Math Skills: Fractions

Lesson 3: Lesson Three--Greatest Common Factor/Reducing Fractions

Lesson Three: Greatest Common Factor (GCF) and using it to reduce fractions, divisiblity rules to determine common factors.

Section One: Finding the Greatest Common Factor to reduce fractions

In this section we will find the Greatest Common Factor (GCF) to reduce fractions to lowest terms, which is sometimes necessary to do to get the correct answer (and ALL answers to work questions must be reduced if necessary).

Reducing fractions to lowest terms using the greatest common factor of both numerator and denominator means to find a number that divides evenly into both numerator and denominator such that the resulting answer doesn't have to be reduced again. That's why the number we seek to find must be the GREATEST number that divides into both numerator and denominator. Since numbers that divide evenly into other numbers are called FACTORS, and since we want factors that go into BOTH numerator and denominator, we want COMMON factors for both.

The thing is, while most fractions can be reduced, some fractions cannot be reduced. Fractions with 1 in the numerator (like 1/3, 1/56) cannot be reduced since you really can't get smaller than 1. But other fractions like 3/7, 4/19, 9/61 cannot be reduced either. There are no numbers OTHER THAN 1 that go into both numerator and denominator of these fractions.

I am going to get a bit technical here, but this info is important. You have prime numbers and composite numbers. Composite numbers have any number of factors besides 1 and itself. Numbers like 4, 9, 15, 18 and so on are composite because they all have other factors besides 1 and itself: the factors of 4 are 1, 2, 4; the factors of 9 are 1, 3, 9; the factors of 15 and 1, 3, 5, 15; the factors of 18 are 1, 2, 3, 6, 9, 18.

But numbers like 2, 3, 5, 7, 11, 13 are prime because the only factors of these numbers are 1 and that number itself.

But with fractions you're dealing with two numbers, numerator and denominator, and you're not always going to have two composite or two prime numbers to work with, and even if the two numbers are composite it won't mean it can be reduced. For instance, 4/9 cannot be reduced even though they are both composite!

In 4/9, the 4 and the 9 are "relatively prime", which means they have no common factors other than 1. If 1 just happens to be the "GCF", the numerator and denominator are "relatively prime" and cannot be reduced.

So, if the fraction has the following attributes:

1. Either (or both) numerator or denominator is prime, the fraction cannot be reduced. 2. If the GCF of numerator and denominator is 1 (they are relatively prime to each other), the fraction cannot be reduced.

For a list of prime numbers to 100, go toList of Prime and Composite Numbers and see if you can pick out the prime numbers.

All other fractions can be reduced.

Using the GCF, I will show you how to reduce a fraction.

Let's say the fraction 6/10.

First of all, since 6 and 10 are both even (divisible by 2) they are both composite and both divisible by 2. But are there any numbers greater than 2 they are both divisible by? We can figure this out by getting all the factors of these numbers.

Factors of 6: 1, 2, 3, 6 Factors of 10: 1, 2, 5, 10.

The common factors are 1 and 2. The greatest of these is 2, so 2 is the GCF. Now that we have the GCF, we can reduce 6/10 by dividing the 6 and 10 by 2:

6 divided by 2 = 3, and 10 divided by 2 = 5, so 6/10 = 3/5.

Lets do one with even and odd composite numbers with a common factor more than 1:

27/63.

Factors of 27: 1, 3, 9, 27 Factors of 63, 1, 3, 7, 9, 21, 63 The common factors are 1, 3, 9, with the GCF of 9. So we divide both 27 and 63 by 9 and get:

27/63 = 3/7 (since 3 and 7 are both prime, the fraction cannot be further reduced).

Let's do one more that is more difficult since not all fraction will be as easy as those two:

72/240.

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240

Whew!

The common factors of 72 and 240 are 1, 2, 3, 4, 6, 8, 12, 24 with the greatest being 24, so the GCF is 24. Divide 72 and 240 by 24 and you get:

72/240 = 3/10.

If you know your times tables and Divisibility Rules, you can reduce ANY fraction that can be reduced. And click here for the only website I have found that gives you the rules for divisibility by 7!

For more about GCF, click here, and for more about reducing fractions, click here.