Working With Word Problems

Word problems are the bane of many college students. They can be confusing because they give you lots of information and no real guidance on how to go about solving them. Here are some techniques to help you demystify the word problem and go about getting the correct answers.

1. Remember, word problems usually give you more information than you need. Whether the intent is to make sure you know what you're talking about, or whether it's simply a trick to catch you unawares, word problems will almost always include information that has no bearing on the solution. For example, recall the riddle about the Road to St. Ives (as included in the third Die Hard movie):

    As I was going to St. Ives,
    I met a man with seven wives;
    Each wife had seven sacks,
    Each sack had seven cats,
    Each cat had seven kits.

    Kits, cats, sacks, wives,
    How many were going to St. Ives?

The answer, of course, is one — I was going to St. Ives. All the rest of the stuff is unnecessary information designed to throw you off. Don't let this happen to you!

2. Prepare a formula sheet before you read one question. As you go through your notes preparing for a test, or even as you run across new formulas in class, put all those formulas on a sheet for later studying. When you sit down to your test, before looking at any of the questions, first write out as many of these formulas as you can remember. (Use scrap paper if the instructor allows it, or the back flap of your blue book.) You'll remember more of the formulas this way without the pressure of solving the questions.

3. For each question, write out the known values. Go through the problem and write out on your scrap paper (or underline on the test paper) the known values that you can work with. This will later help you to identify which formula to use to solve the problem. Consider the following example word problem:

    Your backyard is a rectangle 20 yards wide and 40 yards long. Your puppy can run completely around the perimeter of the yard in 2 minutes. How fast is your puppy running, in miles per hour?

In this example, your knowns would be:

  • Rectangle width = 20 yards
  • Rectangle length = 40 yards
  • Rectangle Perimeter = ?
  • Puppy run = 2 minutes

4. Verify what you are looking to solve. A typical reason students get word problems wrong is that they start solving before they understand what the problem wants for an answer. The math could be spot-on, but in the end a math-correct answer that is wrong. Be sure you recognize what the problem is looking for, and include that in your list of known values. In the example above, this would be:

  • Solve for Puppy speed in miles/hour

5. Check through your formulas to see which best solves the problem. Looking through your list of known values, try to pick a formula that will solve for the required answer. A dead giveaway, of course, is formula that sets the required answer equal to some math calculation of values you have 'knowns' for. Otherwise, work backwards from your answer — what do you need to get that value, and then that value, and so on. When all else fails, pick a formula that has all the knowns and unknowns so that you can solve for your unknown. (Again, don't forget Rule #1!)

In the above example, we are looking for puppy speed around the perimeter in miles per hour. The formula for speed is distance/time; we have the time (2 minutes) but not the distance. So what is the formula for the perimeter of a rectangle? Perimeter is 2*(length+width). So that means our speed formula is now:

    Speed = (2*(length+width))/time

6. Double check your units of measurement. An especially devious word problem trick is to mix up units on you, such as giving you speed in feet/second and asking for the answer in miles/hour. This is a valuable lesson in the fact that you can't just give a correct answer, you have to give the correct answer to the question that was asked.

In this example, we find the puppy's speed is:

    Speed = (2*(40 yards + 20 yards))/2 minutes, or
    Speed = (2*60 yards)/2 minutes, or
    Speed = 60 yards/minute

This answer is absolutely 100% mathematically correct, but not what the question asked for. If you give this answer, your instructor will mark it wrong! You need to convert yards/minute to miles/hour:

    Speed = (60 yards/minute) * (60 minutes/hour) * (1 mile/1760 yards), or
    Speed = 2.045 miles/hour

7. Do a sanity check. Check your answer by math and/or common sense. For a math check, use the formula with your answer in it and solve for one of the other variables; your answer is correct if your variable ends up equalling the problem's variable. For common sense, stand back and examine your answer: is it physically possible, or does it make sense? If a problem asks how much of one pie is left, and you got a negative number, that can't be right. Don't let math get in the way of common sense.

These rules will help you navigate your way through the treacherous "word problem waters" and lead you to the safe harbor of better grades.

Keep paddling,
Professor Cram