Multiply Fractions (advanced)

$\require{cancel}$Multiplying fractions is probably the easiest fraction operation to do: you simply multiply the numerator times the numerator and the denominator times the denominator.  For more details, see lesson Multiply Fractions (basic).

But, what do you do when the numbers just get too big?  For example: 

$\dfrac{6}{20}\times\dfrac{8}{12}=\dfrac{6 \times 8}{20 \times 12}=\dfrac{48}{240}$

Although people who know their times tables could reduce this fraction pretty quickly, the big numbers leave lots of room for multiplication errors and students who don't know their time tables well will take forever to simplify fractions of this size by 2s and 3s.  

$\dfrac{48}{240}=\dfrac{48\div 2}{240\div 2}=\dfrac{24}{120}=\dfrac{24 \div 2}{120\div 2}=\dfrac{12}{60}=\dfrac{12 \div 2}{60\div 2}=\dfrac{6}{30}=\dfrac{6\div 2}{30\div 2}=\dfrac{3}{15}=\dfrac{3\div 3}{15\div 3}=\dfrac{1}{5}$

That process can be torture to watch.

Moreover, when students start multiplying fractions with varibales and rational expressions, they can get really long and onerous. 

So, we want to teach students to cross cancel.  Cross cancelling means reducing the fractions before they get multiplied together. You can reduce any fraction by dividing the numerator and the denominator by the same number (see lesson Simplifying Fractions). The commutative property of multiplication means no matter what order the factors are multiplied in, you come out with the same answer. So, we can simplify fractions at any point in the process. When we reduce fractions before we multiply, the numbers are smaller and easier to manage.

When multiplying fractions, not only can you reduce individual fractions, but, because you are going to multiply the numerators and denominators together, you can reduce a numerator by the other fraction's denominator, and vice versa. This process is called cross-cancelling.

To cross cancel, you find numbers that are diagonal from each other (one should be a numerator and one a denominator) that can be divided evenly by the same number.  Divide them both by that number and replace them with the answers to those division problems. Voila! Smaller numbers to multiply! And remember, you can simplify any fraction in a fraction multiplication problem. So, bottom line, if any numbers on the top of a fraction multiplication problem can be divided by the same number as any number in the bottom of a fraction multipication problem -- reduce by that number!

Example: 

$$\eqalign{\dfrac{6}{20}\times\dfrac{8}{12}=&\\\dfrac{\overset{1}{\bcancel{6}}}{20}\times\dfrac{8}{\underset{2}{\bcancel{12}}}=& \quad &&\text{Divide 6 and 12 by 6}\\\dfrac{1}{\underset {5}{\bcancel{20}}}\times \dfrac{\overset{2}{\bcancel{8}}}{2}=& &&\text{Divide 20 and 8 by 4}\\\dfrac{1}{5}\times\dfrac{\overset{1}{\bcancel{2}}}{\underset{1}{\bcancel{2}}}=& &&\text{Reduce the second fraction by dividing the top and bottom by 2}\\\dfrac{1}{5}\times\dfrac{1}{1}=&\dfrac{1}{5}}$$

So, by cancelling out common factors between numerators and denominators, you can lower the numbers a LOT!

Multiplying fractions is easy!  There are just a couple of things to look out for:

• When you multiply fractions the numerators and denominators can get big, fast, which leads to a lot of simplifying (see lesson Simplifying Fractions).  Cross-cancelling can really cut down on simplifying and is good for students to learn as soon as they master simple multiplication of fractions (see lesson Multiply Fractions (advanced).
• When working with mixed numbers (whole numbers and fractions), students must convert mixed numbers to improper fractions BEFORE they multiply. See lesson Multiply Fractions - with Mixed Numbers.