Study Guide

Word Problems - Word Problems

Word Problems

We've already done a lot of word problems, many of which had straightforward translations into mathematical symbols. When word problems contain more words, we naturally need to work a little harder to find good translations. Some may go on for so long that you'll wonder why they aren't referred to as "paragraph problems" or "novel problems."

We'll give you some general tips for doing word problems, but first we'll do a short problem to warm up. Then we'll finish with a cool-down on the elliptical.

Sample Problem

Ben has twenty-five coins in his pocket, all dimes and nickels. He's going to be disappointed when he discovers that the washing machine only takes quarters. He has half again as many dimes as he has nickels. How much money does Ben have?

First of all:

(number of coins Ben has) = (number of nickels Ben has) + (number of dimes Ben has)

The problem tells us the number of dimes Ben has in terms of the number of nickels he has. Let n be the number of nickels. Then Ben has  dimes, since he has half again as many dimes as nickels. Note the difference here: he doesn't have half as many dimes, he has half again as many. Therefore, we'll be taking half the number of nickels and adding that to the total number of nickels to find the number of dimes. Make sense? If not, nod your head and smile. We can't take the rejection.

We can now finish translating our first equation into symbols:

Now we have an equation to solve. No more words. Hallelujah. Take a stab at solving the equation and see how you do. See the section on equations if you need a refresher.

Hopefully, you got n = 10. If you did, give yourself a pat on the back. If you can't reach it, ask a friend to help you.

The above is all fine and dandy, but "10 nickels'' doesn't answer the question, "how much money does Ben have?'' To find out how much money he has, we add the amount of dough Ben has in nickels and the amount of money he has in dimes. Let's see what ol' Moneybags is worth.

(number of nickels Ben has)(0.05) + (number of dimes Ben has)(0.10)

If Ben has 10 nickels, then he has  dimes, so the total amount of money he has is:

10(0.05) + 15(0.10) = $2.00

Excellent. That's almost enough for half a soda from the vending machine.

Now that we're all warmed up, it's time for the promised general tips about what to do with these monsters. The tips come in two flavors:

  1. How to Read Word Problems
     
  2. How to Do Word Problems

We used to offer a third flavor, Rum Raisin, but it never got a lot of love so we discontinued it.

  • How to Read Word Problems

    We recommend reading any word problem at least three times. By the conclusion of the third reading, you should be magically whisked away to your home in Kansas.

    Read it once...

    ...to get a general idea of what's going on. Is the problem about money? Height and width? Distances? Money? Yeah, we already said money, but it's important. Can you draw a picture in the margin that helps you visualize the problem?

    Twice...

    ...to translate from English into math. Read the problem carefully this time, figuring out which pieces of information are important and which ones aren't. Then ignore the unnecessary bits. This may sound a bit harsh, but they're big boys. They'll get over it.

    Three times...

    ...a lady. Wait, that's not it.

    ...to make sure you answered the right question. To be sure, read the question one last time before drawing a box around your final answer. Even if this method of double-checking saves you only once out of every 100 times, it'll help you in the long run. One out of 100 is better than zero out of 100, but now we're moving into some advanced mathematics.

    Sample Problem

    The local department store was having a sale. Holla! Gabe bought a pair of shoes for $21, although they would've been cheaper if he'd bought penny loafers. He also bought some shirts that were on sale for 25% off their normal retail price of $18 each. Gabe, always the bargain hunter, spent $75 total. How many shirts did he buy? Also, does he really think any of them will go with those shoes?

    Read the problem:

    Once...

    ...for a general idea of what's going on: Gabe went shopping and spent money. Sounds like an old familiar story. Time for an intervention, friends of Gabe. Preferably before he maxes out his Diner's Club card.

    Now we need to figure out how many shirts he bought, so we read the problem:

    Twice...

    ...to translate from English into math. Remember, we can translate a bit at a time, sort of like how Gabe pays for some of his major purchases when he has them on layaway. Sheesh, Gabe. Get a hold of yourself.

    (total amount Gabe spent) = (amount he spent on shoes) + (amount he spent on shirts)

    We know he spent $75 total, of which $21 was spent on shoes. This gives us the equation:

    $75 = 21 + (amount he spent on shirts)

    The problem has gotten smaller. It's depressing when that happens with cake, but great when it happens to a word problem. Now all we need to do is come up with a symbolic expression for how much Gabe spent on shirts, or the cost per shirt times the number of shirts:

    (amount he spent on shirts) = (cost per shirt)(number of shirts)

    Since the shirts are 25% off their normal price of $18, they cost $18 – 0.25(18) = $13.50 each. What a deal, and real polyester, too!

    We need to introduce a variable for the number of shirts; s will do the trick nicely. 

    (amount he spent on shirts) = 13.5s

    When we plug this into the earlier equation, we find that:

    75 = 21 + 13.5s

    Finally, we've reduced this thing to a super-simple-looking equation! Good riddance, vestiges of language! Begone, nouns and verbs!

