We all love to cross reduce! Why? Well, its simple, it makes our problems easier!

There is a problem with cross reducing, however. The problem is, that since we love to use it, we often use it when it is not appropriate. Believe it or not, there really is only 1 situation in which cross reducing can be used.

We will take a look at this process called "cross reducing". We will see how, when, and why it works!

To begin, this question must be answered. Reducing is the process of dividing the numerator and denominator of a fraction by one of their common factors. A fraction is completely reduced when the numerator and denominator are relatively prime (i.e. their greatest common factor is 1). To completely reduce a fraction in one step, we divide the numerator and denominator by their greatest common factor (GCF).

In the situation above, we attempt to reduce the fraction by 4, that is, we divide the numerator and denominator by 4 (4 is a common factor to both 24 and 36). You should notice that what we are really doing is breaking the fraction down into the product of two fractions - in this case . Since is equal to 1, and multiplying by 1 does not change the value of the other fraction, we know that (this is an application of the identity property of multiplication). Although this reduces the fraction, the results can still be reduced, for 6 and 9 have a common factor of 3. We could have reduced this completely by reducing by the GCF - the greatest common factor of 24 and 36 is 12. This is shown below.

Cross reducing is the process of reducing, but with numerators and denominators that are not in the same fraction. That is, you divide a numerator and a denominator by one of their common factors. You must note, however, that this process can only be used in situations where the numerator and denominator to be reduced could, in some valid way, be made part of the same fraction. If the two values cannot be made part of the same fraction, then cross reducing may not be used.

To illustrate this, we will show how to cross reduce using the one situation where it is valid - when multiplying fractions.

Notice that neither fraction can be reduced. However, notice that we did reduce the 6 and 9 by their common factor 3.
Now, remember, this process can only be used when the numerator and denominator can be part of the same fraction. Therefore, if our example above is correct, the 6 and 9 should be able to be a part of the same fraction. The truth is, they can! This is demonstrated below.
Notice, when we multiply fractions, we multiply the numerators together and the denominators together (first step). However, if we do not multiply the numerators and denomintators immediately, we have the opportunity to change their orders using the commutative property. According to the commutative property, 6 times 10 is equal to 10 times 6. If we change the order of 6 and 10, then separate the fractions (as if we did not want to multiply), we are left with a fraction that can be reduced.

Please keep in mind that it is in this situation alone where cross reducing is possible, you cannot use it when adding and subtracting fractions, or in proportions (two equal fractions) - only when multiplying fractions. We can use this process when dividing fractions, but only after the problem has been changed to a multiplication problem (remember, to divide fractions, we multiply the first fraction by the reciprocal of the second fraction).