Say What? Turning Words into Numbers - Getting a Handle on Whole Numbers - Basic Math & Pre-Algebra For Dummies

Basic Math & Pre-Algebra For Dummies, 2nd Edition (2014)

Part II. Getting a Handle on Whole Numbers

Chapter 6. Say What? Turning Words into Numbers

In This Chapter

arrow Dispelling myths about word problems

arrow Knowing the four steps to solving a word problem

arrow Jotting down simple word equations that condense the important information

arrow Writing more-complex word equations

arrow Plugging numbers into the word equations to solve the problem

arrow Attacking more-complex word problems with confidence

The very mention of word problems — or story problems, as they're sometimes called — is enough to send a cold shiver of terror into the bones of the average math student. Many would rather swim across a moat full of hungry crocodiles than “figure out how many bushels of corn Farmer Brown picked” or “help Aunt Sylvia decide how many cookies to bake.” But word problems help you understand the logic behind setting up equations in real-life situations, making math actually useful — even if the scenarios in the word problems you practice on are pretty far-fetched.

In this chapter, I dispel a few myths about word problems. Then I show you how to solve a word problem in four simple steps. After you understand the basics, I show you how to solve more-complex problems. Some of these problems have longer numbers to calculate, and others may have more complicated stories. In either case, you can see how to work through them step by step.

Dispelling Two Myths about Word Problems

Here are two common myths about word problems:

· Word problems are always hard.

· Word problems are only for school — after that, you don't need them.

Both of these ideas are untrue. But they're so common that I want to address them head-on.

Word problems aren't always hard

Word problems don't have to be hard. For example, here's a word problem that you may have run into in first grade:

· Adam had 4 apples. Then Brenda gave him 5 more apples. How many apples does Adam have now?

You can probably do the math in your head, but when you were starting out in math, you may have written it down:

· 4 + 5 = 9

Finally, if you had one of those teachers who made you write out your answer in complete sentences, you wrote “Adam has 9 apples.” (Of course, if you were the class clown, you probably wrote, “Adam doesn't have any apples because he ate them all.”)

Word problems seem hard when they get too complex to solve in your head and you don't have a system for solving them. In this chapter, I give you a system and show you how to apply it to problems of increasing difficulty. And in Chapters 13, 18, and 23, I give you further practice solving more difficult word problems.

Word problems are useful

In the real world, math rarely comes in the form of equations. It comes in the form of situations that are very similar to word problems.

Whenever you paint a room, prepare a budget, bake a double batch of oatmeal cookies, estimate the cost of a vacation, buy wood to build a shelf, do your taxes, or weigh the pros and cons of buying a car versus leasing one, you need math. And the math skill you need most is understanding how to turn the situation you're facing into numbers that you calculate.

Word problems give you practice turning situations — or stories — into numbers.

Solving Basic Word Problems

Generally, solving a word problem involves four steps:

1. Read through the problem and set up a word equation — that is, an equation that contains words as well as numbers.

2. remember_4c.eps Plug in numbers in place of words wherever possible to set up a regular math equation.

3. Use math to solve the equation.

4. Answer the question the problem asks.

Most of this book is about Step 3. This chapter and Chapters 13, 18, and 23 are all about Steps 1 and 2. I show you how to break down a word problem sentence by sentence, jot down the information you need to solve the problem, and then substitute numbers for words to set up an equation.

When you know how to turn a word problem into an equation, the hard part is done. Then you can use the rest of what you find in this book to figure out how to do Step 3 — solve the equation. From there, Step 4 is usually pretty easy, though at the end of each example, I make sure you understand how to do it.

Turning word problems into word equations

The first step to solving a word problem is reading it and putting the information you find into a useful form. In this section, I show you how to squeeze the juice out of a word problem and leave the pits behind!

Jotting down information as word equations

Most word problems give you information about numbers, telling you exactly how much, how many, how fast, how big, and so forth. Here are some examples:

· Nunu is spinning 17 plates.

· The width of the house is 80 feet.

· If the local train is going 25 miles per hour …

You need this information to solve the problem. And paper is cheap, so don't be afraid to use it. (If you're concerned about trees, write on the back of all that junk mail you get.) Have a piece of scrap paper handy and jot down a few notes as you read through a word problem.

For example, here's how you can jot down “Nunu is spinning 17 plates”:

· Nunu = 17

Here's how to note that “the width of the house is 80 feet”:

· width = 80

The third example tells you, “If the local train is going 25 miles per hour... .” So you can jot down the following:

· local = 25

remember_4c.eps Don't let the word if confuse you. When a problem says “If so-and-so were true …” and then asks you a question, assume that it is true and use this information to answer the question.

