Why Do We Rationalise the Denominator? Standard Notation While Using Surds/ Radicals
Adding fractions
Fraction arithmetic
When trying to perform basic arithmetic with fractions, many people struggle as, at first glance, the usual rules of arithmetic don't seem to apply. In this guide, we will look at how to add, subtract, multiply and divide fractions, as well as look into why these methods work.
How to add fractions
The first thing we need to be able to do with fractions is add them together. This is the one that most people struggle with and the one that seems least logical. We will start by looking at a simple fraction addition using pictures.
Let's try 1/3 + 1/3.
Let's try 1/3 + 1/3.
Addition example: 1/3 + 1/3
Addition example: 1/3 + 1/3
Looking at the picture above, we can see that by adding 1/3 of the first rectangle and 1/3 of the second rectangle, we end up with 2/3. All that has happened here is that we have added the numerators (the top numbers) together while keeping the denominator (the bottom number) the same.
1/3 + 1/3 = (1+1)/3 = 2/3
If we start with two identical denominators, this method always works. To remember it, try saying it out loud. If you have one third add another one third, you must have two thirds, in exactly the same way as one banana add one banana makes two bananas. Just resist the urge to add the denominators together!
Similarly 2/9 + 4/9 = (2+4)/9 = 6/9 and so on.
This is great, but what about when the denominators are different?
1/3 + 1/3 = (1+1)/3 = 2/3
If we start with two identical denominators, this method always works. To remember it, try saying it out loud. If you have one third add another one third, you must have two thirds, in exactly the same way as one banana add one banana makes two bananas. Just resist the urge to add the denominators together!
Similarly 2/9 + 4/9 = (2+4)/9 = 6/9 and so on.
This is great, but what about when the denominators are different?
Addition example: 1/2 + 1/4
Addition example: 1/2 + 1/4
We can see quickly from the picture above that 1/2 + 1/4 = 3/4. This seems simple when provided with an image to help us, but how do we find this answer without drawing a picture. The answer lies with equivalent fractions.
By looking at the left-hand square above, we can see that 1/2 = 2/4. These are equivalent fractions as they are exactly the same size, but written with different numbers. To quickly convert a fraction into an equivalent fraction, we simply multiply both the numerator and the denominator by the same amount. In this example we multiplied top and bottom by 2 to convert 1/2 into 2/4.
Our sum now changes from 1/2 + 1/4 into 2/4 + 1/4. We can now see how many quarters we have by adding the numerators together, but keeping the denominator the same as before.
2/4 + 1/4 = (2+1)/4 = 3/4
This method works for all fractions. We make the denominators the same using equivalent fractions and then add the new numerators together over this denominator.
By looking at the left-hand square above, we can see that 1/2 = 2/4. These are equivalent fractions as they are exactly the same size, but written with different numbers. To quickly convert a fraction into an equivalent fraction, we simply multiply both the numerator and the denominator by the same amount. In this example we multiplied top and bottom by 2 to convert 1/2 into 2/4.
Our sum now changes from 1/2 + 1/4 into 2/4 + 1/4. We can now see how many quarters we have by adding the numerators together, but keeping the denominator the same as before.
2/4 + 1/4 = (2+1)/4 = 3/4
This method works for all fractions. We make the denominators the same using equivalent fractions and then add the new numerators together over this denominator.
Addition example: 2/5 + 1/6
Addition example: 2/5 + 1/6
In the picture above, you can see that we're now attempting a more difficult addition, 2/5 + 1/6, but still using the same method of converting the fractions into equivalent fractions with the same denominator.
The best way to convert the denominators is to look for the lowest common multiple of both original denominators. In this case, we have 5 and 6, and the lowest common multiple of these is 30 (the smallest number that is in both the 5 and 6 times tables).
We can now convert our fractions by multiplying top and bottom by whichever number is required to make the bottom into 30. In this case we will multiply the 2/5 by 6 top and bottom, and the 1/6 by 5 top and bottom.
2/5 → 2×6 / 5×6 = 12/30 and 1/6 → 1×5 / 6×5 = 5/30
We can see in the picture above that 2/5 does indeed equal 12/30 and likewise 1/6 = 5/30.
Now add the numerators together as before:
2/5 + 1/6 = 12/30 + 5/30 = 17/30
The best way to convert the denominators is to look for the lowest common multiple of both original denominators. In this case, we have 5 and 6, and the lowest common multiple of these is 30 (the smallest number that is in both the 5 and 6 times tables).
