Fractions Basic Priciples.
1.05.01 Adding and subtracting fractions
The number on the top line of a fraction is called the numerator. The number on the bottom line is called the denominator.
If we want to add or subtract two or more fractions that have the same denominator the procedure is very simple. We add or subtract the numerators as required to find the numerator (top number) of the answer. The denominator (the bottom number) remains the same.
- 1/4 + 2/4 = 3/4
- 5/11 – 3/11 = 2/11
- 4/6 + 1/6 = 5/6
- 7/32 – 5/32 = 2/32
1.05.02 Equivalent fractions
The same size fraction can be written using different numerators and denominators.
e.g. 4/8, 10/20, 3/6, 5/10, 1/2 are all basically the same size fraction they all represent a half. These are all equivalent fractions.
If we multiply both the numerator and the denominator of a fraction by the same integer (whole number) the result is an equivalent fraction.
Similarly if we can divide both the numerator and denominator of the fraction by the same integer (whole number) to produce new integers for the both the numerator and denominator this is again an equivalent fraction.
Expressing fractions in there lowest form
In the first examples of equivalent fractions we looked at e.g. 4/8, 10/20, 3/6, 5/10 & 1/2.
1/2 is the lowest form of expressing the fraction “a half”. There is no number other than one which can divide evenly into both the numerator and the denominator.
If there is no integer greater than 1 that can divide evenly into both the numerator and the denominator of a fraction then the fraction is in its lowest form.
Examples of fractions in their lowest form. 2/3, 5/7, 1/3, 3/22, etc.
1.05.03 Adding and subtracting fractions with different denominators
When you are required to add or subtract two or more fractions with different denominators you must first convert each of the fractions into an equivalent fraction. Each of the equivalent fractions must have the same (common) denominator.
Once this has been done we can simply add or subtract the equivalent fractions as shown in the previous section.
Finally we then convert the answer to its lowest form.
- 1/4 + 1/12
- 1/4 can be written as an equivalent fraction of 3/12 which has the same (common) denominator as the second fraction.
- therefore 1/4 + 1/12 is the same as 3/12 + 1/12
- 3/12 + 1/12 = 4/12
- 4/12 written in its lowest form is 1/3
- therefore 1/4 + 1/12 = 1/3
The following sections first show three methods for finding a common denominator.
Then you will be shown how to find the numerators for each of the equivalent fractions
Finally you will be shown two methods for reducing the answer to its lowest form.
Of the three methods used to find common denominators:-
The first two of these methods are easy to apply and are commonly used when dealing with fractions with fairly small denominators.
The second method in particular is important because it is the basis for dealing with algebraic fractions (where letters are used for the numerator and denominator). Algebraic fractions occur often in scientific equations.
The third method is not used very commonly because it is time consuming and most calculators have the ability to add or subtract fractions quickly.
Even if you intend to always use a calculator to add and subtract fractions you should learn methods 1 and 2. Method 1 just gives you a basic appreciation of what you are doing. Method 2 is needed to deal with some type of scientific equations.
Of the two methods used to reduce a fraction to its lowest form method one is the one that is most commonly used.
1.05.04 Finding common denominators: Method 1 “spotting a common denominator”
This method is just based on experience of working with numbers.
In some cases it is easy to spot a common denominator from the denominators given.
e.g. suppose you were asked to add the following fractions 7/12 + 1/6 + 3/4.
The denominators are 12, 6 and 4. It is easy to see that these numbers all divide evenly into 12.
This means 12 is a common denominator for these fractions.
In this case the equivalent fractions would be 7/12 + 2/12 + 9/12
Note: You will be shown how to determine the numerators of the fractions later
- 2/3 – 2/9 – 1/6
- 3, 9 & 6 all divide evenly into 18 so 18 would be a suitable common denominator
- the equivalent fractions would be 12/18 – 4/18 – 3/18
- 4/5 + 2/3 – 1/10
- 5, 3 & 10 all divide evenly into 30 so 30 would be a suitable common denominator
- the equivalent fractions would be 24/30 + 20/30 – 3/30
This is the quickest and simplest method when it can be applied. But it is not always easy to spot the common denominator. The next two methods can be used to produce a common denominator when you cannot easily spot one.
1.05.05 Finding common denominators: Method 2 multiplication
If you cannot spot a common denominator you can find one by multiplying the denominators of the fractions you are trying to add or subtract.
- 2/5 + 1/4
- 5 x 4 = 20 so 20 is a suitable common denominator
- the equivalent fractions would be 8/20 + 5/20
- 5/6 – 1/3
- 6 x 3 = 18 so 18 is a suitable common denominator
- the equivalent fractions would be 15/18 – 6/18
(You will be shown how to find the numerators for the equivalent fractions later)
This method is quite easy to use when there are only a few fractions with fairly small denominators. For more than three fractions or for larger denominators the common denominator produced by the multiplication tends to get quite large. This makes the work of finding the numerators for the fractions and reducing the final answer to its lowest from more difficult.
1.05.06 Finding common denominators: Method 3 prime factorisation
This is a more complicated method but it always gives the lowest common denominator and so makes the subsequent work of finding the numerators of the equivalent fractions easier.
