Study Guide

Word Problems - Expressions

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Expressions are strings of mathematical symbols that describe quantities. Want to know how many apps there are for the iPhone? There's an expression for that.

Expressions consist of mathematical operations (+ , – , ×, ÷), numbers (1, 2, 3, etc.), parentheses (( )), and variables (x, y, z, etc.). Many different English phrases translate into each of these mathematical symbols. You'll find as we go along that there is a distinct relationship between between math and English. It's like they're married: they have a lot in common and you'll often find them together, but each speaks a totally different language.

English is ambiguous in that one word can have two or more meanings, while mathematics is precise and unambiguous. In other words, if we write ×, we mean ×. If you come across the word "times" in English, it could be talking about the time of day, measuring someone's sprinting speed, or referring to a newspaper. English can sometimes be harder to keep up with than a frazzled 6-year-old. Calm down, Billy. Now tell us, where did Shaggy dig up a human femur again?

We can think of translating English into mathematical symbols as clearing away the ambiguity so we can tell what's going on. Lifting the fog. Rubbing the side of our fist across the shower door.

  • Addition

    Any English phrase that makes it sound like we're adding things together means we're...well, adding things together. Thing + thing + thing + thing. We can abbreviate any such phrase using a + sign.

    Here are some phrases that translate to addition. We don't need to think hard to translate these, since the + symbol is "the plus symbol." We told you we'd start small. Enjoy it while it lasts.

    • The sum of x and two is x + 2.
    • Five plus one is 5 + 1.

    Here are some more phrases that translate into addition. For these, it can be helpful to think of the number line. In fact, it can often be helpful to think of the number line even when you're not doing anything math-related. Why, we often take walks through the park, throwing bread crumbs to the birds and gazing around, taking in the scenery and thinking of the number line. But we digress. 

    • Three added to four means 4 + 3.

    Even though the phrase 3 + 4 gives us the same result, we can't use it because we're starting with 4 and adding 3. Sheesh. Talk about nitpicky.

    We start with four, and then add three:

    Here are some phrases that are less obvious, but which make sense if you think of baskets of apples. Not only baskets of apples, of course. That would be silly. Although you can never have too many apples. We love apples; apples are the best. Not that we're necessarily pro-apple or that we're endorsing them on behalf of a major apple conglomerate. Uh, this is awkward. You kind of backed us into a corner on this one.

    • Six combined with four means 6 + 4.

    If you have a basket of 6 apples and a basket of 4 apples, and you combine the apples in the two baskets into one, you have 6 + 4 = 10 apples. Which of the two baskets do you put them all in? That's the real question. You don't want to be accused of favoritism.

    •  Six together with four also means 6 + 4.

    A basket of 6 apples together with a basket of 4 apples gives you 6 + 4 = 10 apples. These apples won't always be together, however. We see an emotional break-up in their near future.

    •  The total of six and four means 6 + 4.
  • Subtraction

    Any English phrase in which one quantity is taken away from another translates into a mathematical expression with a minus sign. You've seen those self-help books about dealing with loss? Those are probably riddled with minus sign implications.

    Here are some phrases that can be abbreviated with the – sign.

    • Five minus three means 5 – 3.
    • Ten take away four means 10 – 4. If you have ten brownies and you take away (eat) four, you have 10 – 4 = 6 left. Good luck explaining this to your nine brothers and sisters.
    • Five subtract two means 5 – 2.
    • Sixty-four decreased by (or diminished by or made smaller by or reduced by) two means 64 – 2. On the number line, we start at 64 and move two places to the left. In time with the music, if you can.  

    • The difference between 8 and 6 means 8 – 6. 

    When talking about subtraction in English, we can play with the order in which we talk about the quantities involved. We can first say how many things we start out with ("start with five pencils, take away two''), or we can first say how many things we'll be taking away ("take away two of the five pencils''). In English, either order is fine as long as we're consistent with what we're saying. English is flexible like that. You should see it do a center split.

    In mathematics, however, the rules are stricter. English is the lenient substitute teacher who wants to make it through the day alive, while math is a regular teacher who won't stand for you flying paper airplanes across the room. In every mathematical expression, we must begin by saying how many things we start out with, and then how many things we'll take away.

    (amount we start out with) – (amount to take away)

    Except with more numbers and fewer words. You get the picture.

    Below are some English phrases that mean "subtraction." Not ones you'll hear as often as "have a nice day" or "do you want fries with that?" but English phrases nevertheless. Each starts by saying how many things we'll be taking away. The number line can be helpful for figuring out which number should be written first in the mathematical translation. Oh, number line. You're always there for us when we need you. 