We can now convert our fractions by multiplying top and bottom by whichever number is required to make the bottom into 30. In this case we will multiply the 2/5 by 6 top and bottom, and the 1/6 by 5 top and bottom.
2/5 → 2×6 / 5×6 = 12/30 and 1/6 → 1×5 / 6×5 = 5/30
We can see in the picture above that 2/5 does indeed equal 12/30 and likewise 1/6 = 5/30.
Now add the numerators together as before:
2/5 + 1/6 = 12/30 + 5/30 = 17/30
How to subtract fractions
Subtracting fractions works in an almost identical way to adding, except this time we subtract the second numerator from the first one.
Subtraction example: 5/9 - 2/9
Subtraction example: 5/9 - 2/9
As with our first addition example above, think of this question out loud. We have 5 of something and we are subtracting 2 of them, hence we must end with 3 of them. This can be seen in the image above.
Just like adding, we keep the denominator the same.
5/9 − 2/9 = (5−2) / 9 = 3/9
Just like adding, we keep the denominator the same.
5/9 − 2/9 = (5−2) / 9 = 3/9
Subtraction example: 5/6 - 2/3
We now have an example with two fractions with different denominators. Again, we can act just like with addition of fractions and use our equivalent fractions to make the denominators the same. As 3 goes into 6, we only need to convert the second fraction.
2/3 = 2×2 / 3×2 = 4/6
Therefore
5/6 − 2/3 = 5/6 − 4/6 = (5−4)/6 = 1/6
2/3 = 2×2 / 3×2 = 4/6
Therefore
5/6 − 2/3 = 5/6 − 4/6 = (5−4)/6 = 1/6
Subtraction example: 7/8 − 5/12
Let's start by converting the denominators. The lowest common multiple of 8 and 12 is 24, so we will start by multiplying top and bottom of the first fraction by 3 and the second fraction by 2 to get our denominators equal to this.
7/8 = 7×3/8×3 = 21/24
5/12 = 5×2/12×2 = 10/24
We can now subtract:
7/8 − 5/12 = 21/24 − 10/24 = 11/24
7/8 = 7×3/8×3 = 21/24
5/12 = 5×2/12×2 = 10/24
We can now subtract:
7/8 − 5/12 = 21/24 − 10/24 = 11/24
Subtraction example: 6 - 3/5 (subtracting fractions from whole numbers)
To subtract a fraction from a whole number, we must first convert the whole number into a fraction with the same denominator. We start by writing our whole number in the form of a fraction with 1 as the denominator and then converting it into an equivalent fraction as before.
6 = 6/1 = 6×5 / 1×5 = 30/5
We can now subtract as before:
6 − 3/5 = 30/5 − 3/5 = 27/5
Note that we would do the same thing with the whole number if we were dealing with a fraction minus a whole number as well.
6 = 6/1 = 6×5 / 1×5 = 30/5
We can now subtract as before:
6 − 3/5 = 30/5 − 3/5 = 27/5
Note that we would do the same thing with the whole number if we were dealing with a fraction minus a whole number as well.
How to multiply fractions
Multiplying fractions is probably the easiest of the four. When we use the word 'times', as in 2/3 times 4/5, we can think of this as meaning 'of', and so we get 2/3 of 4/5.
To actually compute this, we do exactly as gut instinct would tell us to do; multiply the numerators together and multiply the denominators together.
To actually compute this, we do exactly as gut instinct would tell us to do; multiply the numerators together and multiply the denominators together.
Multiplication example: 2/3 × 4/5
Let's complete the example above, 2/3 × 4/5. We will multiply the numerators 2 and 4 together, and put this above the product of the denominators 3 and 5.
2/3 × 4/5 = 2×4 / 3×5 = 8/15
2/3 × 4/5 = 2×4 / 3×5 = 8/15
Multiplication example: 5/8 × 2/9
Simply multiply the numerators 5 and 2, and multiply the denominators 8 and 9.
5/8 × 2/9 = 5×2 / 8×9 = 10/72 = 5/36
In this example, we have simplified our answer by noticing that 10 and 72 are both even and hence multiples of 2. We can divide numerator and denominator by 2 to get the equivalent fraction 5/36. As 5 and 36 don't share any more factors (other than 1) we can't simplify any further.
