Properties of Fractions This section on fractions is split into two pieces. The first deals mainly with the basics properties of fractions and operations on fractions. The second gets into applications of fractions and how to convert between fractions, percents, decimals, and ratios. To begin, let’s start by talking about the definition of a fraction. Definition: A fraction is any number that is written using division of two values a and b. The value of a is called the numerator and b is called the denominator. If a and b are both integers then we call this fraction a rational number. (Note also that b can never equal 0 – more on this later) This could look like a or a / b or a ÷ b . These are all symbols for division of two values. We will be b
using the first notation a throughout this tutorial since it is easier to see the clear difference between b
the numerator and denominator. Notice something very important about this definition. FRACTIONS ARE JUST ANOTHER WAY TO WRITE DIVISION! Many people get really frustrated and worked up about fractions but you can lay your fears to rest because we are just doing something you already know how to do: division. For example, if I write 6 this could be read as the fraction “six‐fifteenths” or you could just think about 15
it like 6 divided by 15. If I write 16 this could be read as the fraction “sixteen‐fourths” or you could 4
think about it as simply 16 divided by 4. This brings us to what is quite possibly the most important property of fractions and division that makes all operations on fractions possible. If you multiply or divide the numerator and denominator of a fraction by the same amount, the value of the fraction does not change. For example, if you have $12 and you want to divide it evenly between 4 friends, you would calculate 12
which gives $3 to each person. Suppose you doubled the amount of money to $24 but also 4
doubled the amount of people to 8. This is proportionally the same problem and you would now calculate 24
which still gives $3 to each person. THE RESULT IS THE SAME! 8
In general a ain
=
as long as n and b do not equal 0. b bin
Another way to think about this is by using something called the multiplicative identity property. In other words, If you multiply something by 1, it always stays the same. So if you multiply 5 i 1 you will get 5. Now here’s the trick for fractions – think about other ways that you could write the number 1. How about 2
9
1
−26
643.123
or or or or ? They are all the same! They all equal 1. 2
9
1
−26
643.123
So another way to write 5 i 1 might be 5 i
2
9
−26
or 5 i or 5 i
and so on. −26
2
9
Now let’s go back to the previous example with the $12 split into 4 groups or 12
. 4
If we multiply the top by 2 and the bottom by 2 it is the same thing as multiplying by 2
which is just 1! 2
12
= 3
4
12 2
24
i =
= 3 4 2
8
→
Notice what else we just did here. By multiplying the 12 by 2 and multiplying the 4 by 2 we just multiplied the two factions 12
2
and . This brings us to our first operation on fractions: 4
2
To multiply fractions you multiply the two numerators together and multiply the two denominators together. a c
aic
i
=
b d bid For example 4 3
4i3
12
15 2
15 i 2
30
or i
=
=
i =
=
7 10
7 i10
70
3 8
3i8
24
It’s that easy….you just multiply straight across the top and straight across the bottom. No need for common denominators or any conversions, you just multiply. Now there is a reason why we started with multiplication first. It’s the easiest operation for fractions and it helps us better understand all the other steps needed to do division, addition, and subtraction. Let’s look at division next since it is so closely related to multiplication. Take a look at the following example: Suppose I want to divide 5 by 3. I could write this in different ways depending on whether I look at this as a multiplication problem or a division problem. (Remember, multiplication and division are the same basic operation – just in different directions) Division 5
3
Multiplication 5 1
i 1 3
Can you see how these are the same thing since 5 1
5 i1
5
i
=
= ? 1 3
1i 3
3
Consider another example of dividing 6 by 28: Division 6
28
Multiplication 6 1
i
1 28
The reason for bringing this up is because this is the way we need to think about division when we work with fractions. When you divide by a value, it is equal to multiplying by its inverse. a÷b= ai
1
b So let’s say you wanted to divide 32 by 5. You could write 32 / 5 or 32 ÷ 5 or 1
