Cancelling Fractions: Rules

Transcription

1 Cancelling Fractions: Rules The process of cancelling involves taking fractions with large numerators and denominators (top and bottom numbers) and rewriting them with smaller numerators and denominators whilst ensuring that the value of the fraction remains the same. Another way to say that a fraction is cancelled is to say that it is in its simplest form or expressed in its lowest terms. Before getting into the rules of cancelling let s briefly review the important subject of factors. A factor is any number that divides into another number wholly with a remainder of zero. For example the factors for 6 are 1, 2, 3 and 6 because 1, 2, 3 and 6 will each divide into 6 with a zero remainder. The factors of 8 are 1, 2, 4 and 8. The factors of 12 are 1, 2, 3, 4, 6 and 12. Also, the other terms we will use are: Numerator: The top number in the fraction. Denominator: The bottom number in the fraction. Now let s turn our attention to the process of cancelling. There are different ways to cancel and it always involves looking at the numerator and denominator of the fraction in question. If these numbers are both small (say about 30 or less) we can use the Highest Common Factor (HCF) method as shown immediately below. Otherwise if the numerator and denominator are large we can take a slightly alternative approach and use either the 2, 3, 5 or 10 Rule as explained further down in these notes.

2 Highest Common Factor (HCF) Method Rules for using the HCF method to cancel a fraction: Find the HCF of the numerator and denominator. Divide the numerator and the denominator by the HCF to create the new numerator and denominator of the cancelled fraction. Note: This process works well for fractions in which the numerator and denominator are relatively small otherwise your list of factors can get very long Example Reduce 8 12 to its lowest terms. The steps for cancelling a fraction are as follows: 1. Look at the numerator (the top number of the fraction) and the denominator (the bottom number of the fraction) and find the Highest Common Factor (HCF) of these two numbers: The Highest Common Factor (HCF) can be found by listing the factors of the two numbers and selecting the highest number that appears in both lists: 8: 1, 2, 4, 8 12: 1, 2, 3, 4, 6, 12 As you can see, the HCF of 8 and 12 is 4 since this is the largest number in both lists.

3 2. Divide the numerator by the HCF and divide the denominator by the HCF to create two new numbers: 8 You are trying to cancel. 12 You must divide the numerator (8) and the denominator (12) by the HCF (which in this case is 4): Numerator: 8 4 = 2 Denominator: 12 4 = 3 3. These new numbers are the numerator and the denominator of the final cancelled equivalent fraction: Your equivalent fraction is (in its simplest form): 2 3 Example Cancel 6 9 : You can say that 8 12 = The HCF of 6 and 9 is found by listing the factors of each: 6: 1, 2, 3, 6 9: 1, 3, 9 The HCF of 6 and 9 is Now divide both the numerator and the denominator by this HCF of 3 to give: Numerator: 6 3 = 2 Denominator: 9 3 = 3

4 3. Your equivalent fraction is (in its simplest form): 2 3 You can say that 6 9 = 2 3 Cancelling Larger Fractions You may often be required to cancel a fraction that has a relatively large numerator and/or denominator. This may be the case when you are trying to calculate an infusion rate in ml per hour and you are given a large volume (such as 1000 ml) over a long period of time (24 hours). When the numerator and denominator are large it is not practical to use the HCF method (and write a list of all the factors) as the list would take a long time to compose. Instead you can revert to some general rules to perform the cancelling process. This will involve breaking down the fraction in lots of small steps using the 2, 3, 5 and 10 Rules. Dividing by Factors of 10 (The 10 Rule) The 10 Rule: You can remove one or more zeros from the end of the numerator AND denominator provided that the same number of zeros are removed from both numbers and provided that those zeros that are removed are at the end of both numbers. Note: Once you have applied the 10 Rule the fraction may still not be in its lowest terms in which case you should continue to cancel it by using any one of the cancelling rules.

5 Example Reduce to its lowest terms. The numerator is 5300, which ends in a 0. The denominator is 10, which ends in a 0. You are allowed to remove one 0 from the end of the numerator as long as you remove one 0 from the end of the denominator. You cannot remove both 0 s from the end of the numerator since there is not a second 0 at the end of the denominator. Removing a single zero from the end of the numerator and also from the end of the denominator: 5300 will become will become 1 Replacing the old numbers with the new, your fraction will become: This fraction cannot be reduced any further and so you can say that it is in its lowest terms. You can say that = Example Cancel

6 The numerator is 2200, which ends in a 00. The denominator is 300, which ends in a 00. The same number of zeros can be removed from the end of both numbers will become will become 3 Replacing the old numbers with the new, your fraction will become: 22 3 This fraction cannot be reduced any further and so you can say that it is in its lowest terms. You can say that = 22 3 Dividing by 5 (The 5 Rule) The 5 Rule: If the numerator AND denominator both end in a digit of 0 or 5 you can divide both the numerator and the denominator by 5 to produce a new numerator and denominator. Note: Once you have applied the 5 Rule the fraction may still not be in its lowest terms in which case you should continue to cancel it by using any one of the cancelling rules.

