Study Guide

Word Problems - In the Real World

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In the Real World

These are some weighty topics, and we know that sometimes it's enough to think clearly about why you got up from the couch and went into the kitchen, but knowing how to translate word problems into mathematics can help you think more clearly about non-math questions that appear in everyday life. English is ambiguous, and sometimes people don't speak clearly. Translating into math involves removing ambiguity and figuring out what's going on. We're like number detectives, but without the deerstalker cap and calabash pipe.

We don't recommend translating your mother's questions about the day into mathematical expressions at the dinner table, but the word problem practice might help you pick out the important bits in her conversation. Like what will happen to you if you don't eat all your vegetables. Brussels sprouts again, Ma? For reals?


Mathematics is used to help with a lot of big, real-world stuff. It can be used to get an idea of how diseases spread, model how computer network traffic behaves, and all sorts of other important things. In order to use math for such useful, noble purposes, researchers first need to find a way to turn the question they want to answer into a question that's more math-y, and less English-y.

  • I Like Abstract Stuff; Why Should I Care?

    The Theory of Notation

    We've been translating English to math using what's called infix notation. You didn't even know that, did you? It's true. Using infix notation means that we put operators in between the numbers they're acting on. For example, to add 3 and 5 we write a plus sign in between them:

    3 + 5

    Infix isn't the only way to write mathematical expressions, though. There are also ways called prefix notation and postfix notation. "Pre" means before and "post" means after, so we'll bet you can guess what these entail. In case you're not in a guessing mood, we'll tell you.

    In prefix notation we write the operator before the numbers. To add 3 and 5, we write + 3 5. In postfix notation we write the operator after the numbers. To add 3 and 5, we write 3 5 +. You may never have occasion to use these, but it's a nice tidbit to keep in the back of your mind in case you ever do. Better to learn it pre you needing it rather than post.

    These different types of notation are studied in some computer science classes. If you plan on spending much of your life in cyberland, it'll benefit you to know them. There are some calculators that use postfix notation, also called reverse Polish notation, because that notation makes it easier to design the internal workings of the calculator.

    More Research!

    A lot of the mathematics we study in school has been around since Euclid, when the years were still counting down to zero. No wonder he was so good with negatives. Although you might not believe it to see all the math textbooks that exist, mathematicians today are still doing research and learning new things about math. If Euclid were alive today, our advancements would blow his mind.

    A large part of mathematical research involves figuring out how to translate the questions you want answered into problems that can be tackled with mathematics. Translating the question "what shapes can a soap bubble take?'' into mathematics is no easy task.

    When you've got a problem where the answer isn't known by anybody, it's called an open problem. We're not talking about something that is baffling merely you and your eighteen classmates. If your teacher knows the answer or it's in a book somewhere, that puppy is closed.

    There are some famous open problems, some of which are even worth a million dollars. So if you're looking for something to work toward in your spare time, and you could use a million dollars...

  • How to Solve a Math Problem

    There are three steps to solving a math problem.

    1. Figure out what the problem is asking.
    2. Solve the problem.
    3. Check the answer.

    Let's go through a problem in long and excruciating detail. If it gets too bad, you can take a nap.

    Sample Problem

    A cashbox has one-, five-, ten-, and twenty-dollar bills. There are 3 more five-dollar bills than one-dollar bills, half as many ten-dollar bills as one-dollar bills, and two more twenty-dollar bills than one-dollar bills. There is $365 total in the cashbox. That's almost a dollar a day! How many one-dollar bills are in the cashbox? Also, who are the wiseguys who insisted on paying their theater ticket in singles?

    • Figure out what the problem is asking. 

    After an initial read-through, we can grasp that, in general, we're being asked to find the number of one-dollar bills in the cashbox. Plus something about wiseguys that seems significantly less important.

    • Solve the problem. 

    Now we do a second, more careful reading of the problem, translate from English to math, and solve whatever equation or inequality we get.

    First we need to translate the problem from English into math. Let x be the number of one-dollar bills we have. Then we also have:

    x + 3 five-dollar bills,


    and x + 2 twenty-dollar bills.

    We have $365 total, so $365 = amount in ones + amount in fives + amount in tens + amount in twenties.

    The amount we have in ones is just x. We don't want to assign variables to any other amount, because that'll make things too confusing. We're trying to go toward the light, not away from it.

    The amount we have in fives is 5(x + 3) + (5 times the number of 5 dollar bills), and so on. So here's our equation for the total value of cash in the cashbox:

    And we can simplify that a bit:

    365 = x + 5x + 15 + 5x + 20x + 40
    365 = 31x + 55

    Phew! Our equation still looks long, but we got rid of all parentheses and fractions, and from here it's a matter of shifting all your x terms to one side and everything else to the other. Stop hitting the "shift" key. That won't help you.

    After solving this equation, we have x = 10.

    • Check the answer.

    If we have 10 one-dollar bills, then we have 13 fives, 5 tens, and 12 twenties. The value of our money adds up to:

    10(1) + 13(5) + 5(10) + 12(20)

    Yep, that's $365, as it should be. All is right in the universe.

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