First you write each of the denominators as a multiple of its prime factors.
A prime number is just a number that is only evenly divisible by itself and 1, e.g. 1, 2, 3, 5, 7, 11 etc are prime numbers.
A factor of a number is just an integer that can be multiplied by another integer to give the number
e.g. 4 x 5 = 20 so both 4 and 5 are factors of 20.
5 is a prime factor because it is also a prime number.
Example 3/12 + 1/6 + 1/4
you can probably spot that 12 is a common demominator but we will prove this by writing each denominator as a multiple of its prime factors.
- The prime factors which multiply to give 12 are 1 x 2 x 2 x 3
- ie 1 x 2 x 2 x 3 = 12 and each of the numbers 1,2, & 3 are prime numbers
- The prime factors which multiply to give 6 are 1 x 2 x 3
- The prime factors which multiply to give 4 are 1 x 2 x 2
From above we can see that:
- When 12 is expressed as a multiple of its prime factors there is one 1, two 2s and one 3
- When 6 is expressed as a multiple of its prime factors there is one 1, one 2 and one 3
- When 4 is expressed as a multiple of its prime factors there is one 1, two 2s
Note the greatest number of times each prime factor appears for each of the denominators
- The largest count of 1s in any of the prime factors for each denominators is 1 1
- The largest count of 2s in any of the prime factors for each denominators is 2 2, 2
- The largest count of 3s in any of the prime factors for each denominator is 1 3
Finally you multiply these numbers together to give the lowest common denominator.
This gives 1 x 2 x 2 x 3 = 12
ie the lowest common denominator is 12
- 5/12 – 2/10
- 1 x 2 x 2 x 3 = 12
- 1 x 2 x 5 = 10
- largest count of 1s = one 1
- largest count of 2s = two 2 2
- largest count of 2s = one 3
- largest count of 5s = two 5
- 1 x 2 x 2 x 3 x 5 = 60
- 60 is the common denominator
equivalent fractions would be 25/60 – 12/60 = 13/60
therefore 5/12 – 2/10 = 13/60
The fraction 13/60 is already in its lowest form (there is no number that divides evenly into both the numerator and denominator)
1.05.07 Finding the numerators of the equivalent fractions
Once you have found the common denominator you must find the numerator for the equivalent fractions.
To do this divide the common denominator by the original denominator of each fraction and multiply the numerator by this value.
ie for 2/7 + 3/5 using a common denominator of 35
- The first denominator is 7
- 35/7 = 5
- The first numerator is 2
- 2 x 5 = 10
- therefore 2/7 = 10/35
- The second denominator is 5
- 35/5 = 7
- The second numerator is 3
- 7 x 3 = 21
- therefore 3/5 = 21/35
Therefore 2/7 + 3/5 can be written using equivalent fractions as 10/35 + 21/35.
10/35 + 21/35 = 31/35
(this fraction is in its lowest form)
1.05.08 Expressing the factor in its lowest common form
Method 1 spotting common divisors.
Sometimes you can quite easily spot a number that will divide evenly into both the numerator and the denominator. By carrying out this division you will simplify the fraction.
The process may have to be repeated simplifying the fraction in stages until it is in its lowest common form.
- e.g 6/18 both 6 and 18 can be divided evenly by 3 giving 2/6
- in this case we can see that both 2 and 6 can be divided by 2 giving 1/3
- Now there is no whole number that will divide evenly into both 1 and 3 other than 1 so the fraction is in its lowest form
If you cannot spot an obvious divisor try 2 or 3 to begin with.
1.05.09 Expressing a fraction in its lowest form
Method 2 prime factorisation
write the numerator and the denominator as a multiplication of their prime factors. e.g. for 6/18
- 6 = 1 x 2 x 3
- 18 = 1 x 2 x 3 x 3
- therefore 6/18 = 1 x 2 x 3/ 1 x 2 x 3 x 3
- cancelling out common factors in the top and bottom line leaves 1 x 1 x 1 / 1 x 1 x 1 x 3
The highest factor in the numerator is 1 and the highest factor in the denominator is 3 so the lowest form of the fraction is 1/3
(When we cancel out numbers we divide pairs of numbers in the numerator and denominator by a common divisor ( as these are prime numbers the divisor will be the number itself!) e.g there is a 2 in both the numerator and denominator 2 divided by 2 = 1, similarly the 3 on the numerator can be cancelled with one of the 3s in the denominator 3 divided by 3 = 1.)
Note as the numbers in the numerator and denominator get larger this method becomes more difficult and method 1 is likely to be quicker.
1.05.10 Improper fractions
An improper fraction is one where the numerator is larger than the denominator e.g. 22/5, 3/2 etc. It is ok to write improper fractions this way. However sometimes you are required to convert them into proper fractions ie to express them as a whole number plus any fraction left over.
e.g. 11/7 = 1 4/7
To do this you just divide the numerator by the denominator.
The number of times the denominator dives evenly into the numerator gives you the whole number.
The remainder gives you the numerator of the fraction.
- 22/5 = 4 2/5
- (5 goes into 22 four times remainder 2)
- 3/2 = 1 1/2
- (2 goes into 3 once remainder 1).