Cross-cancelling

[sjsfan01]

If you cross-cancel when multiplying fractions, will your answer always be in simplest form?

 

[lev888]

If you cross-cancel when multiplying fractions, will your answer always be in simplest form?

Example? Yes, you can cancel same terms in the numerator and denominator to simplify the fraction. But you are not stating the question clearly enough to understand what the fractions look like. E.g. you can eliminate x here: 4/x * x/2 = (4x)/(2x) = 4/2. But you can simplify it further.

 

[Otis]

Hi sjsfan. I would say no, although many beginning exercises are created to turn out that way. Try canceling the 3 and 4, before multiplying the following fractions, and show us what you get for the product. Is it simplified?

\(\displaystyle \frac{3}{4} × \frac{80}{15}\)

?

 

[Dr.Peterson]

If you cross-cancel when multiplying fractions, will your answer always be in simplest form?

First, "cross-canceling" implies that you are only canceling between one numerator and the other denominator; in that case, the result will only be in simplest form if the two given fractions were! (In Otis' example, the second is not; in lev888's, not all cross-canceling has been done.)

But really there is no need to distinguish between "cross-canceling" and canceling within the same fraction; you can just cancel any factor anywhere in the numerator with the same factor anywhere in the denominator. And if you "fully cancel" in that way, then by definition the result is in simplest form.

 

[lev888]

First, "cross-canceling" implies that you are only canceling between one numerator and the other denominator; in that case, the result will only be in simplest form if the two given fractions were! (In Otis' example, the second is not; in lev888's, not all cross-canceling has been done.)

But really there is no need to distinguish between "cross-canceling" and canceling within the same fraction; you can just cancel any factor anywhere in the numerator with the same factor anywhere in the denominator. And if you "fully cancel" in that way, then by definition the result is in simplest form.

I assumed that cancelling was removing identical terms from numerator and denominator. Which means in my example cancelling was done fully since 4 was not represented as 2*2. Am I wrong, and cancelling is synonymous with simplifying?

 

[Dr.Peterson]

I assumed that cancelling was removing identical terms from numerator and denominator. Which means in my example cancelling was done fully since 4 was not represented as 2*2. Am I wrong, and cancelling is synonymous with simplifying?

Since it's not a proper mathematical term, it may well be used in different ways in different places, but I understand canceling (in fractions) to mean dividing a numerator and a denominator by the same thing, which is equivalent to removing common factors even if they are "hidden" (e.g. removing a 3 from 9). A quick search shows this to be typical: https://www.mathbootcamps.com/multiplying-fractions-cross-cancelling/

 

[Steven G]

using your method, \(\displaystyle \frac{80}{4} × \frac{3}{15}\) would become \(\displaystyle \frac{240}{60}\) which is not in reduced form.

using your method,\(\displaystyle \frac{10}{2}\) can't be reduced.