    • Four less than seven means 7 – 4. On the number line, we start at 7 and move 4 spaces to the left:

    •  Four fewer then seven also means 7 – 4.
    • Two smaller than five means 5 – 2. However, if someone ever phrases something that way, you have our permission to smack them. With your eyes. We start at 5 on the number line and move two spaces to the left:              
    •  Two subtracted from five also means 5 – 2.

    Whenever we're translating English into mathematics and subtraction is involved, we need to identify

    • which number is the amount we start out with, and
    • which number is the amount we'll take away. 

    There's a big difference between starting with 10 cookies and having someone take 9 of them away, and starting with 9 cookies and having someone take 10 of them away. We hate it when someone leaves us with negative cookies. How will we ever put on our winter weight?

  • Multiplication

    Some English phrases are clearly talking about multiplication. You hear that, phrases? We can read through your subtext.

    • The product of 3 and 4 is 3 × 4.
    • Three times five is 3 × 5.
    • Four multiplied by seven is 4 × 7.

    Also, any English phrase that talks about taking multiple copies of a number is actually talking about multiplication. Doesn't matter if they're in color or in black and white, stapled or otherwise.

    • If we double the number 4 we get 2 × 4.
    • If we triple the number four we get 3 × 4.
    • If we quadruple the number four we get 4 × 4.

    "Quadruple'' isn't used as much as "double'' or "triple,'' but if you do see it, now you'll know. "Quad'' means "four,'' like in the words "quadruped'' (four-legged animal), "quadriceps" (four-part muscle), or "quadrimom" (four-headed mother).

  • Division

    Sometimes we have a direct translation from English to math, since the symbol ÷ abbreviates the phrase "divided by.'' None of that shady "together with" or "product of" business.

    Like so: five divided by three means 5 ÷ 3.

    Here are some phrases that aren't quite as direct, but still straightforward. In other words, they may not make eye contact with you, but you still believe what they're saying.

    With these phrases, the first number mentioned goes on top of the line, while the second number mentioned goes below the line. (The second number would totally win in a limbo competition.)

    • The ratio of six and seven is .
    • The quotient of seventy and thirteen is .

    We can think of the word of as meaning either multiplication or division. How's that for confusing? Or, put another way, how's of that of for of confusing?

    Sample Problem

    What is one-third of seven?

    If we translate of as multiplication, we get .

    If we translate of as division, we get .

    Thankfully,  and  mean the same thing. As the saying goes,  of one,  of another. Although these two translations look different, they give us equivalent expressions.

    When translating the word of, look at the numbers and other words involved to decide if it's more appropriate to translate as multiplication or division. Yes, you'll need to use some logical reasoning here; it won't always be spelled out for you. When words like one-half, one-third, or one-fourth are floating around next to the of, you can think of this as multiplication by a fraction, or you can choose to translate it as a division problem. We'll even spell it out for you: A D-I-V-I-S-I-O-N P-R-O-B-L-E-M.

  • Parentheses

    When evaluating algebraic expressions, we evaluate things inside parentheses first. Think of the parentheses as two bananas that are quickly ripening, and you need to down them before they turn brown and go bad. You can think of them as apples, too, but they look much more like bananas. When translating from English into math, we need to be careful to put parentheses in the correct places. If anyone else ever tells you they know where you can put your parentheses, they're probably being rude.

    If an English phrase says to perform first one operation and then another, we can put parentheses around the operation that's performed first. We need to take care of our browning bananas, you see.

    Sample Problems

    If we triple a number and then add eight, we get (3x) + 8.  

    This is an example of a situation where we can get away with ditching the parentheses, since multiplication is performed before addition anyway. The answer 3x + 8 is also fine. The parentheses just make it look so much more official.

    If we add eight to a number and then triple it, we get 3(x + 8).  

    Here we do need the parentheses, or we would get 3x + 8 again, which is different from 3(x + 8).

    Often, a problem won't come right out and say "first you do this, then you do this other thing.'' Some of them are quite shy and will be tight-lipped even when you plead with them to tell you. Therefore, we need to reason out for ourselves which operation is performed first. Thanks a lot, problem. Some help you are.

  • Variables

    When a problem contains a word for an unknown number or quantity, we use a variable to represent that unknown number or quantity. Just replace those unknowns with x, or y, or z, or just about any other letter. We could also replace them with some other variable, but these are the three that get the most play. They must have great agents.

    Here are some English phrases translated into mathematical symbols.

