Apply Cancellation to Units

Now, let’s consider some simple multiplication problems that include units.

Four boxes contain 8 marbles each. How many marbles are there altogether?

This problem statement gives us two pieces of information. We will represent each as a fraction.

4 boxesFour boxes 1

8 marblesEight marbles per box 1 box

Now we can multiply these two fractions:

4 boxes 8 marbles 4 boxes * 8 marbles 1 1 box 1 * 1 box

Next, we can group the units together:

4 boxes 8 marbles 4 boxes * 8 marbles 4 * 8 boxes marbles 1 1 box 1 * 1 box 1 * 1 box

=

= =

*

*( )

( ) )(

)(

Apply Cancellation to Units

The next step is to use cancellation to reduce this fraction. Note that in this particular example, the numerical part of the fraction is already 1, so we only need to concentrate on reducing the units. We see that the unit “box(es)” is common to both the numerator and the denominator, so we can cancel them.

4 boxes 8 marbles 4 * 8 boxes marbles 4 * 8 marbles 1 1 box 1 * 1 box 1

Let’s walk through one more demonstration. Cookies are on sale, priced at 3 cookies for 2 dollars. If we spend 6 dollars, how many cookies can we buy?

Use fractions to represent the information in the problem:

3 cookies3 cookies for 2 dollars 2 dollars

6 dollars6 dollars 1

Now, we multiply our two fractions:

* = = = 32 marbles( ) )(

Apply Cancellation to Units

3 cookies 6 dollars 3 cookies * 6 dollars 2 dollars 1 2 dollars * 1

3 * 3 * 2 cookies dollars 2 dollars

9 cookies 1

The goal of the following exercises is not merely to correctly answer the question. Instead, the key is to correctly deal with fractions and units.

1 ) There are eight bottles in the pantry. There are two liters of soda in each bottle. How many liters of soda are in the pantry?

Use fractions to represent the information in the problem:

__8 bottles____ ___2 liters____ 1 bottle

* =

=

= 9 cookies

( ) )(

=

Apply Cancellation to Units

Multiply the fractions, carrying the units through the calculation and cancelling appropriately.

_8 bottles__ _2 liters___ 8 * 2 bottles liters 1 1 bottle 1 bottle

2) There are three bunches of four bananas each. How many bananas are there altogether?

Express the problem information as fractions:

__3 bunches____ __4 bananas____ 1 1 bunch

Multiply the fractions, carrying the units through the calculation and cancelling appropriately.

3 bunches _4 bananas_ 3 * 4 bunches bananas 1 1 bunch 1 bunch

* =( ) )( = 16 liters

* =( ) )( = 12 bananas