How To Cross Simplify Fractions

What is cross-cancellation?

Cross-counterfoil is a shortcut that y'all can employ to make multiplying fractions easier. Sometimes y'all have to simplify fractions after doing arithmetic with them. Cantankerous-cancellation is a simplification that can be done before. That's great, because simplifying before means when you do multiply, you'll have smaller numbers, and smaller numbers are easier to work with.

Did you take hold of that? Cross-cancellation makes fractions easier.

But what operation can you use cross-cancellation on? Multiplication of fractions.

In mathematics, nosotros use the word operation for something basic that you do to a number. The most common examples of operations are adding, subtracting, multiplying, and dividing (in that location's others, but let's non get into that right now).

$Cross-cancellation is a great shortcut you can take to make multiplying and dividing fractions easier.$

Cantankerous-cancellation and Division

In addition to multiplication, cross-cancellation can be used to brand math easier when fractions are divided. That's because y'all can always catechumen a fraction division problem to a fraction multiplication problem. Just take the reciprocal of the divisor. Taking the reciprocal of a fraction just means swapping the numerator and the denominator (put the summit on the lesser, and the bottom on the top). The divisor is the number that you are dividing past.

How practise y'all cross-abolish?

You can cross-abolish systematically, but there's also some shortcuts. Here's the systematic manner to cross-abolish:

If a division problem is being converted into multiplication, first rewrite information technology converting the divisor into its reciprocal.
Observe the prime factorization of both the numerator and denominator of both fractions.
If either gene was not in simplest class, perform the simplification now.
Cancel whatever factors found both in a numerator and the opposite denominator. In fact, it'south because these factors are located diagonally from each other, that nosotros call it cross-counterfoil. If multiple copies of any factor are present, deal with them by canceling only as many copies as are present in the opposite location (the same equally when simplifying fractions).
Repeat the previous footstep on the other diagonal.
Multiply all factors on the top of both fractions to class the numerator of the answer, and multiply all factors on the lesser of both fractions to form the denominator of the reply.

$A fraction division problem (30/88 divided by 63/24) is solved in full detail, with all steps shown.$

Step-by-step cantankerous-counterfoil example

Practice you lot ever have to do all that? No, not always. Did you notice where nosotros used the discussion systematic to draw that step-by-step way of doing cross-counterfoil? Some people probably ignored that word, and some other people might have wondered what information technology meant, and had a sinking feeling that we're using fancy words, and things might soon end making sense. Please don't worry. What we meant by systematic was that anyone who can follow those steps will get the right answer at the end. When you're familiar with the process, you tin can ordinarily combine or skip steps, employ alternative techniques, or accept short cuts.

Cross-cancellation with the GCF Shortcut

You can save work if the numbers are relatively small and you lot can see some common factors. In such a case you don't accept to find the complete prime factorization. Instead, we usually cantankerous out numbers across a diagonal and write replacement numbers next to them with the Greatest Mutual Factors removed. Here'southward an example of what this technique looks like.

$An example fraction multiplication problem solved by factoring out GCFs on the diagonals.$

Example of using GCFs to simplify a fraction

Cross-cancellation and Fraction Simplification

Cross-cancellation is really a special version of simplifying fractions. You tin can but accept advantage of information technology when multiplying or dividing fractions. Information technology's worth practicing simplifying fractions first in club to get a larger perspective, and to get an appreciation for how it's useful. In fact, the advantage of cross-cancellation over multiplying then simplifying is that before the numbers are multiplied they are smaller and easier to piece of work with. You get the aforementioned reply if you multiply first, and so simplify, merely it can be a lot more work. Cantankerous-counterfoil is just a shortcut, only a good one.

How do y'all Get Good at Cross-cancelling?

Getting skillful at nigh things takes practice. We have created a game, Bubbly Primes, with the purpose of giving students lots of practice factoring so they gain the ability to see common factors like in the GCF technique. It'due south a fun way to cultivate skills that traditionally come from spending lots of time on worksheets and quizzes. Nosotros encourage you to spend some fourth dimension playing the game.

What'south the difference between cantankerous-cancellation and cross-multiplication?

Sometimes students confuse cross-cancellation with cross-multiplication. It's easy to do, because the names are like, both have to do with fractions, and furthermore, both have to do with multiplication. Yet:

Applycross multiplication to algebraically manipulate an equality involving fractions on either side of the equal sign, usually to solve for a variable.
Utilize Cantankerous-cancellation as a simplification technique when two fractions are existence multiplied.

Usually the 2 techniques are for entirely unlike kinds of issues. Hither's an example of each.
Examples of a cross multiplication and a cross-cancellation problem shown side-by-side in order to illustrate their differences.

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