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Solving Math
Word Problems

 

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There are two steps to solving math word problems:

  1. translate the wording into a numeric equation
  2. solve the equation!

Usually, once you get the math equation, you're fine.
But getting to the equation can seem difficult.
These strategies may help you translate,
but practice will determine your success.

  • Read the problem entirely
    Get a feel for the whole problem
  • List information and the variables you identify
    Attach units of measure to the variables (gallons, miles, inches, etc.)
  • Define what answer you need,
    as well as its units of measure
  • Work in an organized manner
    Working clearly will help you think clearly
    • Draw and label all graphs and pictures clearly
    • Note or explain each step of your process;
      this will help you track variables and remember their meanings
  • Look for "key" words
    Certain words indicate certain mathematical operations:

Addition

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

Vocabulary

  • "Per" means "divided by"
    as "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon"
    Also 30 miles/gallon

  • "a" sometimes means "divided by"
    as in "When I tanked up, I paid $3.90 for three gallons, so the gas was 1.30 a gallon, or $1.30/gallon

  • "less than"
    If you need to translate "1.5 less than x", the temptation is to write "1.5 - x".  DON'T!  Put a "real world" situation in, and you'll see how this is wrong:  "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

  • "quotient/ratio of" constructions  
    If a problems says "the ratio of x and y",
    it means "x divided by y" or x/y or x÷y

  • "difference between/of" constructions
    If the problem says "the difference of x and y",
    it means "x - y"

    Examples

Wording

Math expession

What is the sum of 8 and y?

 8 + y

4 less than y

  y - 4
y multiplied by 13 13y
the quotient of y and 3 y/3
the difference of 5 and y 5 - y
the ratio of 9 more than y to y (y + 9)/y
nine less than the total of a number (y) and two (y + 2) - 9 or y - 7
The length of a football field is 30 yards more than its width.  Express the length of the field in terms of its width y y + 30
Twenty gallons of crude oil were poured into two containers of different size.  Express the amount of crude oil poured into the smaller container in terms of the amount y poured into the larger container."  The expression they're looking for is found by this reasoning:   There are twenty gallons total, and we've already poured y gallons of it.   That means that there are X gallons left. 20 - y

The
Purplemath web site has created some word problems for you to solve:

"Age" problems, involving figuring out how old people are (or will be)
"Area/volume/perimeter" problems, involving very basic geometric formulas
"Coin" problems, involving figuring out how many of each type of coin you have
"Distance" problems, involving speed/rate, distance, time, and the formula "d = rt".
"Investment" problems, involving investments, interest rates, and the formula "I = Prt".
"Mixture" problems, involving combining elements and find prices (of the mixure) or percentages (of, say, acid or salt).
"Number" problems, involving "Three more than two times the smaller number..."
"Percent of" problems, involving finding percents, increase/decrease, discounts, etc.
"Work" problems, involving two or more people or things working together to complete a task, and finding how long they took.

Related topics in the Purplemath web site:  canceling units, percent of, solving equations.  Purplemath index of lessons:  http://www.purplemath.com/modules/modules.htm

Also recommended:
"How to Solve Word Problems in Algebra", by Mildred Johnson

This guide has been adapted from Purplemath (http://www.purplemath.com/index.htm) web site, with permission of the author, Elizabeth Stapel ©2000-2001. 

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The Study Guides and Strategies web site was created and is maintained by Joe Landsberger,
academic web site developer at the University of St. Thomas (UST), St. Paul, Minnesota.  It is collaboratively maintained across institutional and national boundaries, and  last revised September 04, 2002 . 

Permission is granted to freely copy, adapt, print, transmit, and distribute
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