After reviewing this unit, you will be able to:
Equivalent fractions are fractions that may look different, but are equal to each other. Two equivalent fractions may have a different numerator and a different denominator. (A fraction is also equivalent to itself. In this case, the numerator and denominator would be the same.)
Let's take a moment to demonstrate the concept of equivalent fractions. Follow the steps below.






The shaded portion of the paper does not change, so the fraction of the paper shaded does not change. The fractions 2/3 and 4/6 are equivalent.
Equivalent fractions can be created
by multiplying or dividing both the numerator and denominator
by the same number. This number is referred to as a multiplier. We can do this because,
if you multiply both the numerator and denominator of a fraction
by the same nonzero number, the fraction remains unchanged in
value. In the demonstration above, we could get the fraction 4/6
by multiplying both the top and bottom of 2/3 by 2.
Find two fractions that are equivalent to the fraction 1/2.
Two fractions equivalent to 1/2 are 3/6 and 9/18.
Show that the fraction 8/12 is equivalent to the fraction 2/3.
If you multiply both the numerator and denominator of 2/3 by 4,
you get the fraction 8/12. Therefore, the two are equivalent.
We typically want to put fractions in their simplest form. Fractions can be simplified when the numerator and denominator have a common factor in them.
Fundamental Theorem of
Arithmetic Every composite number can be expressed as a product of prime numbers. These are referred to as prime factors. 
Finding the prime factors of a composite number is done by dividing out the prime factors. For example, if we wish to find the prime factors of 24, we can start by dividing 24 by 2:





Now we can look at our sequence of division and list all the prime factors of 24. To review our sequence of division we have:
From this we can see that 2, 2, 2, and 3 are all the prime factors of 24. A number can be written as the product of its prime factors, so 2 x 2 x 2 x 3= 24. When we report the prime factors of 24, we must list each occurrence of a number.
How do we know where to start when looking for prime factors? To some extent this is trial and error, but there are some rules to help you find prime factors.
Divisible By  Test  Example 


The number is an even number.  2248 is an even number. It will have 2 as a prime factor. 

The sum of its digits is divisible by 3.  951 has 3 as a prime factor since 9 + 5 + 1= 15, which is divisible by 3. 

The number ends in 5 or 0.  33505 ends in a 5, so it has 5 as a prime factor. 
The first fifteen primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.
One way to help you keep track of all occurrences of prime factors is creating a factor tree.
The factor tree for 24 is shown at the right. The number circled at the end of each branch is a prime factor of our original number. Compare this with the sequence of division shown earlier. The factor tree makes it easy to identify the prime factors of a number. 

Now let's look at an example of finding the prime factors of a number.
What are the prime factors of 112?
The prime factors of 112 are 2, 2, 2, 2, and 7, so 2 x 2 x
2 x 2 x 7= 112.







If we wrote this as a sequence of division, it would look like:
Using either of these techniques, we can find the prime factors. From the factor tree, just write down all the prime factors that are at the ends of branches. From the division sequence, we write down all the divisors and then the end result. Thus, we find the prime factors of 112 are: 2, 2, 2, 2, and 7.
Take a moment now to try a practice problem for this first part of the unit. When you are done you move on to the next section on simplifying fractions.





