


Equivalent fractions  Comparing
fractions  Example 1: Folded paper model of equivalent fractions Fractions which represent the same value are called equivalent fractions. We can see from the movie that 1/2, 2/4 and 4/8 all represent the same amount of piece of paper, therefore they are equivalent fractions. Each of these fractions represents the same number, it is just the way the number is named that changes. Example 2: 2/3 = 8/12
Example 3: 2/3 = 10/15
Recognising patterns Equivalent fractions enable us to rename fractions so that we can compare, add and subtract them. We can generate equivalent fractions by recognising the patterns they form. Example 4: What are the missing numbers in this sequence? ....
The fraction wall pictured below illustrates that 1/3 is the same as 2/6 which is the same as 3/9 and so on. Example 5: What are the missing numbers in this sequence? The diagram below shows the pattern of these equivalent fractions. Can you fill in the missing numbers and then find the next equivalent fraction in the sequence?
Example 6: What are the missing numbers? In this example there is no obvious pattern to follow, but we can still work out the equivalent fractions using the same method. The diagram below shows the equivalent fractions.
Can you find 2 more greater equivalent fractions for one half?
Click here to see the explanation of this result at Justification for a way to find equivalent fractions. Example 7: Which of these fractions is the largest, 1/9, 7/8, 1/2? We can find the answer to this problem by roughly visualising the size of each of these fractions.
In some cases, we can visualise common fractions and can recognise the difference between fractions without the need to use more formal mathematical procedures to compare them. Example 8: Which fraction is largest, 2/3, 3/4 or 5/6? Use the rectangle above to estimate your answer. It is possible to mentally picture each of these fractions using the rectangle and estimate which is largest, but another way is to change each fraction so that they all have the same denominator. We call fractions which have the same denominator like fractions which means they have a common denominator. The next section explains methods we can use for Finding common denominators. In order to compare the fractions 2/3, 3/4 and 5/6 we can find an equivalent fraction for each one all with the same denominator. This will make the comparison easy. To do this, we simply list the sequence of equivalent fractions for each number: 2/3 = 4/6 = 6/9 = 8/12 = 10/15 = 12/18 = 14/21 = 16/24 = 18/27 = .. 3/4 = 6/8 = 9/12 = 12/16 = 15/20 = 18/24 = 21/28 = 24/32 = .. 5/6 = 10/12 = 15/18 = 20/24 = 25/30 = 30/36 = 35/42 = .. Twelve occurs as a denominator on all three lists. Thus 2/3, 3/4 and 5/6 can be rewritten as 8/12, 9/12, 10/12. Clearly, it can be seen that 10/12 represents the largest fraction. Hence 5/6 (which is equivalent to 10/12) is larger than 2/3 or 3/4. Each fraction has a denominator of 12. Twelve is a common denominator for these three fractions. The list shows us that 24 is another common denominator for these three fractions, and there are others  can you find them? (Hint: continue the lists and look for a pattern). Because 12 is the lowest of all the common denominators for these three fractions, it is called the LOWEST COMMON DENOMINATOR. When we are comparing fractions we usually try to use the lowest common denominator so that we are working with smaller numbers, but this is not essential. Any common denominator can be used. Another way of finding a common denominator is to multiply all the denominators together  can you explain why this works and can you find out when this method gives the lowest common denominator? Finding the lowest common denominator To find the lowest common denominator for the fractions of 2/3, 3/4 and 5/6 we could have used either of these methods:
Lowest common denominators can also be systematically found by considering the prime factors of the denominators. This is the best method for large numbers, but the methods above are easy to carry out in most cases and quick to teach. Using multiplication of fractions to find equivalent fractions If you are familiar with how to multiply fractions you can use this method to find equivalent fractions. In order to find equivalent fractions you need to understand that the value of a fraction is unaltered when it is multiplied by one. One can be written as any number divided by itself (1 = n/n). For example, 1 is the same as 2/2, 5/5 or 89/89, so we can multiply 2/3 by any of these fractions without changing its value. 2/3 x 2/2 = 4/6 = 2/3 2/3 x 5/5 = 10/15 = 2/3 2/3 x 89/89 = 178/267 = 2/3 Example 9: How do we change 1/2 into the equivalent fractions 3/6 or 5/10?
Justification for a way to find equivalent fractions
Quick quiz
To
view the answers to the quiz, click
here.


© University of Melbourne 2003 