. Can you see how this is the same? 5
think about it in terms of multiplying by saying 32 i
32 i
32
. Or you could 5
1
32 1
32
=
i
=
= 32 ÷ 5 5
1 5
5
And this is how we approach division with fractions. Instead of dividing, we just multiply by the inverse. 2 7
÷ . There’s not really a clean, easy way to divide something 3 4
4
by seven‐fourths so instead we multiply by the inverse instead. 7
For example, consider the problem 2 7
2 4
2i4
8
÷
=
i
=
=
3 4
3 7
3i 7
21
As another example, let’s try 15 1
1
7
÷ . Instead of dividing by we multiply by 4
7
7
1
15 1
15 7
15 i 7
105
÷
=
i
=
=
4
7
4
1
4 i1
4
To divide fractions, you instead multiply by the inverse. (Notice that we only invert the second value, not the first) a c
÷
b d
=
a d
i
b c
=
aid
bic
This now brings us to the last two basic operations: addition and subtraction. The nice thing about these operations is that the rules for addition apply for subtraction as well. However, adding and subtracting fractions is completely different than multiplying and dividing. When adding and subtracting you have to make sure you are working with common denominators. Consider the following example. Using the area diagram below, suppose we want to add 1
2
+ 1
2
and 2
5
2
5
So how do we combine these shapes together? = ? There are 3 items, but what do they represent? Can we call this 3 halves? 3 fifths? You can see that issues arise here since we are trying to combine two things that are totally different. It’s like trying to add 1 orange and 2 apples. That doesn’t give us 3 oranges or 3 apples or even three apple‐oranges. We need to do something to these values before we add them in order to make sure we are combining two of the same things. This is where the idea of a common denominator comes in. We need to split these pieces up so that they are measuring the same value. To do this, we will go back to those original shapes and cut them up differently so instead of 2 pieces versus 5 pieces, they each have the same number to begin with. For this example we will choose 10. 1
2
+ 2
5
5
10
+ 104 Now this is easy to calculate: 5 tenths + 4 tenths = 9 tenths This is the way we need to approach all addition and subtraction with fractions. You CANNOT add or subtract fractions if the denominators are different. 4
1
− . There is no way to find this value since we are trying to compare 6
8
3
3
3
6ths with 8ths. You can’t just take 4 ‐1 and call it or or even . We need to do a little work in 6
8
48
Consider another example: the beginning before we can add or subtract. We need to turn that 6 and 8 into the same value. Remember back to the very first rule about fractions: if you multiply or divide the top and bottom by the same amount, then the fraction doesn’t actually change. We need to use this to our advantage here. Let’s think about what we could multiply by 6 or by 8 so that they end up being the same value. Don’t 6 and 8 both go into 24? So let’s change these both into 24ths. 4 4 16
i =
6 4 24
and 1 3
3
i =
8 3
24
Now we can do our original problem: 4
1
16
3
− can be rewritten as −
and now we can do a 6
8
24
24
fair comparison since these values are measuring the same thing. Now we just simply do 16 – 3 and our final answer would be 13
. 24
To add or subtract fractions, you must always find a common denominator first. Then you combine only the numerators, leaving the common denominator on the bottom. a c a+c
a c a−c
+ =
− =
b b
b
b b
b
Let’s look at a few more examples. Consider 1 3
+ . We know that 9 and 4 both go into 36, so let’s 9 4
use that as our common denominator. 1 4
4
3 9
27
i
=
and i
=
9 4
36
4 9
36
1 3
4
27
31
So + is the same as +
which equals 9 4
36
36
36
What if you are dealing with whole numbers? Let’s try 7 +
2
. 23
Now if you remember mixed numbers, we can easily just combine this into 7
2
and be finished. Let’s 23
stick with our current process, though, to get the practice and understand why this works. The value 7 is the same as 7
7 2
so our problem really says +
. 1
1 23
Making a common denominator here is easy since we can just multiply the first value by 23
23
This gives us 7 23
161
161 2
i
=
which makes our original problem change to + 1 23
23
23 23
which equals 163
2
or 7
23
23
For the final example, let’s look at 8 14
. If you look at the denominators 3 and 2, we need to find −
3 2
something that they both go in to. We could pick 6, 12, 18, 24, 30, and on and on. Let’s stick with the smallest option to keep things easy – we will change both fractions into 6ths. 8 2 16