    Things look much nicer now, right? No worrying about shirts or shoes or prices or Gabe's uncontrollable shopping addiction. For the moment, we can forget about the word problem and solve the equation. The answer is s = 4, by the way. In case you were interested.

    Three times...

    ...to make sure we're answering the right question. We want to know how many shirts Gabe bought. Is that the answer we arrived at? We found that s = 4, and s was the number of shirts Gabe bought, so we're all done. Four new shirts for Gabe, and three of them feature a Hawaiian pattern. Gabe, if you insist on buying far more clothes than you need, can't you at least have a decent fashion sense?

    When translating from English into math, some information can be ignored. We don't care that "the local department store was having a sale.'' Gabe might, but we certainly don't. We care about statements that tell us numbers, and statements that tell us what the question is. Any extraneous information has been placed there simply as a decoy. We're not going to fall for that. Wait a second...two for one? We'll grab our jacket and meet you there.

    Some people find it helpful to underline the important pieces of information in a word problem. You're like an actor highlighting in a script the lines that are important for him to remember. Unlike an actor, however, you can always look back at the original problem if you draw a blank. Also, you don't need to wear any stage makeup.

    In the problem we just did, the important bits might look something like this:

    The local department store was having a sale. Gabe bought a pair of shoes for $21 and some shirts that were on sale for 25% off their normal retail price of $18 each. If Gabe spent $75 total, how many shirts did he buy?

    As you practice, you'll become better at figuring out which parts of the word problem you can ignore and which parts are important.

  • How to Become a Word Problem Expert

    If you want to be better at word problems, you need to practice doing word problems.

    There's no way around this unless you know of some way to download that skill onto a microchip and implant it in your brain. Of course, even if you do know of a way to do that, you'll need to wait for the download, schedule a time for the operation, two to three weeks of recovery time, and so on. Too much hassle. Better to learn it our way.

    As with any other acquired skill in the world, it's all about practice. If you want to become fluent in Spanish, you need to practice speaking Spanish. If you want to become fluent in math, you need to practice doing math. If you want to become fluent in doing a Christopher Walken impression, you need to start practicing your Christopher Walken impression. By the way, you get 0 points for originality with that one. We would've been much more impressed if you impersonated someone slightly more obscure like Nick Offerman.

    Who? Exactly.

    Math is also a lot like art, which incidentally takes a ton of practice as well. You can watch an artist draw a tree and understand that she's shading to indicate where the shadow is, unless she's drawing one of those shadowless trees, which are so much easier to draw albeit less grounded in reality. In order to draw a good tree yourself, though, you'll probably need to get out a pencil and practice first. Don't be alarmed if your first several attempts look more like spears of broccoli.

    Watching someone else, such as your teacher, a tutor, or Shmoop, work out a problem is a good start toward understanding mathematical concepts. Most of us learn best by example, which is our story and we're sticking to it. But we're going off-topic.

    While having access to examples is nice, when it comes time for the test, it's whether you can do the problems on your own that counts. Understanding a problem when someone else goes through the steps is different from being able to do the problem yourself. See how we repeated that point and bolded it twice? Hm. Might be an important distinction.

    While we're going through examples, have your paper and pencil at the ready—or your pen, if you're feeling cocky. Try to work out the examples on your own. Then, if your answers match ours, you can rest assured that you know your stuff, because we're pretty sure we did these right.

    As you do more word problems, you'll notice similarities between some of them. You'll find that many word problems can be grouped into "types'' that are solved in similar ways. Not every word problem will fit obviously into a certain type, and some will fit into more than one type. Unlike with people, you can quickly pinpoint what type of word problem you're dealing with. Stereotyping someone in real life like this is a terrible, terrible idea, but fortunately, it works great with word problems. Besides, some of our best friends are word problems.

  • Geometry Problems

    Geometry problems are great, because they often come with pictures. Who doesn't like pictures? We especially like the ones by the late, great Maurice Sendak, but you won't see any of them in this section. Sorry.

    Sample Problem

    Find the area of the shape shown below, which consists of a triangle and a semicircle:

    (Area of shape) = (Area of triangle) + (Area of semicircle).

    We know how to find the area of a triangle:

    The height of the triangle is 5, and the base is 10 (the diameter of the circle). Therefore, the area of the triangle is:

    We also know how to find the area of a semicircle. We know how to find the area of a circle, and the area of a semicircle is half that, right? Like a semi is half a truck? Wait a second...

    Since the radius of this semicircle is 5, here's its area:

    That makes the area of the entire shape:

    (Area of shape) =

    That's as nice as this answer will ever be if we don't want to round things. As it happens, we're not in much of a rounding "mood" at the moment, so yeah, we're done.

    When geometry problems don't come with pictures, we can often draw our own pictures anyway. Our lines may not be as perfectly straight as those generated by a computer, but at least they'll have that personal touch that may be otherwise lacking. In fact, you may want to sign and date it in case your mom wants to post it on the fridge for a few months.

