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How to change repeating decimals to fractions proof

January 23, 2012 by Simon 11 Comments

Repeating decimals to fractions..

.. is a favourite GCSE question, usually grade 7+ and worth around 2/4 marks.

A repeating decimals to fractions question is a little more ‘abstract’ but, once you understand the principles, it should be relatively straightforward. The answer is to use algebra to show a formal proof. As with most mathematics aim for a logical progression with the equals sign in the centre of your working.

The reason for this type of question is that fractions, decimals and percentages can all represent the same information and it’s good to be able to swap between them – particularly for presenting or comparing information.

Actually, any number can be written in “decimal form” and there three different types:

  • Exact (sometimes called terminating) – a decimal where you can write down all its digits ie. 16.125. These are the most common types of decimals found in exam questions, apart from…
  • Irrational – a decimal that doesn’t repeat such as pi or Euler’s number e (to 50 decimal places 2.71828182845904523536028747135266249775724709369995… try and find the repeats!)

However, there’s another type of decimal form called

  • Recurring (sometimes also called repeating) – a decimal which goes on forever and some of the digits are repeated forever i.e. 7.142142142142142… (142 is repeated) Sometimes recurring decimals are written with a bar over the digits which are repeated, or with dots over the first and last digits that are repeated.
Converting repeating decimals to fractions involves a little bit of mental gymnastics – but it’s also quite satisfying to follow the proof through. I hope you enjoy the video!
Watch on YouTube
How to change recurring or repeating decimals into fractions proof
You might also like to visit these other posts:
Convert recurring decimals to fractions; GCSE maths level 7+
How to add, subtract, multiply, divide decimals

Comments

  1. TheWendyBird says

    June 10, 2015 at 12:33 pm

    Thank you! Ugh my exam is tomorrow so I am cramming – I have a feeling I am
    going to be using your videos quite a lot today :’) Thank you so much, you
    explain things very quickly and simply.

    Reply
  2. matt selwob says

    May 6, 2015 at 8:34 pm

    Thanks

    Reply
  3. leah38521 says

    February 3, 2015 at 6:30 pm

    This was a really helpful video thankyou for uploading it

    Reply
  4. rove lopez says

    October 2, 2014 at 8:29 pm

    thank you so much for you help i was so straggle with it but now i know
    how to do

    Reply
  5. Jake Joyce says

    March 10, 2014 at 9:01 am

    Thank you 🙂

    Reply
  6. Simon Deacon says

    November 5, 2013 at 9:48 pm

    That’s great and thanks for letting me know – all best S

    Reply
  7. Emma Balmer says

    November 5, 2013 at 8:02 pm

    This is great!! Thank you so much! Brilliant!!

    Reply
  8. Simon Deacon says

    February 15, 2013 at 3:37 pm

    Hi – no, it’s usually non calculator. Rgds S

    Reply
  9. Ali Abokar says

    February 15, 2013 at 3:22 pm

    is this for the calculator paper?

    Reply
  10. Vay Skabrin says

    June 16, 2012 at 9:40 pm

    you’re sooo helpful thanks so much !!

    Reply
  11. Misznovember says

    June 3, 2012 at 1:21 am

    thanks 🙂 i was stuck on that exact question! (the last one)

    Reply

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