Word Problems… !#$%@;*!!

Word problems can be… frustrating.  Most of their reputation arises from their use of words to describe a quantitative problem.  And if the problem’s author did not choose their words very carefully, you’ve got Trouble (with a capital T). So why are so many word problems assigned? Because they are more similar to the quantitative problems you might encounter in life than many of the practice problems in your textbook: you have to supply some insight and organization in order to arrive at a solution.

Just about every level of mathematics, not to mention chemistry and physics, seems to send periodic spasms of word problems your way.  The SAT makes liberal use of them as well.  Since you cannot avoid them, you had might as well learn how to solve them once and for all.

I have watched many people work on word problems, and most seem to spend 25% of their time translating the word problem into mathematical notation, then 75% of their time solving the resulting math problem. When I solve a word problem, I think my time ratio is the reverse: I spend closer to 75% of my time trying to make absolutely sure I have interpreted the words correctly and fully understand the nature of the question, then 25% of my time using math to find the answer.  I mention this mostly to encourage students not to rush the wording interpretation phase – if you mis-interpret the wording, you are very likely to get the wrong answer – so why even bother starting to “work the math” until you are certain you have fully understood the problem?

I recommend the following generic steps to solving any word problem:

1) Read the last sentence first.  Much of the time, the last sentence will tell you exactly what you need to find. By reading about this first, you are more likely to interpret the rest of the words correctly the first time.

2) Based on the question being asked (usually the last sentence), decide on what your variable(s) must be to conveniently answer the question. On your work paper (which I hope is scrap paper), write each variable with a short description following it. The description should be more than one word, and should (if possible) include the units that are used for the variable (inches, square meters, dollars, count, etc.). For example, if the word problem involves Nickels and Dimes, you could easily write down something like: “N = Nickels, D = Dimes”. But this is not specific enough… in fact I made this error years ago, and proceeded to use N as the number of nickels in one equation, then the value of the nickels in the next. My description of N was not specific enough to help me catch my inconsistent interpretation of what N stood for. If I had written either “N = Nickels Count, D = Dimes Count” or “N = $ value of Nickels, D = $ value of Dimes”, I would have had a better chance of solving the problem correctly.

3) Now that you know what your objective is, and have decided on and described the variable(s) you will be using (with units), you are ready to begin reading the word problem from the start. As you read through the problem, “doodle” the information in the problem on your work paper. Draw a picture that illustrates the situation being described, note each quantity mentioned on your paper, draw lines or arrows, build tables for the data… in short, do whatever will help you process the information in the problem the best.

At this stage, you are NOT seeking to solve the problem. You are seeking to understand the problem as fully as possible. I tend to stop reading after each tidbit of information (often only a sentence fragment), then re-read to make sure there was only one way of interpreting the language used, then doodle the information on my paper, then resume reading. I repeat this process as many times as needed until I have read the entire problem. My goal is to understand as much about the problem as I would if I had just experienced the situation in real life. Visualize the situation, draw a picture of it, think it through, then replay it in your mind to see if every fact you have been given fits into your understanding of the problem. You should also make sure that every fact in the problem is reflected in your doodle, so that (ideally) you need not refer to the words again.

4) You are now ready to bring in any additional information that is needed. This is the “insightful” part of the process, where your overall understanding of the situation is most important. The more you understand the nature of the situation and the question being asked, the easier it will be to figure out what additional formula(s), if any, are needed.

Typical formulas that are useful include
Distance = Speed x Time
Force = Mass x Acceleration
Density = Mass / Volume
Area of a Rectangle = Length x Width
Downwind Speed = Plane Speed + Wind Speed
… etc …

I recommend always writing any such formulas down in their general form first – you can substitute values into the formula later. Writing them down clearly in their general form at this stage will help to avoid errors.

5) Review everything you have written to verify that each of the numbers in the problem plays a role in the equation(s) you are about to solve.

Most word problems don’t give you any more information than needed to solve the problem, so it is good to verify that you have used all the data provided. Having said that, some will give you either more information than needed (leaving it up to you to decide which data are needed to solve the problem), or less information than needed (so, you cannot solve the problem). Such wrinkles add a layer of complexity, however the more word problems you have done, the more easily you will handle them.

6) Review everything you have written to verify that you have a solvable problem. If you have more than one variable to solve for, do you have at least as many equations as you do variables (a system of equations)? Do the equations you have written look solvable (do they look like algebra problems you have done before)? Do the equations you have written contain all the (relevant) data given to you in the problem, in one way or another?

Don’t ignore the units used to measure each quantity in the problem. Every quantity should be written with its units after it, then treat the units as if they were variables. The units you expect for your answer should “survive” the algebraic solution process, and any other units in the problem should cancel each other out. If they don’t, you have not solved the problem correctly.

If it is a geometric situation, can you remember the theorems you will need to use to get from the givens to what is asked for? Will they get you all the way to the information you need, or are there more theorems you will need to rely on as well?

If the problem does not quite look solvable at this point, you may need to return to step 1 and think each step through again to double check that you have indeed not been given enough information to solve the problem (some texts have more of these than others). However if everything looks good, you have finished the “word” part of the problem, and are now ready to treat the rest of this problem as an algebra or geometry problem.

7) When you have finished solving the algebra or geometry, and believe you have your answer, there is one last crucial step: re-read the question (usually the last sentence) and make absolutely sure that your answer answers the question. Watch out for situations where you might have calculated the Number of Nickels, but the question asked for the Value of the Nickels. Or questions which ask for two numbers, but you have only solved for one of them. It is really frustrating to have all your work be correct, but then lose points because you only provided a partial or intermediate answer to the problem.

As you build your base of experience solving word problems, you will discover that you typically only see a few word problem types at a time (typically between 2 and 4 types). Once you have mastered how to recognize each of the problem types, and know what formulas are relevant in each type of situation, they will begin to feel easy… until you run into one that was ambiguously worded. If you run into ambiguous wording on a quiz or test, don’t guess at its meaning – ask the teacher about it. Your entire class will be glad you did, as will the teacher.

Reading and understanding the process above is not enough. You have to do a bunch of word problems, probably two to three times as many as you expect, before you will truly have slain this dragon. So: use scrap paper, doodle, read very carefully, take your time, and practice, practice, practice.  Once you master them, word problems become fun.

By Whit Ford

Math tutor since 1992. Former math teacher, product manager, software developer, research analyst, etc.

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