Fractions

Procedures for multiplying fractions

How do we think of multiplication problems?

In these examples we can see that x and of have the same meaning. Thinking of multiplication as of can help us picture a model when solving multiplication problems.

we could say, 3 groups of 6 = 18

we could write, 3 x 6 = 18

we could say, a half of 6 is 3

we could write, 1/2 of 6 = 3 or

1/2 x 6 = 3

we could say, a quarter of a dozen

we could write, 1/4 of 12 = 3 or

1/4 x 12 = 3

Multiplying a whole number by a fraction

Example 2: 3 x 4/7

If we count the green shaded areas below we can see that we have 12 sevenths

Using the area model to multiply a fraction by a fraction

Example 1: Using the area model to illustrate 1/5 x 3/4= 3/20

The method used in this example is sometimes referred to as the unit square method.

Click image to see the slideshow

Multiplying mixed numbers & improper fractions

Example 3: 1 1/2 x 5 = 7 1/2

If I need 5 times 1 and a 1/2 metres of ribbon to make a dress, how much ribbon do I have to buy?

We can use 3 methods as set out below to solve this problem.

Question	Setting Out	Thinking
5 x 1 1/2	method 1: = 5 x 1 + 5 x 1/2 = 7 1/2	I think of the fractions and the whole numbers separately, 5 x 1 1/2 = 5 x (1 + 1/2) First I multiply the whole number and then I multiply the fractional part. This is a use of the distributive law.
	method 2: = 5/1 x 3/2 = 15/2 = 7 1/2	I change the 5 into a fraction, 5/1 I change the mixed number, 1 1/2, into the improper fraction, 3/2, 1 = 2/2, 2/2 +1/2 = 3/2 I then multiply the fractions = (5x3)/(1x2) = 15/2
	method 3: = 5 x 3/2 = 15/2 = 7 1/2	I change the mixed number, 1 1/2, into the improper fraction,3/2, 1 = 2/2, 2/2 + 1/2 = 3/2 I think of the question as '5 groups of 3 halves', (5 x 3)/2 I then multiply the whole number by the improper fraction of 3/2

Therefore, I need to buy seven and a half metres of ribbon.

Below are some more examples of multiplying mixed numbers.

Example 4: 2 1/2 x 1 3/4 = 4 3/8	Example 5: 1 1/3 + 2 1/5 = 2 14/15
Click image to see the movie	Click image to see the movie

General rule for the multiplication of fractions

Steps for multiplication of fractions

To multiply fractions:

Step 1: Convert mixed numbers to improper fractions, if applicable

Step 2: Multiply the numerators together, multiply the denominators together

Step 3: Simplify the answer if required.

Advanced Procedures

Multiplication with cancelling:

Cancelling is a procedure which we can use to make multiplication of fractions easier for us. We do not have to use cancelling - it just makes the process more efficient. The word cancelling is not really a helpful description of what takes place and can lead the unsuspecting into indiscriminately crossing out numbers!

Cancelling actually means dividing the numerator and denominator by the same number until a fraction is renamed in its simplest form.

Example 6: 5/2 x 2/3 = 5/3

Question

Setting Out

Thinking

using cancelling:

I can see that the numerator and the denominator both contain a factor of 2. I can cancel the 2s: in effect, this step divides both the numerator and the denominator by 2 . This does not change the value of the fraction.

without using cancelling:

I multiply the fractions as usual. I can see that the answer (10/6) is equivalent to and can be simplified to (5/3). If a factor is not cancelled out before multiplication occurs, it will need to be cancelled to simplify the final answer.

Example 7: multiplication with cancelling, 8/3 x 15/6 = 6 2/3

Click the image to see the movie

Quick quiz

1.	Use the unit square method to find the answer to 2/3 x 5/8. (Think of the question as 2/3 of 5/8 and use a unit model to solve it)

2.	a)	b)	c)
	d)	e)	f)

3.	The cancelling technique will make it easier to answer these questions. Use this technique if you feel confident.
	a)	b)	c)
	d)	e)	f)

To view the quiz answers, click here.