Equivalent Fractions

Objective

After reviewing this unit, you will be able to:

Write equivalent fractions.
Find the prime factors of a number.
Write a fraction in its simplest form.

Equivalent Fractions

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.

Take a sheet of paper and fold it twice, creating three equal sections.
Now shade two of them. This shaded portion represents 2/3.
Fold the paper again, in the other direction, but down the center of the paper. The shaded portion is now 4/6.

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 non-zero 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.

Example

Find two fractions that are equivalent to the fraction 1/2.
Two fractions equivalent to 1/2 are 3/6 and 9/18.

and

Example

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.

Prime Factorization

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.

A factor of a number is a number that can be divided into the original number evenly (meaning there is no remainder). For example, 4 is a factor of 8. That means 8 can be divided by 4 and there is no remainder (8 ÷ 4 = 2). This means that 2 is also a factor of 8.
A prime number is a number that has only two factors, 1 and itself. For example, the number 2 can be divided evenly only by itself and 1, therefore, it is a prime number. The five smallest prime numbers are 2, 3, 5, 7, 11, and 13.
Numbers that are not prime numbers are referred to as composite numbers. The number 8 is a composite number since it has factors of 2 and 4.

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:

When we divide 24 by 2 we get a result of 12. Since 2 goes into 24 evenly, it is a factor of 24. It is also a prime factor since it can only be divided by itself and 1. So, 24 has 2 as a prime factor and 12 as a composite factor.

If we want to find all the prime factors of 24, we must continue by finding the factors of 12.

We can also divide 12 by 2. When we do this we find that 2 and 6 are factors of 12, with 6 being another composite factor.

When we further divide 6 by 2 we get a result of 3. Finally, we have a result that is a prime number. So both 2 and 3 are prime factors of 6.

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
2	The number is an even number.	2248 is an even number. It will have 2 as a prime factor.
3	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.
5	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.

Example

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.

The first thing we notice is that 112 is an even number. This means it is divisible by 2.

If we set this up as the beginning of our factor tree, we have the diagram shown at the right.

From here, we have one branch with a prime factor, 2, and the other still has a composite number, 56. This composite number is even and therefore divisible by 2. So, we add this on to our tree.

We keep going down the factor tree dividing out the branches that have a composite number until we have all prime factors at the ends of our branches.

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.


[practice]	[additional practice]	[next section]

When we divide 24 by 2 we get a result of 12. Since 2 goes into 24 evenly, it is a factor of 24. It is also a prime factor since it can only be divided by itself and 1. So, 24 has 2 as a prime factor and 12 as a composite factor. If we want to find all the prime factors of 24, we must continue by finding the factors of 12.
We can also divide 12 by 2. When we do this we find that 2 and 6 are factors of 12, with 6 being another composite factor.
When we further divide 6 by 2 we get a result of 3. Finally, we have a result that is a prime number. So both 2 and 3 are prime factors of 6.

The first thing we notice is that 112 is an even number. This means it is divisible by 2.
If we set this up as the beginning of our factor tree, we have the diagram shown at the right.
From here, we have one branch with a prime factor, 2, and the other still has a composite number, 56. This composite number is even and therefore divisible by 2. So, we add this on to our tree.
We keep going down the factor tree dividing out the branches that have a composite number until we have all prime factors at the ends of our branches.