do you want to write equations and solve or do you want to step through the problem using basic time increments to see what happens? People often hate doing the equations but also aren’t confident that using time increments will be enough to get down to just one answer. Notice how close the answer choices were here, and yet we only had to test two timeframes (12:45p and 1p) in order to get to one answer. It really does work, every time! Just make it real. : )

* copyright ManhattanPrep