5/8 × 2/9 = 5×2 / 8×9 = 10/72 = 5/36
In this example, we have simplified our answer by noticing that 10 and 72 are both even and hence multiples of 2. We can divide numerator and denominator by 2 to get the equivalent fraction 5/36. As 5 and 36 don't share any more factors (other than 1) we can't simplify any further.
Multiplication example: 2/5 × 4 (Multiplying fractions and whole numbers)
To multiply a fraction by a whole number, think of the whole number as a fraction. Any whole number can be converted into a fraction of itself over 1. For example 7 = 7/1, 39 = 39/1 and so on.
By doing this and following our method, we can see that the whole number will multiply the numerator, while the denominator will remain the same as it is effectively being multiplied by 1.
2/5 × 4 = 2×4 / 5 = 8/5
By doing this and following our method, we can see that the whole number will multiply the numerator, while the denominator will remain the same as it is effectively being multiplied by 1.
2/5 × 4 = 2×4 / 5 = 8/5
How to divide fractions
Dividing fractions is a strange one, but the method itself is fairly simple.
To start with think about what happens when we divide by a unit fraction (1/n).
If we divide 6 by 1/3 we are saying how many thirds go into 6. As there are 3 thirds in 1, there must by 6 × 3 = 18 thirds in 6. We have ended up multiplying 6 by the denominator, 3, to get our answer. With a bit of thought we can see this must always be the case; to divided by 1/n, we multiply by n.
If our fraction has a numerator other than 1, it must go into our first number fewer times (as it is larger than 1/n now). If the numerator is twice as large, it will go in half as often; if it is three times as large, it will go in a third as often and so on. Again we can expand this to see that we must divide by the numerator.
A quick method to multiply by the denominator and divide by the numerator is to flip the second fraction upside down and then multiply using our method from earlier.
To start with think about what happens when we divide by a unit fraction (1/n).
If we divide 6 by 1/3 we are saying how many thirds go into 6. As there are 3 thirds in 1, there must by 6 × 3 = 18 thirds in 6. We have ended up multiplying 6 by the denominator, 3, to get our answer. With a bit of thought we can see this must always be the case; to divided by 1/n, we multiply by n.
If our fraction has a numerator other than 1, it must go into our first number fewer times (as it is larger than 1/n now). If the numerator is twice as large, it will go in half as often; if it is three times as large, it will go in a third as often and so on. Again we can expand this to see that we must divide by the numerator.
A quick method to multiply by the denominator and divide by the numerator is to flip the second fraction upside down and then multiply using our method from earlier.
Division example: 2/5 ÷ 3/8
We flip the second fraction and multiply:
2/5 ÷ 3/8 = 2/5 × 8/3 = 16/15
2/5 ÷ 3/8 = 2/5 × 8/3 = 16/15
Division example: 10/17 ÷ 1/2
10/17 ÷ 1/2 = 10/17 × 2/1 = 20/17
Division example: 6/13 ÷ 3 (Dividing a fraction by a whole number)
In this case we just need to divide the numerator by the whole number (think of it as taking six items and splitting them three ways).
6/13 ÷ 3 = 6÷3 / 13 = 2 /13
If the numerator won't divide by the whole number, then we need to convert the whole number into a fraction over 1, flip it upside-down and then multiply as before.
6/13 ÷ 3 = 6÷3 / 13 = 2 /13
If the numerator won't divide by the whole number, then we need to convert the whole number into a fraction over 1, flip it upside-down and then multiply as before.
Division example: 5 ÷ 2/3 (Dividing a whole number by a fraction)
To divide a whole number by a fraction, convert the whole number into a fraction over 1 as before and use the usual division method.
5 ÷ 2/3 = 5/1 ÷ 2/3 = 5/1 × 3/2 = 5×3 / 1×2 = 15/2
5 ÷ 2/3 = 5/1 ÷ 2/3 = 5/1 × 3/2 = 5×3 / 1×2 = 15/2
Summary
Now that we've seen a few examples of fraction arithmetic, let's remind ourselves of the methods used.
Addition: Use equivalent fractions to ensure the denominators are the same, add the new numerators and put over the new denominator.
Subtraction: As with addition, but subtract the numerators.
Multiplication: Put the product of the numerators over the product of the denominators.
Division: Flip the second fraction upside down and then multiply as above.
Addition: Use equivalent fractions to ensure the denominators are the same, add the new numerators and put over the new denominator.
Subtraction: As with addition, but subtract the numerators.
Multiplication: Put the product of the numerators over the product of the denominators.
Division: Flip the second fraction upside down and then multiply as above.
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