Word Problems

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#### When I'm given a 'word problem', where do I start?

Start by understanding what the problem is asking. What is the exact question you need to answer? Next, figure out what information you already know, and what you need to know in order to answer the question.

While there is no ONE BEST way to solve word problems, we can work through an example to illustrate the typical steps involved.

Harry Potter has asked his friend Hermione for a potion to turn them and their friend Ron into birds. (The flying car is in for repairs, so they need to make the trip to Diagon Alley as birds.)

Diagon Alley is 9 miles away and a dose of Hermione's potion lasts 50 minutes. They only have enough potion for one dose each. If they can go 24 miles an hour as birds, and they start at 4:30 p.m., can they get to Diagon Alley and back to Hogwarts again before the potion runs out at 5:20 p.m.? If so, how much time will they be able to spend in the Alley?

So, where do you start? Here are a few pointers to keep in mind.

First, it's important to understand what the problem is asking. What exactly is the question you need to answer? For this problem, it's really two questions:

Can the kids get there and back in 50 minutes?
...and...
how long can they stay?

The next thing to do is to figure out what information you already know, and what you need to know in order to answer the question(s).

Don't be distracted by "extra" information! In this problem, it doesn't matter what time the kids leave and come back. It only matters whether they get there and back in 50 minutes. So you should ignore the "4:30 p.m." and "5:20 p.m.," but stay within the 50 minute timeframe.

There are many different ways to handle this: you could draw a picture to represent the different places, and put in the pieces of information you have. Here's a drawing for our problem, where each  -|  is a mile:

```
0                 9                             24
|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
^                 ^
|Hogwarts         |Diagon Alley
|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
^
distance they can travel in an hour as birds|

```

In this example, the children need to travel back and forth, 9 miles each way, for a total of 18 miles. We need to know whether they can do that in 50 minutes. What do you think, based on your drawing? Now's a good time to estimate.

Or you might want to rewrite the problem in your own words, and work with that information. You might say,

As birds, the children can go 24 miles in one hour. They want to go 9 miles twice, or 18 miles total, in 50 minutes. How much time, if any, do they have to spare?

Or you might use a formula you already know. Using the formula Distance = Rate * Time, we could say:

We want to find the time it takes to go 9 miles twice (or 18 miles in all) at 24 miles/hour, so:
```          18 miles = 24 mph * Time,
```
We want to know if the time is less than 50 minutes (or 50/60, or 5/6 of an hour), and how much less if so.

Next, you need to figure out how to get from the information you're given to the answer you want. What can you do with your given information? What other tools might you need? A different formula? The ability to change fractions to decimals or vice versa?

It's a good idea to do the work one step at a time if it's a big problem, so you don't try to solve the whole thing at once. It's much easier to keep the pieces of the problem in order if you work this way, and much easier to avoid mistakes.

It's important to check your work as you go, and, after you finish, to make sure you're getting results that make sense. How do your results compare with the estimate you made earlier? For instance, if you came up with 3 hours as the round-trip time from Hogwarts to Diagon Alley and back, and you plugged 3 hours in with the 24 mile-per-hour flying rate, that would seem pretty farfetched, right? 72 miles there and back, when the problem tells us it's only 9 miles? Or if you come up with a negative number as an answer for round-trip time, that's not possible. The kids can't get back to Hogwarts before they leave!