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## Fractions Help

Right, so I’m in the process of putting together a site for our A level students. In it I am going to include some pages to help with their studies. I first thought I should cover some of the basics that every year students arrive from GCSE struggling with. This is the first draft of the fractions help page and I’d love to hear your thoughts on it. Is it clear? Does it need amending? Have I overcolicated things? Have I missed anything? I’d also love suggestions of other topics to cover. In the first instance I’m thinking to of an LCM post to act as a companion to this one. An indices post, a quadratics post and perhaps one around trigonometry. Anyway, here’s the fractions post:

Ok, so you should really be able to handle fractions by now. The fact you’re on the A level course means you’ve got a decent GCSE pass, which in turn implies a grasp of the basics, but we know that fractions can be confusing and every year we encounter A Level students who have trouble working with them.

What are fractions?

I hope you know this, but just incase, a fraction is a rational number. It can be expressed as a/b. The denominator (that’s the bottom number) tells you the number of equal parts you split 1 into to find 1 of them, and the numerator (top number) tells you how many you have. I.e. 1/4 means you split 1 into 4 equal parts and have 1 of it, so 0.25 as a decimal.

And the top has to be smaller right?

NO, NO, NO, NO, NO. Improper fractions are absolutely fine, in fact they are infinitely more useful than mixed numbers and I’ve no idea why you need to spend so much time changing improper fractions to mixed numbers at GCSE. Now you’re doing A Level you should avoid mixed numbers like the plague and always change them to improper fractions. Also, stop changing everything to decimals. Fractions are tge more exact form and decimals can give rise to rounding errors. Fractions are much, much better.

How do I add 1 to a fraction?

To add 1 to a fraction you just need to add the amount in the denominator to the numerator. To add 1 to 3/4 add 4 to the numerator to get 7/4. To add 1 to 3/2 add 2 to the numerator to get 5/2. Simples.

This is because anything over anything is equal to 1. 4/4 = 1 5/5 =1 x/x=1 etc. This is the reason why equivalent fractions work:

Equivalent Fractions

You can multiply or divide the numerator and denominator by the same number to get an equivalent fraction. So 4/8 = 1/2 (numerator and denominator divided by 4) that’s because 4/4 is 1 so you are essentially dividing the fraction by 1.

Adding fractions with the same denominator is easy, you just add the numerators. If you gave a fifth, then add another you have two fifths. (1/5 + 1/5 = 2/5). Subtraction is the inverse of addition so it works the same.

The problem comes when we need to add fractions with different denominators. We do this using equivalent fractions to make two fractions with the same denominator. It’s usually best to use the lowest common denominator, this isn’t too important when using numbers, as you can easily simplify later, but when dealing with algebraic fractions it’s always better to find the lowest common denominator as otherwise you may end up with a high order polynomial.

The lowest common denominator is the lowest common multiple of of the denominators. So we would change 1/4 + 1/3 to 3/12 + 4/12, and 1/(x + 1) + 1/(x -1) would become (x – 1)/(x + 1)(x – 1) + (x + 1)/(x + 1)(x – 1)

Multiplying Fractions

This is easy to do, you just multiply the numerators together and the denominators together. So 2/5 x 3/7 = 6/35. The problem is many people forget this, possibly because they never bothered to really understand why. The denominator of a fraction is how many equal parts 1 is broken up into to get the “unit” we are dealing with. So multiplying the units together gives us how many equal parts our answer will deal with. If you break up 1/4 into thirds you get 1/12 (as 3/12 = 1/4). The same works for the numerators:

Ed: I’m going to include a diagram here, and maybe a few others throughout the post.

Dividing Fractions

This is often a source of confusion. And to really understand it you need to really understand what a fraction is. We know the denominator is the amount of equal parts 1 is split into, and the numerator is how many of those parts we have. Which is the same as saying a fraction is what we’d get is we split the numerator into the amount of equal parts specified in the denominator. So 3/4 is 3 divided by 4.

If we imagine dividing 1 by a half, it’s the same as saying how many halves are there in 1. As we get a half by splitting 1 into 2 equal parts, the answer must be 2. Dividing and multiplying are inverse operations, and that is why dividing by 2 is the same as halving something. Multiplying by somethings reciprocal is the same as dividing by it. This works with fractions too, so if we want to divide by 3/4 we can multiply by its reciprocal (4/3). Remember, reciprocal means 1/ and the reciprocal of a fraction is found by switching the numerator and denominator around, or “flipping” it.

Remember, these rules work for all fractions, whether you know the numbers or have them in algebraic form.