    • "A number plus four'' translates to x + 4.
    • "A quantity doubled'' translates to 2x.
    • "One third of an amount'' is .
    • "Two less than a value'' is x – 2.

    Sometimes a problem already contains a variable and asks for some other quantity in terms of that variable. In that case, we don't need to plug in and solve anything. Instead, we only need to describe the quantity using an expression that has the variable in it. It comes in handy when your mother storms into your room demanding to know how many chocolate chip cookies you ate, and you can tell her, "x – 3." She can't possibly deduce from that remark how many you ate, and you'll be off the hook. We promise.

    Sample Problem

    Alice has x dollars. Tanya has 3 dollars more than Alice. How many dollars does Tanya have?

    Tanya has (the amount Alice has) + 3, which is x + 3 dollars.

    This may actually be good news for Alice. She'll be in a lower tax bracket.

    Sometimes a problem doesn't come right out and say there's a variable, but there's still one hiding in there somewhere. When all else fails, check under the bed and in the closet.

    Sample Problem

    Dwight has some brownies, but Bill has three times as many brownies. Lucky Bill. Express the number of brownies Bill has in terms of the number of brownies Dwight has. Not to make Dwight feel bad about it or anything, just so's we know.

    Bill has 3(the number of brownies Dwight has). That's kinda obnoxious to write out, though.

    We need a variable to express the number of brownies that Dwight has. Since the problem didn't tell us what letter to use, we can use whatever letter we want. Let's use D for "Dwight." Bet you didn't see that coming. Then we can answer the question like this:

    "Let D be the number of brownies Dwight has. Then Bill has 3D brownies.''

    Wow. They're so real-looking you could almost reach out and touch them.

    When a problem doesn't tell us what letters to use for variables, we can pick our own. This situation is great for us, because we have control issues. When we do this, it's very, very, very important to write down—somewhere, anywhere—the meanings of the letters that we're using. Otherwise, we might arrive at the end of the problem and not remember if x is height, speed, or the number of peanuts per mile. One would hope that you'd never forget something as important as the variable that stands for peanuts per mile, but stranger things have happened.

    If a student answered the previous problem by saying "Bill has 3D brownies'' but never bothered to say what D was, someone could be confused. Either the teacher trying to read and grade the homework, or the student trying to use the homework to study for a test later. With enough time (and brownies) you can probably figure it out, but why leave it to chance?

    This is one of those little details that can make math way easier than it would be otherwise. You'll be on your way to the next problem, while your friends are busy trying to figure out what x means. Man, there's nothing you love more than leaving your friends in the dust.

    Be Careful: Whenever you introduce a new variable, write down what that variable means so you can remember when you come to the end of the problem. Don't trust your brain to remember. It has failed you before, and it will fail you again.

  • Translating Slowly

    Don't you hate it when someone's translating something from another language and they go so quickly that you can't keep up? We hate that, too. We also hate it when we need to go from English to math in a hurry. We prefer going slowly, piecing together a little bit at a time. We don't need to go from a paragraph straight to a symbolic expression, and it's often easier not to. Ne comprenez-vous?

    Sample Problem

    What is the surface area of a rectangular box with a lid?

    Let's think about this with common sense. Fortunately, we kept some in reserve for exactly this moment. To find the surface area, we need to add the surface area of the top, the surface area of the bottom, and the surface areas of the four sides. Writing this partially in symbols, we get:

    (surface area of top) + (surface area of bottom) + (surface areas of sides)

    To find the surface areas of the top, bottom, and sides, we'll need variables for the dimensions of the box. Let's use h for height, w for width, and l for length. It's a good idea to label these in the picture, too. Don't worry that this will mess up the box. We weren't planning to enter it into any art contests anyway.

    The surface area of the top and the surface area of the bottom are each lw, which comes less from common sense and more from the memorization of an uber-useful formula. We can now translate a bit more into symbols:

    lw + lw + (surface areas of sides)

    All we have left to worry about are the surface areas of the sides. Worry we will, until we have it figured out. We're perfectionists like that. Also, we're Yoda. You had no idea.

    The two sides on the left and right ends each have surface area wh, and the front and back sides each have surface area lh. Now we can finish translating to get:

    lw + lw + wh + wh + lh + lh

    Nice. We didn't even need to use Google Translate. Finally, we tidy up a little, because this equation is a mess. Where was it raised, a barn?

    The surface area of the box is:

    2lw + 2wh + 2lh

    Translating from English to math a bit at a time can make the work take a little longer, but if it helps you find the right answer consistently, it's probably worth it. When we say "probably," we mean "definitely." We were being sarcastic. There's a first time for everything.

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