When you jot down information this way, you're really turning words into a more useful form called a word equation. A word equation has an equals sign like a math equation, but it contains both words and numbers.

Writing relationships: Turning more-complex statements into word equations

When you start doing word problems, you notice that certain words and phrases show up over and over again. For example,

· Bobo is spinning five fewer plates than Nunu.

· The height of a house is half as long as its width.

· The express train is moving three times faster than the local train.

You've probably seen statements such as these in word problems since you were first doing math. Statements like these look like English, but they're really math, so spotting them is important. You can represent each of these types of statements as word equations that also use Big Four operations. Look again at the first example:

· Bobo is spinning five fewer plates than Nunu.

You don't know the number of plates that either Bobo or Nunu is spinning. But you know that these two numbers are related.

You can express this relationship like this:

· Bobo + 5 = Nunu

This word equation is shorter than the statement it came from. And as you see in the next section, word equations are easy to turn into the math you need to solve the problem.

Here's another example:

· The height of a house is half as long as its width.

You don't know the width or height of the house, but you know that these numbers are connected.

You can express this relationship between the width and height of the house as the following word equation:


With the same type of thinking, you can express “The express train is moving three times faster than the local train” as this word equation:


remember_4c.eps As you can see, each of the examples allows you to set up a word equation using one of the Big Four operations — adding, subtracting, multiplying, and dividing.

Figuring out what the problem's asking

The end of a word problem usually contains the question you need to answer to solve the problem. You can use word equations to clarify this question so you know right from the start what you're looking for.

For example, you can write the question, “All together, how many plates are Bobo and Nunu spinning?” as

· Bobo + Nunu = ?

You can write the question “How tall is the house” as:

· height = ?

Finally, you can rephrase the question “What's the difference in speed between the express train and the local train?” in this way:

· express − local = ?

Plugging in numbers for words

After you've written out a bunch of word equations, you have the facts you need in a form you can use. You can often solve the problem by plugging numbers from one word equation into another. In this section, I show you how to use the word equations you built in the last section to solve three problems.

Example: Send in the clowns

Some problems involve simple addition or subtraction. Here's an example:

· Bobo is spinning five fewer plates than Nunu. (Bobo dropped a few.) Nunu is spinning 17 plates. All together, how many plates are Bobo and Nunu spinning?

Here's what you have already, just from reading the problem:

·  Nunu = 17

· Bobo + 5 = Nunu

Plugging in the information gives you the following:

· Bobo + 5 = 17

If you see how many plates Bobo is spinning, feel free to jump ahead. If not, here's how you rewrite the addition equation as a subtraction equation (see Chapter 4 for details):

· Bobo = 17 − 5 = 12

The problem wants you to find out how many plates the two clowns are spinning together. So you need to find out the following:

· Bobo + Nunu = ?

Just plug in the numbers, substituting 12 for Bobo and 17 for Nunu:

· 12 + 17 = 29

So Bobo and Nunu are spinning 29 plates.

Example: Our house in the middle of our street

At times, a problem notes relationships that require you to use multiplication or division. Here's an example:

· The height of a house is half as long as its width, and the width of the house is 80 feet. How tall is the house?

You already have a head start from what you determined earlier:


You can plug in information as follows, substituting 80 for the word width:


So you know that the height of the house is 40 feet.

Example: I hear the train a-comin’

Pay careful attention to what the question is asking. You may have to set up more than one equation. Here's an example:

· The express train is moving three times faster than the local train. If the local train is going 25 miles per hour, what's the difference in speed between the express train and the local train?

Here's what you have so far:


Plug in the information you need:


In this problem, the question at the end asks you to find the difference in speed between the express train and the local train. Finding the difference between two numbers is subtraction, so here's what you want to find:

· express − local = ?

You can get what you need to know by plugging in the information you've already found:

· 75 − 25 = 50

Therefore, the difference in speed between the express train and the local train is 50 miles per hour.

Solving More-Complex Word Problems

The skills I show you previously in “Solving Basic Word Problems” are important for solving any word problem because they streamline the process and make it simpler. What's more, you can use those same skills to find your way through more complex problems. Problems become more complex when

· The calculations become harder. (For example, instead of a dress costing $30, it costs $29.95.)

· The amount of information in the problem increases. (For example, instead of two clowns, you have five.)

Don't let problems like these scare you. In this section, I show you how to use your new problem-solving skills to solve more-difficult word problems.

