Peter is presenting a poster about beetles for science class. The average dung beetle is \begin{align*}\frac{1}{2}\end{align*} to 1 inch long. He is also researching the Goliath beetle. This is a huge beetle at \begin{align*}4 \frac{1}{2}\end{align*} inches long. Peter wants make a comparison of the two beetles. What is the difference between their lengths?

In this concept, you will use the problem solving strategy: choose an operation.

### Problem Solving Plan with Choosing an Operation

When solving a problem, it is very important to read and understand the problem in order to figure out which operation you need to find a correct answer. This is especially important with word problems. You can identify key words in a word problem that let you know which operation to use. The four operations are addition, subtraction, multiplication and division. Here is a description of each and some key words that you can look for when reading a problem.

**Addition** – a problem where two quantities are combined. Key words: sum, total, in all, combined, and altogether

Let's look at an example.

John has 8 books and Mary has four books, how many books do they have in all?

"In all" are the key words that tells you to use addition.

**Subtraction** – a problem that compares two quantities. One quantity is being removed from another quantity. Key words: left, left over, take away, difference, how many more, how many less, more than, less than, compare, greater than, increased by, and decreased by

Let's look at an example.

Karen had fifteen records, but she gave three to her friend Louise. How many does she have left?

"Left" is the key word that tells you to use subtraction.

**Multiplication** – word problems using repeated addition. The key words are similar to addition, but pay attention to the addition of the same group multiple times. Key words: total, in all, each, every, per, how much, at this rate, and of

Let's look at an example.

What is one-half of 18?

"Of" is the key word that tells you to multiply.

**Division** – Division problems generally involve a situation in which a single quantity is split up into many equal-sized parts. Key words: split, divide, shared, equal size, average, groups, and per

Let's look at an example.

Chris has fourteen marbles. He has divided them into two groups. How many marbles are in each group?

In addition to the word "divide," "group" is the key word that tells you to divide.

### Examples

#### Example 1

Earlier, you were given a problem about Peter and the beetles.

Peter is comparing the difference between the size of the dung beetle, between \begin{align*}\frac{1}{2}\end{align*} and 1 inch long, and the Goliath beetle, \begin{align*}4 \frac{1}{2}\end{align*} inches long. The key word "difference" tells Peter that he will need to subtract to find the difference.

\begin{align*}4 \frac{1}{2} - \frac{1}{2} = 4\end{align*} inches differences for a small dung beetle

\begin{align*}4 \frac{1}{2} - 1 = 3 \frac{1}{2}\end{align*} inches difference for a large dung beetle

The difference in size ranges from \begin{align*}3 \frac{1}{2}\end{align*} inches to 4 inches.

#### Example 2

Identify the key words, the operation used, and the solution to the problem.

Kyle has fourteen nickels. He found four more nickels in his pocket. How many nickels does he have? How much money are the nickels worth in all

There are two questions in this problem. The first questions is looking for the total number of nickels. The second question is looking for the value of all the nickels.

Start with the first question. The key words "in all" tells you to add up the nickels.

14 nickels + 4 nickels = 18 nickels

Then, there is a second part of the problem. A nickel is worth 5 cents. To find the value of 18 nickels, you can add 5 cents 18 times. However, notice that this involves repeated addition. Multiply the value of a nickel by 18.

5 \begin{align*}\times\end{align*} 18 \begin{align*}=\end{align*} 90

The key word is "in all." The operations used were addition and multiplication. Kyle has 18 nickels, which is equal to 90 cents.

#### Example 3

Identify the operation indicated by the key words: "in all."

The key words indicate addition or multiplication.

#### Example 4

Identify the operation indicated by the key words: "left over."

The key words indicate subtraction.

#### Example 5

Identify the operation indicated by the key words: "split up."

The key words indicate division.

### Review

Identify the key words, the operation used, and the solution to each problem.

- Clara has 30 dollars. If she splits it into 5 equal groups, how many dollars will each group have?
- If Clara doubles this number and then splits it into 5 equal groups, how many dollars will each group have?
- What if she starts with 24 dollars? How does this change the division?
- In 5 weeks, Bo made 300 dollars. What was the average amount Bo made per week?
- In 10 weeks, at this rate, how much money will Bo have made in all?
- If Bo made 600 dollars in five weeks, what is the average amount made per week?
- Bob has 45 dollars. Clara has 23 dollars. What is the difference between the amount of money Bo and Clara have?
- Lakshmi reads 20 pages per hour. At this rate, how many pages will she read in 7 hours?
- Bonnie has 85 hair barrettes. Clara has 43 hair barrettes. What is the total number of hair barrettes?
- If Bonnie decided to give 5 of her hair barrettes away to Joanne, how many would Bonnie and Clara have left?
- Each box has 12 bottles. How many bottles are in 15 boxes?
- Five equal-sized boxes weigh 40 pounds. How much does each box weigh?
- Magda had 42 fish. She gave 16 of them to Peter. How many fish did Magda have left?
- Yusef has a \begin{align*}37 \frac{2}{3}\end{align*} inch long board. If he cuts it into 9 equal sized pieces, how long will each piece be?
- If Yusef cuts the same board into 18 pieces, how long will each piece be?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.18.