  • Averages

    Averages often appear on word problems. To find the average of a collection of numbers, we add all the numbers and then divide by how many numbers there are. If someone calls you average and they haven't done all the legwork, you can disregard their judgment of you.

    Sample Problem

    Find the average of the five smallest positive integers.

    The five smallest positive integers are 1, 2, 3, 4, and 5. Remember, integers are counting numbers, and if you're counting down to a shuttle launch, you'll rarely use fractions. Unless there's some sort of mechanical malfunction and you're stalling for time. "Two...uh, one and a half..." 

    To find the average of these, we add them and divide by 5, since that's how many numbers we have.

  • Percents

    Go here to review percents if you're feeling a little fuzzy about them. If you're still feeling fuzzy after reviewing that section, you should see a doctor. You may have a concussion. Hopefully it wasn't algebra-induced.

    Sample Problem

    Fifteen percent of what number is six?

    This is a fairly straightforward question, so the first thing we need to do is to make sure we're interpreting it correctly. We're not looking for 15% of 6. Instead, we're looking for a number that will give us 6 when we take 15% of it. In this example, "what number'' translates to x, and "15% of'' means "0.15 times." If you take an algebra test and you get it back with a red "0.15" scrawled across the top, you'll know how well you did. Answer: not so well.

    Now we solve the equation.

    0.15x = 6

    Divide both sides by 0.15 to get x = 40. We can backtrack and double-check our work to make sure that's right. Sure enough, 15% of 40 is 6. Hot diggity.

  • The Word "Per"

    The word "per'' is a signal that it's time to do division. It means roughly "for every." Think to yourself, "For every numerator there is a denominator." We're sorry we don't have this for you in acronym form, unless FENTIAD helps you somehow.

    Sample Problem

    Luisa drove 150 miles and used 6 gallons of gas. How many miles per gallon does her car get? Also, when will she start caring about the environment and go electric?

    To find the number of miles per gallon, we take the number of miles divided by the number of gallons:

    so Luisa's car gets 25 miles per gallon. Oh, good. She can keep driving away from the Earth's problems.

  • Coin Problems

    There are some problems where we need to be careful to differentiate between the number of something we have and the value of something. If someone asks how much change you've got on you and you say 75, does that mean you have 75 cents or 75 coins? If it's the latter, you could have $75 if they're all silver dollars. Either way, you should probably tell your friend you only have 75 cents, so he doesn't take advantage of you.

    Sample Problem

    Lois has two more nickels than she has quarters. She has one dollar in all. How many nickels does she have?

    First of all, here's what we know:

    (amount of money Lois has) = (amount of money Lois has in nickels) + (amount of money Lois has in quarters)

    We also know that:

    (amount of money Lois has in quarters) = 0.25(number of quarters Lois has) = 0.25q

    ...where q is the number of quarters Lois has. Where did we find that number, you ask? A quarter is worth 25 cents, you see. Ask a silly question, get a silly answer.

    Since Lois has two more nickels than she has quarters, we can also write:

    (amount of money Lois has in nickels) = 0.05(q + 2)

    Lois has one dollar total. Putting all our information together, we can rewrite our first equation:

    1 = 0.05(q + 2) + 0.25q

    Solving for q, we find q = 3. Since "she has three quarters'' doesn't answer the question "how many nickels does Lois have?'' we need to do one more step: Lois has two more nickels than quarters, so she has 3 + 2 = 5 nickels. See how it helps to take one last look at what the problem's asking you to find? Phew. We almost had a national crisis on our hands.

  • Written Inequalities

    Some word problems lead us to inequalities rather than equations, which is good, because algebra would be awfully boring if we never got to mix it up. We know what you're thinking, but don't say it.

    When we're solving an inequality to answer a word problem, we need to think about the answer before we write it down, especially for problems where answers need to be integers.

    Sample Problem

    Liam has $20. He wants to buy some comic books that cost $1.75 each. He doesn't like comic books, but his mom is forcing him to start a hobby so he doesn't spend all day, every day watching television. How many comic books can he afford to buy?

    Liam only has $20, so we need this inequality to be true:

    (1.75)(the number of comic books Liam buys) ≤ 20

    The above is true because we're assuming Liam won't be able to sweet-talk his way into any additional comic books, and that he won't go back into his wallet for a credit card. Probably a safe assumption to make.

    Let x be the number of comic books Liam buys, and translate the inequality into symbols as:

    1.75x ≤ 20

    Solving this inequality gives us some horrible non-integer between 11 and 12. It's approximately 11.4, but actually a number that causes much more of a headache to look at. Liam can't buy 12 comic books, because he doesn't have enough money. He also can't buy part of a comic book, as they usually won't let you tear out a bunch of individual pages, so he can buy 11 whole comic books. Plus, he'll still have a few cents leftover to throw into a fountain and wish he were back at home watching TV.

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