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Translating Word Problems: Examples (page 2 of 2)

Sections: Keywords, Worked examples

  • Translate "the sum of 8 and y" into an algebraic expression.

    This translates to "8 + y"

  • Translate "4 less than x" into an algebraic expression.

    This translates to "x – 4"

Remember? "Less than" is backwards in the math from how you say it in words!

  • Translate "x multiplied by 13" into an algebraic expression.

    This translates to "13x"

  • Translate "the quotient of x and 3" into an algebraic expression.

    This translates to " x/3"

  • Translate "the difference of 5 and y" into an algebraic expression.




    This translates to "5 – y"

  • Translate "the ratio of 9 more than x to x" into an algebraic expression.

    This translates to "(x + 9) / x"

  • Translate "nine less than the total of a number and two" into an algebraic expression, and simplify.

    This translates to "(n + 2) – 9", which then simplifies to "n – 7"

Here are some more wordy examples:

  • The length of a football field is 30 yards more than its width. Express the length of the field in terms of its width w.

    Whatever the width w is, the length is 30 more than this. Recall that "more than" means "plus that much", so you'll be adding 30 to w.

    The expression they're looking for is "w + 30".

This one is important: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

  • 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 g 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 g gallons of it. How many gallons are left? There are 20 – g gallons left. They want the answer "20 – g".

This is the "how much is left" construction: You will be given some total amount. Smaller amounts, of unspecified sizes, are added (combined, mixed, etc) to create this total amount. You will pick a variable to stand for one of these unknown amounts. After having thus accounted for one of the amounts, the remaining amount is whatever is left after deducting this named amount from the total.

  • They may tell you that a trip took ten hours, and that the trip had two legs. You might name the time for the first leg as "t", with the remaining time for the second leg being 10 – t.
  • They may tell you that a hundred-pound order of animal feed was filled by mixing products from Bins A, B, and C, and that twice as much was added from Bin C as from Bin A. Let "a" stand for the amount from Bin A. Then the amount from Bin C was "2a", and the amount taken from Bin B was the remaining portion of the hundred pounds: 100 – a – 2a.

I'm making a big deal about this "how much is left" construction because it comes up a lot and tends to cause a lot of confusion. Make sure you understand this one!

Once you've learned to translate phrases into expressions and sentences into equations, you are ready to dive into word problems. Of course, there are infinitely-many possible word problems (physics is all word problems; business math is all word problems; "real life" can feel like an essay question...). The following links lead to explanations and examples of some basic types of word problems that you can expect to see in your classes:

"Age" problems, involving figuring out how old people are, were, 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 speeds ("uniform rates"), 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.
Quadratic word problems, such as projectile motion and max/min questions.
"Work" problems, involving two or more people or things working together to complete a task, and finding how long they took.

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Cite this article as:

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



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