Find practice problems and solutions for these concepts at: The Keywords of Word Problems Practice Problems.
Now that you've completed the pretest, we're ready to start breaking down some word problems. Often, the hardest part of a word problem isn't the computation—it's figuring out what operation to use. In this chapter, we'll look at the keywords that signal addition, subtraction, multiplication, and division.
In math problems, a keyword is a word that signals what operation to use. A problem may contain more than one keyword, especially if the word problem requires more than one operation to solve.
Fuel for Thought
An operation is a procedure that is applied to one or more numbers or variables. Addition, subtraction, multiplication, and division are all operations. The symbol for each operation is called an operator. The symbols +, –, ×, and ÷ are all operators. Most operations involve two or more operands. An operand is a number or variable used in an operation. In the number sentence 4 + 5 = 9, 4 and 5 are operands, + is the operator, and the operation is addition.

Addition
What kinds of words signal addition? How would you describe addition to someone who had never heard of it? In order for a word problem to be about addition, the author of the word problem needs to describe addition in words, so that we'll know to use that operation. If you can think of how to describe addition in words, you'll likely use some of the same words a test maker may use to write a word problem.
An addition problem is made up of two or more addends and a sum. The addends are the numbers that are being added, and the sum is the result of adding the addends together, or the answer. In the number sentence 3 + 1 = 4, 3 and 1 are the addends, and 4 is the sum.
You could say, "Addition is combining two or more numbers together to form a total." Or you may say, "Addition is the sum of a few numbers." You may also say, "Addition is putting one quantity and another together," or "Addition is one number plus another number." Each of these sentences contains one or more keywords.
Here are those sentences again, with the keywords highlighted:
Addition is combining two or more numbers together to form a total.
Addition is the sum of a few numbers.
Addition is putting one quantity and another together.
Addition is one number plus another number.
The words combine, together, total, sum, and, and plus are all keywords that represent addition. When you thought about how to describe addition, you may have used these words or words like altogether, increase, both, or more.
Example
Heather's cat weighs 8 pounds, and Serge's cat weighs 11 pounds.
How much do the two cats weigh in total?
This word problem asks us to find how much the cats weigh in total. Total is a keyword that signals addition, so we must add the weights of the two cats. 8 + 11 = 19 pounds. The two cats weigh 19 pounds in total.
Subtraction
Now let's think about subtraction. A subtraction problem is made up of a minuend, a subtrahend, and a difference. The number from which we are subtracting is the minuend, the number being subtracted is the subtrahend, and the result of or the answer to the subtraction problem is the difference. In the number sentence 3 – 1 = 2, 3 is the minuend, 1 is the subtrahend, and 2 is the difference.
How would you describe subtraction? Here are a few possible definitions:
Subtraction is taking away one number from another.
Subtraction is the difference between two numbers.
Subtraction is one number minus another number.
Subtraction is the operation by which you decrease a number or quantity.
The keywords that signal subtraction are highlighted in the previous sentences. Your definition may have used those words or words like left, more than, less than, fewer, or remain.
Example
Heather's cat weighs 8 pounds, and Serge's cat weighs 11 pounds.
What is the difference between the weight of Serge's cat and the weight of Heather's cat?
This example uses the same numbers as the last example, but the keyword difference tells us that, this time, we need to use subtraction. The difference between the weight of Serge's cat and the weight of Heather's cat can be found by subtracting the larger weight from the smaller weight: 11 – 8 = 3 pounds.
Inside Track
When working on a word problem, underline the keywords. Some word problems can be long either because they contain information not needed to solve the problem or because they require more than one operation to solve. By underlining the keywords, we can help ourselves remember which operation(s) we need to solve the problem.

Multiplication
What keywords or phrases signal multiplication? Write two different sentences that describe multiplication.
You may have written a sentence or two like the following:
Multiplication is one number times another number.
Multiplication is the product of two or more numbers.
Multiplication is a factor times another factor.
A multiplication problem is made up of two or more factors and a product. The numbers that are being multiplied are the factors, and the result of the multiplication problem is the product. In the number sentence 4 × 5 = 20, 4 and 5 are the factors, and 20 is the product.
Caution!
The keyword increase can also signal multiplication, even though it can signal addition, too. Later, we'll look at how to tell when increase means addition and when it means multiplication.

Multiplication can be a tougher operation to spot than addition or subtraction. When a quantity is used for each or for every amount of another quantity, we'll need to use multiplication.
Example
If Heather's cat sleeps for 13 hours each day, how many hours does it sleep over five days?
The problem tells us how many hours Heather's cat sleeps in one day, or each day. Since her cat sleeps 13 hours in one day, to find how many hours it sleeps over five days, we must multiply the number of hours it sleeps in one day by the number of days: 13 × 5 = 65 hours.
Caution!
Increase isn't the only keyword that could mean addition or multiplication. That last example could have been trickier if it had read: "If Heather's cat sleeps for 13 hours each day, how many total hours does it sleep in five days?" That sentence uses the word total, even though we need to use multiplication and not addition to solve the problem. In this example, we had the keyword each to help us determine that multiplication was the right operation. Remember, multiplication is like repeated addition. 13 × 5 could also be written as 13 + 13 + 13 + 13 + 13. That's why multiplication sometimes seems like a total.

