In the previous lesson we learned how to write variable and numerical expressions involving addition and subtraction. In this lesson we learn how to use multiplication and division to write expressions.

Variable expressions with multiplication and division

Numerical expressions with multiplication and division

$\frac{k}{4}$

$\frac{5m}{7}$

6•12

$\frac{36}{4}$

$\frac{5•21}{7}$

Here is a table of key words that indicate multiplication and division.

Operation	Verbal expressions (key words)	Variable expression
$\large~\times~$	2 times a number 2 multiplied by a number the product of 2 and a number twice as big as a number	$2m\mbox{ or }2\cdot~m$
$\large~\div~$	6 divided into a number a number divided by 6 the quotient of a number and 6 a number cut into 6 equal parts	$a\div~6\mbox{ or }\frac{a}{6}$

Example 1:

A grocery sells apples by the bag. There are 4 apples in each bag. How many apples will you get if you purchase n bags?

To find the total number of apples in n bags, make a table an fill in some values.

Number of bags	Total number of appl
2	8
5
9
n

If n = 7, how many apples are there altogether? [There would be 28 apples.]

If n = 12, how many apples are there altogether? [There would be 48 apples.]

Can you see that the total number of apples is always 4 times the number of bags?

If there were n bags, then we would have "4 times n" number of apples.

"4 times n" is written as 4n.

NOTE: A number right next to a variable is assumed to represent multiplication.

Example 2:

A stick is x inches long and is cut into 4 equal pieces. How long is each of the pieces?

To understand this question, first suppose you had a 78-inch long stick and you cut it into 6 equal-sized pieces. How long would each piece be?

Dividing 78 by 6 would give the lenth of each piece. $$\frac{78}{6}=13$$

Each piece would be 13 inches long.

Now replace the "78" with x and 6 with 4.

$$\frac{x}{4}\mbox{ is the length of each of the four pieces.}$$

EXPLORE!

Move the red point up and down the number line and observe how the blue point moves in relation.
What variable expression describes the blue point, if we consider the red point as $\large~x$?

Self-Check

Question 1

Imagine a sheet of stamps. Each stamp costs 35 cents. Write a variable expression for the cost of n stamps.

[show answer]

Question 2

David has 9 bags. He places an equal number of coins in each bag. If there are n coins total, write an expression for the number of coins in each bag.