Simplifying Fractions

To express a fraction in its simplest form, divide both the numerator and denominator by their greatest common factor (GCF).

Example: Find the simplest form of 10/25.

The GCF of 10 and 25 is 5.
Divide both the numerator and the denominator by 5.
10/25 divided by 5/5 = 2/5

The simplest form of 10/25 is 2/5.

If the numerator of a fraction is bigger than the denominator, the fraction is an Improper Fraction.

You can simplify any improper fraction:
First, divide the numerator and the denominator by the GCF.
Then, change the fraction to a mixed number.

Example: Find the simplest form of 42/12.

The GCF of 42 and 12 is 6.
42/12 divided by 6/6 = 7/2

Change 7/2 to a mixed number by dividing 7 by 2.

The simplest form of 42/12 is 3 & 1/2.

Two fractions are EQUIVALENT if they have the same value in simplest form.

3/12, 2/8, and 1/4 are equivalent fractions because the simplest form of each fraction is 1/4.

Least Common Denominator

To find the least common denominator (LCD) of two fractions, find the least common multiple (LCM) of their denominators.

Example: Find the LCD for 2/3 and 3/4

3 and 4 are the denominators.
Multiples of 3 are 3, 6, 8, 12
Multiples of 4 are 4, 8, 12, 16

The LCD is 12 because 12 is the least common multiple of the denominators 3 and 4.

Least Common Multiple

Least Common Multiple (LCM) - The least common multiple of two or more numbers is the smallest number (excluding zero) that is a multiple of all of them.

Example: Find the LCM of 6 and 8.

The multiples of 6 are 6, 12, 18, 24, 30
The multiples of 8 are 8, 16, 24, 32, 40

24 is the least common multiple (LCM) of 6 and 8. An easy way to find all the multiples of a specific number is to just count by that number. Example, to find the multiples of 5 you would do 5, 10, 15, 20, 25, 30, etc. Easy enough.

Another characteristic of the least common multiple (LCM) of two or more numbers is that it is the smallest whole number that can be evenly divided by each of the numbers.

Another way to find the LCM is to find the PRIME FACTORIZATION of each number.

Example: Find the LCM of 12 and 15.

1. Factor each of the numbers into its prime factors.
12 = 2 x 2 x 3
15 = 3 x 5

2. For each prime, find the MOST number of times it is used in any one factorization.
12 = 2 x 2 x 3
15 = 3 x 5
2 appears twice, 3 appears once, and 5 appears once.

3. Multiply these prime factors together to get the LCM.
LCM = 2 x 2 x 3 x 5 = 60

That's the trick for bigger numbers.

Another side note: The least common multiple (LCM) is t he same as the least (or lowest) common denominator (LCD). The LCD is used to rename or change unlike fractions to like fractions in the addition and subtraction of fractions and mixed numbers.

Greatest Common Factor

Greatest Common Factor - The greatest common factor of two or more numbers is the greatest number that is a factor of each of them.
For short, this little mathmatical tactic is called GCF. Clever, I know.

Example: Find the GCF for 8 and 12.

You would find all factos for 8 and all factors for 12. Then you would compare the lists and find the greatest commin factor of the two numbers. Real life application unknown at this time.
An easier and much more convienent and more simple way to under GCF is to think of the GCF as the biggest whole number that can divide evenly into both numbers. That works better.

A good way to find the GCF for two numbers if you're not using a calculator and randomly trying whole numbers is to find the prime factorization of each number, and then compare. I may have already said this.

Another clever way to find the GCF of two or more numbers is to find all of the factors of the lowest number, and check the higest of those numbers with the other number or numbers, until you find the GCF

Factor

A factor is a number that is divisable by a specifc number. We'll use the number 6 as an example. 1, 2, and 3 are factors of 6 because all three of those numbers could be used in the equasion that makes 6. The greatest factor of 6 is 3.

Prime Factorization

Prime Factorization: Creating a product of prime numbers. Making every composite number a prime number in an equasion. I guess. An equasion of nothing but prime numbers.

To find the prime factors of a number, find just one pair of factors at each step. Factor again any number that is still composite, and stop whne all factors are prime. Example:

300 = 10 x 30
10 = 5 x 2
30 = 5 x 6

Only one composite number is left (6), so we make that 2 x 3 which gives us the grand prime factorization of 300 as: 300 = 2 x 5 x 5 x 2 x 3

Prime and Composite

Prime: A number is prime if it has exactly two factors; 1 and the number itself. 1 isn't a prime number because it only has one factor. @ is the only even prime number becaquse it can be divided by 1 and 2.

Composite: A composite number is the opposite of a prime number. A composite number has more than two factors. It can be divided by more than two numbers.

Multiple

Multiple: A multiple of a number is the product of that number and any other whole number. Example: 6x6=36 - 6 is a multiple of 6