When numbers get serious

A lot of problems that look tough aren't much more difficult than the problems I show you in the previous sections. For example, consider this problem:

· Aunt Effie has $732.84 hidden in her pillowcase, and Aunt Jezebel has $234.19 less than Aunt Effie has. How much money do the two women have all together?

One question you may have is how these women ever get any sleep with all that change clinking around under their heads. But moving on to the math, even though the numbers are larger, the principle is still the same as in problems in the earlier sections. Start reading from the beginning: “Aunt Effie has $732.84 … .” This text is just information to jot down as a simple word equation:

· Effie = $732.84

Continuing, you read, “Aunt Jezebel has $234.19 less than Aunt Effie has.” It's another statement you can write as a word equation:

· Jezebel = Effie − $234.19

Now you can plug in the number $732.84 where you see Aunt Effie's name in the equation:

· Jezebel = $732.84 − $234.19

So far, the big numbers haven't been any trouble. At this point, though, you probably need to stop to do the subtraction:


Now you can jot this information down, as always:

· Jezebel = $498.65

The question at the end of the problem asks you to find out how much money the two women have all together. Here's how to represent this question as an equation:

· Effie + Jezebel = ?

You can plug information into this equation:

· $732.84 + $498.65 = ?

Again, because the numbers are large, you probably have to stop to do the math:


So all together, Aunt Effie and Aunt Jezebel have $1,231.49.

As you can see, the procedure for solving this problem is basically the same as for the simpler problems in the earlier sections. The only difference is that you have to stop to do some addition and subtraction.

Too much information

When the going gets tough, knowing the system for writing word equations really becomes helpful. Here's a word problem that's designed to scare you off — but with your new skills, you're ready for it:

· Four women collected money to save the endangered Salt Creek tiger beetle. Keisha collected $160, Brie collected $50 more than Keisha, Amy collected twice as much as Brie, and together Amy and Sophia collected $700. How much money did the four women collect all together?

If you try to do this problem all in your head, you'll probably get confused. Instead, take it line by line and just jot down word equations as I discuss earlier in this chapter.

First, “Keisha collected $160.” So jot down the following:

· Keisha = 160

Next, “Brie collected $50 dollars more than Keisha,” so write

· Brie = Keisha + 50

After that, “Amy collected twice as much as Brie”:

· Amy = Brie × 2

Finally, “together, Amy and Sophia collected $700”:

· Amy + Sophia = 700

That's all the information the problem gives you, so now you can start working with it. Keisha collected $160, so you can plug in 160 anywhere you find Keisha's name:

· Brie = 160 + 50 = 210

Now you know how much Brie collected, so you can plug this information into the next equation:

· Amy = 210 × 2 = 420

This equation tells you how much Amy collected, so you can plug this number into the last equation:

· 420 + Sophia = 700

To solve this problem, change it from addition to subtraction using inverse operations, as I show you in Chapter 4:

· Sophia = 700 − 420 = 280

Now that you know how much money each woman collected, you can answer the question at the end of the problem:

· Keisha + Brie + Amy + Sophia = ?

You can plug in this information easily:

· 160 + 210 + 420 + 280 = 1,070

So you can conclude that the four women collected $1,070 all together.

Putting it all together

Here's one final example putting together everything from this chapter. Try writing down this problem and working it through step by step on your own. If you get stuck, come back here. When you can solve it from beginning to end with the book closed, you'll have a good grasp of how to solve word problems:

· On a recent shopping trip, Travis bought six shirts for $19.95 each and two pairs of pants for $34.60 each. He then bought a jacket that cost $37.08 less than he paid for both pairs of pants. If he paid the cashier with three $100 bills, how much change did he receive?

On the first read-through, you may wonder how Travis found a store that prices jackets that way. Believe me — it was quite a challenge. Anyway, back to the problem. You can jot down the following word equations:


The numbers in this problem are probably longer than you can solve in your head, so they require some attention:


With this done, you can fill in some more information:


Now you can plug in $69.20 for pants:

· jacket = $69.20 − $37.08

Again, because the numbers are long, you need to solve this equation separately:


This equation gives you the price of the jacket:

· jacket = $32.12

Now that you have the price of the shirts, pants, and jacket, you can find out how much Travis spent:

· amount Travis spent = $119.70 + $69.20 + $32.12

Again, you have another equation to solve:


So you can jot down the following:

· amount Travis spent = $221.02

The problem is asking you to find out how much change Travis received from $300, so jot this down:

· change = $300 − amount Travis spent

You can plug in the amount that Travis spent:

· change = $300 − $221.02

And do just one more equation:


So you can jot down the answer:

· change = $78.98

Therefore, Travis received $78.98 in change.