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Translating Word Problems: Keywords (page 1 of 2)

Sections: Keywords, Worked examples

The hardest thing about doing word problems is taking the English words and translating them into mathematics. Usually, once you get the math equation, you're fine; the actual math involved is often fairly simple. But figuring out the actual equation can seem nearly impossible. What follows is a list of hints and helps. Be advised, however: To really learn "how to do" word problems, you will need to practice, practice, practice.

The first step to effectively translating and solving word problems is to read the problem entirely. Don't start trying to solve anything when you've only read half a sentence. Try first to get a feel for the whole problem; try first to see what information you have, and what you still need.




The second step is to work in an organized manner. Figure out what you need but don't have, and name things. Pick variables to stand for the unknows, clearly labelling these variables with what they stand for. Draw and label pictures neatly. Explain your reasoning as you go along. And make sure you know just exactly what the problem is actually asking for. You need to do this for two reasons:

  1. Working clearly will help you think clearly, and
  2. figuring out what you need will help you translate your final answer back into English.

Regarding (2) above: It can be really frustrating (and embarassing) to spend fifteen minutes solving a word problem on a test, only to realize at the end that you no longer have any idea what "x" stands for, so you have to do the whole problem over again. I did this on a calculus test -- thank heavens it was a short test! -- and, trust me, you don't want to do this to yourself!

The third step is to look for "key" words. Certain words indicate certain mathematical operations. Below is a partial list. Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

    Addition increased by
    more than
    combined, together
    total of
    sum, plus
    added to
    Subtraction decreased by
    minus, less
    difference between/of
    less than, fewer than
    Multiplication of
    times, multiplied by
    product of
    increased/decreased by a
      factor of (this type can
      involve both addition or
      subtraction and
    Division per, a
    out of
    ratio of, quotient of
    percent (divide by 100)
    Equals is, are, was, were, will be
    gives, yields
    sold for

Note that "per" means "divided by", as in "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon". Also, "a" sometimes means "divided by", as in "When I tanked up, I paid $12.36 for three gallons, so the gas was $4.12 a gallon".

Warning: The "less than" construction is backwards in the English from what it is in the math. If you need to translate "1.5 less than x", the temptation is to write "1.5 – x". Do not do this! You can see how this is wrong by using this construction in a "real world" situation: Consider the statement, "He makes $1.50 an hour less than me." You do not figure his wage by subtracting your wage from $1.50. Instead, you subtract $1.50 from your wage. So remember; the "less than" construction is backwards.

Also note that order is important in the "quotient/ratio of" and "difference between/of" constructions. If a problems says "the ratio of x and y", it means "x divided by y", not "y divided by x". If the problem says "the difference of x and y", it means "xy", not "yx".

Now we need to learn to extract the keywords from the word problems....

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Stapel, Elizabeth. "Translating Word Problems: Keywords." Purplemath. Available from Accessed



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