7 Example Cancel The numerator is 125, which ends in a 5. The denominator is 130, which ends in a 0. Since both the numerator and the denominator end in a 0 or a 5 then you can divide both of them by 5. If you do this you will get: = = 26 This result will become the new numerator and denominator. The fraction now becomes: This fraction cannot be reduced any further and so you can say that it has been expressed in its lowest terms. You can say that = Dividing by 2 (The 2 Rule) This rule is one of the most frequently used rules in cancelling. Often (but not always) you will have to chip away at the fraction that is use this rule several times in a row to break the fraction down.

8 The 2 Rule: If the numerator AND denominator both end in an even digit (this means: 2, 4, 6, 8 or 0) you can divide both the numerator and the denominator by 2 to produce a new numerator and denominator. Note: Once you have applied the 2 Rule the fraction may still not be in its lowest terms in which case you should continue to cancel it by using any one of the cancelling rules. Example Reduce to its lowest terms. The numerator is 3000 (which is even since it ends in 0) and the denominator is 28 (which is even because it ends in 8). Because both numbers are even you can divide them by 2: = = 14 The result of this procedure becomes the new numerator and denominator of the fraction. Your fraction has therefore become: You can now repeat the 2 Rule because the numerator (1500) is even (since it ends in a 0) and the denominator (14) is also even (since it ends in 4): = = 7

9 The result becomes the new numerator and denominator. The fraction now becomes: Neither of these two numbers (750 and 7) is even and therefore you cannot apply the 2 Rule and divide them by 2 any further. The fraction happens to be reduced to its lowest terms. You can say that = Note that if you had selected the Highest Common factor (HCF) from the beginning by writing (long) lists for 3000 and 28 (which would not be the recommended method) you would have found it to be 4. Dividing both the numerator (3000) by 4 and the denominator (28) by 4 you would have arrived at exactly the same answer. Dividing by 3 (The 3 Rule) This rule has been included after all of the previous rules simply because it is not used as frequently as the others. It is generally best to try and use this rule only when you have established that all of the other cancelling rules cannot be applied. Please also not that this rule only applies to 3.

10 The 3 Rule: If the sum of the digits in a number is divisible by 3 then the original number is also divisible by 3. This means if other cancelling rules cannot be applied to the fraction: Add the digits of the numerator and check whether the result is divisible by 3 (ie whether 3 divides into that number with a zero remainder). Repeat the test above with the denominator. If the answer is yes in both cases divide the original numerator and denominator by 3 to produce a new numerator and denominator. Example Cancel Look at both of the numbers in the fraction. You cannot apply any of the other cancelling rules since the digits at the end of the numbers are 3 and 5. To try the 3 Rule start by adding the digits in each of the numbers: 123: = 6 225: = 9 If these results (6 and 9) are both divisible by 3 (that means that 3 will divide into both of them with a remainder of zero) then the original numbers (123 and 225) will also be divisible by 3. This means that you can divide the two original numbers by 3. In this case 3 will divide into both 6 and 9 (with a remainder of zero) and you therefore know that 3 will also divide into 123 and 225.

11 This means that you can divide both 123 and 225 by 3 to reduce the fraction: = = 75 The new results become the new values of the fraction: = Now attempt to repeat the cancelling process using any of the previous cancelling rules if they can be applied. In this case you cannot apply any of the cancelling rules on 41 and 75. The fraction happens to be reduced to its lowest terms. Note: If there is a non-zero remainder when you try and divide 3 into the sum of the digits of a number then you should not attempt to use the 3 Rule. For instance, cancel In this case you cannot apply any of the other cancelling rules and so you can try the 3 Rule: 124: = 7 225: = 9 3 will divide into 7 with a remainder of 1 (this is a non-zero remainder) and therefore you know in advance that 3 will not divide into 124 without a remainder. As such you cannot use the 3 Rule on this particular fraction.