If the phrase "word problem" sends a shiver down your spine, you're not alone. A lot of people have trouble with so-called word problems in math. But, believe it or not, these problems usually aren't any harder to solve than non-word problems—they just look very, very different. And they require a slightly different mindset to solve.
Today, I'm going to tell you about my simple 5-step method that will help you solve all your math problems—including those pesky word problems. In particular, we're going to talk about how to turn a word problem into an algebraic equation and then solve it.
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A "Real World" Math Drama
Today's word problem begins with a story about cats and dogs. It goes something like this…
Like all dogs, your dog loves toys. And you love giving them to him. Your cat, on the other hand, does not love your dog and therefore finds it amusing to hide his toys. Being quite clever, you suspect that the cat is the culprit, so you begin to monitor his favorite hiding spot: the pile of towels next to his bed.
But (perhaps being a little too clever for your own good) instead of constantly checking this spot, you decide that you'd like to rig up an ingenious system to automatically report to you exactly how many toys are missing.
The question is: How can you do this?
Step #1: Stop and Think Before Doing Anything
The biggest mistake people make when solving problems is trying to solve them too soon.
The most important thing to do when faced with a problem like this is to stop working on it. Honestly, it sounds paradoxical, but the biggest mistake people make when solving problems is trying to solve them too soon. Instead, stop and think about what you need to do. Make sure you understand exactly what the question is asking and make sure you understand exactly what you are trying to solve for.
In our problem, we should ask ourselves: Can we actually build something that will discern the numer of hidden dog toys? Sure, all we need to do is put the cat's pile of towels on a smart scale that sends its weight to your computer. Whenever the scale senses a weight increase, it can tell your computer that another toy has been hidden. Your computer can then use some as-yet-unknown equation to figure out exactly how many toys are hidden. When that number goes above a certain limit, your computer can sound an alarm to let you know that it’s time to go fetch.
Now that we have a plan, it's time for the big English-to-equation translation.
Step #2: English-to-Equation Translation
The second step in solving word problems is turning the words into one or more mathematical expressions or equations. In our case, we need to figure out how to write an equation that takes the current weight on a scale and gives us back the number of dog toys hidden on it. How can we do it?
Well, let’s take the total weight on the scale, which we’ll call W_total, and subtract the weight of just the towels, which we’ll call W_towels. The difference between these two weights must be equal to the combined weight of all the dog toys, W_toys:
But we don’t actually want to know the weight of the toys, we want to know the number of toys. How can we do that? Well, if we know the total weight of all the toys, W_toys, and we divide that by the weight of a single toy, W_toy (assuming they're all the same weight), we get the total number of toys, Ntoys:
But how did we know the values of W_towels and W_toy? We must have been clever enough to measure them and write them down before we put the towels on our scale. Or, if we didn’t do that, we’d better do it now!
Step #3: Solve for Whatever You’re Interested In
The third step in solving our word problem—or any word problem—is to solve for the variable you’re interested in. This step will often entail going through the procedure outlined in the How to Solve an Equation episode. Our goal is to solve for the total number of dog toys on the scale, N_toys. Combining the two equations we came up with in our English-to-equation-translation, we get just such an equation:
N_toys = ( W_total – W_towels ) / W_toy
We can now turn this abstract solution (abstract in the sense of being written in terms of a bunch of variables) into a numerical solution simply by plugging in numerical values for all the variables on the right side of the equals sign. I should say that plugging in values isn't always necessary. For example, we don’t actually have numerical values to use in our problem. Which makes sense since our equation was only intended to be used to tell our computer how to convert from total weight to total number of dog toys—we weren’t actually looking for a specific answer to a specific problem.
Step #4: Make Sure You Understand the Result
Slow down and take a minute to think about your result.
The fourth step in the problem solving process is closely related to the first. The name of the game here is to slow down and take a minute to think about your result. Ask yourself if it makes sense. If you plugged in numbers and got a negative number, ask yourself if you expected to get a negative number. If you got a huge number, ask yourself if you expected that.
The bottom line is this: Don’t declare that you’re done simply because you got an answer. The only reason you should declare that you’re done is because you understand the answer you got.
Step #5: Use Your Result to Solve Other Problems
The fifth and final step of the problem solving process is to use the result you’ve obtained to solve other problems. Why have I included this as a step? Because we’re talking about solving real world problems here—not just textbook problems. In the real world, many problems you solve will make you think of something related that you need to solve, too. The good news is that doing so is usually easier because you already have a solution to build upon.
And those are all of the steps you should go through when solving real world math problems. Of course, once you get good, you won't actually think about doing each step—they'll just happen. But while you're learning, it's best to be deliberate about things and to actually think your way through each step.
Good luck in all your future problem solving endeavors!
Wrap Up
Okay, that's all the math we have time for today.
Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Problem solving image from Shutterstock.
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