Division
Words like quotient, share, percent, out of, and average are all hints that tell us to use division. Words like per and each can signal division, but can also signal multiplication. Spotting those words can narrow the possible operations to multiplication and division, and then we must use the other words in the problem to help us choose which operation to use.
A division problem is made up of a dividend, a divisor, and a quotient. The dividend is the number being divided, the divisor is the number that divides the dividend, and the quotient is the result of division. In the number sentence = 7, 56 is the dividend, 8 is the divisor, and 7 is the quotient.
Example
Serge bought 90 ounces of cat food. If his cat eats 6 ounces of food per day, for how many days can Serge's cat eat before Serge must buy more cat food?
The keyword per tells us to use either multiplication or division. Which should we use? We're given the total number of ounces of cat food and the number of ounces that are eaten each day. We need to find a number that is less than the total number of ounces, since Serge's cat eats 6 of those ounces every day. Division will give us a number that is less than 90. By dividing the total number of ounces by the number of ounces eaten each day, we can find the number of days the cat food will last: 90 ÷ 6 = 15 days.
If that problem had been worded a little differently, we may have needed to use multiplication instead of division.
Example
Serge's cat eats 6 ounces of food per day. How much cat food must Serge buy to feed his cat for 12 days?
This word problem also contains the keyword per. However, this time, we're given the number of ounces eaten per day and the number of days. We're looking for the amount of cat food eaten for all of those days combined. We need to multiply the number of ounces eaten per day by the number of days: 6 × 12 = 72 ounces.
Pace Yourself
Write a word problem of your own without using any of the keywords you've read in this chapter. What operation is required to solve your word problem? How could a student know what operation to use by reading your word problem? Did you just discover a new keyword?

Deciding Between Multiplication and Division: Which Answer Makes More Sense?
Multiplication and division word problems often sound alike. When trying to decide between multiplication and division, think about fact families. For instance, these facts are all part of the same fact family:
4 × 6 = 24
6 × 4 = 24
24 ÷ 4 = 6
24 ÷ 6 = 4
A fact family is a group of related equations that use the same numbers. A fact family usually pairs addition and subtraction equations, or multiplication and division equations. The number sentences 5 – 3 = 2, 5 – 2 = 3, 2 + 3 = 5, and 3 + 2 = 5 are all members of the same fact family.
We've seen how words like each and per can signal multiplication or division. When you see those words, write a number sentence using multiplication and number sentence using division. Then, solve the multiplication sentence and decide if the answer you find makes sense given the situation described by the word problem.
Example
Michelle gives $24 to her four children to spend at an arcade. If each child receives the same amount of money, how much does each receive?
The only keyword in this problem is each. We are given the numbers 24 and 4. Either we need to multiply 24 by 4, or we need to divide 24 by 4:
$24 × 4 = $96
$24 ÷ 4 = $6
Given the situation in the word problem, which answer makes more sense? Michelle gave $24 to her children. If she had only one child, that child would receive all $24. Since she has more than one child, each child will receive less than $24. The answer to this problem will be less than $24, because $24 needs to be shared evenly among four children. Multiplication leads to a larger number, which doesn't make sense. We must divide to find how much each child receives. $24 ÷ 4 = $6.
Let's look at another word problem that comes from the same fact family.
Example
Six friends each volunteer for four hours at the local community center. How many hours do they volunteer in all?
Again, this word problem uses the keyword each. We are given the numbers 6 and 4. We could divide 6 by 4, or we could multiply 6 by 4:
6 ÷ 4 = 1.5
6 × 4 = 24
Apply the same strategy: which answer makes more sense? Each friend volunteers for four hours, which means that one friend volunteers for four hours, and two friends would volunteer for more than four hours together. We are looking for an operation that leads to a number that is larger than four. Since each friend volunteers for four hours, we must multiply the number of friends by 4 to find how many hours they volunteer in all: 6 × 4 = 24 hours.
Inside Track
When deciding between multiplication and division, think about the given situation in terms of one. What is the cost for one person? What is the amount that one person has? How much will one person receive? Then, think about the quantity given in the problem. Will you use the amount that one person has to use in order to find out how much a group of many has to use? If so, you will need to multiply. If you can't tell how much one person has because you are given a total, you will need to divide.
Look again at the last two examples. The first does not tell you how much one child receives, but gives you a total and asks you to find how much one child receives. Division was the operation to use. The second example gives you how much one friend volunteers, and you can use that number to find how many hours six friends volunteer. Multiplication was the operation to use in that example.

Pace Yourself
Write a multiplication word problem and a division word problem using the following fact family. How would a student know which operation to use for each of your word problems? How would you explain the correct way to solve each problem to a student?
7 × 8 = 56
8 × 7 = 56
56 ÷ 7 = 8
56 ÷ 8 = 7

Summary
Keywords can help us decide which operation to use to solve a word problem. We learned which of the four major operations each keyword signifies. In the next chapter, we'll work out a stepbystep process for solving word problems.
Find practice problems and solutions for these concepts at: The Keywords of Word Problems Practice Problems.