14 3 42
i =
and i =
3 2
6
2 3
6
16 42
So our problem really reads −
which means we must find 16 – 42. This is a negative value, 6
6
26
which is OK. Our answer will just be a negative fraction. Since 16 – 42 = ‐ 26 our final answer is −
6
Other Bits and Pieces Reducing and Equivalent Fractions: If you notice above, in all the examples and problems we worked out, we never simplified or reduced any fractions. We have been introduced to the idea of equivalent fractions (for example 6 3
= ) as this 8 4
comes from our very first rule of fractions: If you multiply or divide the top and bottom by the same value, the fraction stays the same. So using this rule we can see why the above example is true: 3 2 6
i = 4 2 8
Now most teachers and textbooks ask that you always write your fractions in “simplest” form or “reduced” form which just means that you have to keep dividing until you can make the numerator and denominator into the smallest possible whole numbers that you can. For example, take 36
18
. We can divide both of these by 2, giving us . We can divide both of these by 24
12
6
. As long as you can still divide both values by a common number then you can keep 4
3
reducing. We see that 6 and 4 can still be divided by 2 so we now get which is as low as we go 2
3, giving us because dividing by anything else can’t give us a whole numbers. If you notice that we kept picking very small numbers to divide by (2 and 3 each time) but you don’t have to do this. In fact…the higher the value you reduce by, the less steps you will have! For example, notice that 36 and 24 can both be divided by 12 giving us 3
and we’re done! One step! 2
If you can’t see the big factors right away then it’s fine to use small ones, but just keep in mind that the larger value you divide by, the fewer steps. So any time you are faced with a problem involving fractions, just remember that there is more than one way to write every fraction. If you notice easy ways to reduce throughout the problem, go ahead and do it. The more comfortable you become with converting back and forth between equivalent fractions, the more patterns and shortcuts you will start to notice and the easier fractions will start to become. Finding the Best Common Denominator: Recall that when you are adding or subtracting fractions you always have to find a common denominator before you can combine. In the examples given, we just picked the first thing that came to mind…but it also happened to be the smallest possible denominator that could work. Consider 7
5
+
. If we want to turn 8 and 12 into the same value we have a lot of options to choose 8 12
from. Take a look at the following table of multiples and notice all the common values we could choose: 8 16 24 32 40
48
56
64 72 . . . . 12 24 36 48 60
72
84
96 108 . . . .
And it keeps going on forever and ever. So we could very easily turn these both into something out of 72 or 48 or even 24000, but doesn’t it make sense to pick the smallest, most convenient one….24? This value is called the least common multiple and there are lots of ways to find it. What’s really important though, is making sure you understand HOW to find a common denominator first. Begin by just picking whatever is the easiest for you to find. Then, once you start getting really comfortable, you can begin looking for the lowest common multiple. That’s all for this section on the basic operations of fractions. It may feel like there are a lot of different rules and formulas to remember but here are the key things to keep in mind: ‐
The rules for addition/subtraction are COMPLETELY DIFFERENT than the rules for multiplication and division. Don’t overwhelm yourself by trying to do them all at once. Practice one until you feel really confident then move on to another. ‐
This takes a lot of practice and although fractions seem really awful and difficult at first they end up being your best friend down the road. The more comfortable you become with fractions now, the easier math will be later on when you learn more about division, factors, inverses, rational equations, etc. They will keep coming up for the rest of your life so practice, practice, practice! For some reason, fractions are one of the most widely hated parts of math yet are one of the most useful concepts out there. You are not the only one who struggles with them. It takes time and practice, but you can do it! Good luck and don’t give up! www.mathmadesimple.org