Recurring DecimalsChange recurring decimals into their corresponding fractions and vica versa. 
Do not use a calculator. You can earn a trophy if you get at least 9 questions correct and you do this activity online.
This is Recurring Decimals level 2. You can also try:
Level 1
InstructionsTry your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. 




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A decimal with a repeating digit (or set of digits) is called a recurring decimal.
For example \(0.77777777...\) is a recurring decimal and is called "nought point seven recurring"
\(9.247347347...\) is also a recurring decimal and is called "nine point two four seven recurring"
The period of a recurring decimal is the number of digits in the repeating section so for the second example above the period is three.
A more efficient way of writing out a recurring decimal is by only writing the repeating digit once but putting a dot over the first and last number in the repeating sequence. Another method is drawing a line over the repeating digit or digits.Here are some examples.
\(0.333333333... = 0.\dot 3 = 0.\overline 3\)
\(0.76531531531... = 0.76\dot 53\dot 1 = 0.76\overline{531}\)
A fraction can be converted to a decimal using long division; dividing the numerator by the denominator. If the decimal is recurring the repeating pattern of numbers will be spotted in the long division working. The following example shows the repeating patterns when converting \( \frac{7}{11} \) to a decimal:
There are two common methods for converting a recurring decimal to a fraction:
Let the recurring decimal be represented by \(x\)
$$x = 0.8888888...$$Multiply both sides by 10 (as there is one repeating digit)
$$10x = 8.8888888...$$Subtract the first equation from the second
$$9x = 8$$ $$x = \frac{8}{9}$$Let the recurring decimal be represented by \(x\)
$$x = 1.36363636...$$Multiply both sides by 100 (as there are two repeating digits)
$$100x = 136.36363636...$$Subtract the first equation from the second
$$99x = 135$$ $$x = \frac{135}{99}$$ $$x = \frac{15}{11}$$The method is the same but multiply both sides by 1000.
Example: convert \(0.8888888...\) to a fraction.
This method requires you to know that \(\frac19 = 0.1111111...\)
\(0.8888888...\) is exactly eight times \(0.1111111...\)
$$\therefore 0.8888888... = \frac{8}{9}$$Example: convert \(0.45454545\) to a fraction in its lowest terms.
This method requires you to know that \(\frac{1}{99} = 0.01010101...\)
\(0.45454545...\) is exactly forty five times \(0.01010101...\)
$$\therefore 0.45454545... = \frac{45}{99}$$ $$0.45454545... = \frac{5}{11}$$Example: convert \(0.\dot 61\dot 2\) to a fraction in its lowest terms.
This method requires you to know that \(\frac{1}{999} = 0.\dot 00\dot 1\)
\(0.\dot 61\dot 2\) is exactly six hundred and twelve times \(0.\dot 00\dot 1\)
$$\therefore 0.\dot 61\dot 2 = \frac{612}{999}$$ $$0.\dot 61\dot 2 = \frac{68}{111}$$Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can doubleclick the 'Check' button to make it float at the bottom of your screen.
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Transum,
Wednesday, August 30, 2017
"Here's something to think about, discuss with your friends and share with your teacher:
What is the difference between \(0.\dot 9\) and one?
"
Ann Roberts, London
Sunday, September 27, 2020
"Most fractions convert to recurring (repeating) decimals. Think of ⅓ converting to 0.333...
There are some fractions that convert to terminating decimals. Think of ½ converting to 0.5.
Which fractions convert to terminating decimals? I'll tell you. Write your fraction in its lowest terms n/d. Calculate the prime factorisation of the denominator d. Your fraction will only convert to a terminating decimal if: the prime factorisation of the denominator d contains ONLY 2s and/or 5s. In other words, the prime factorisation might include only 2s, or only 5s or a combinations of 2s and 5s. No other number is allowed.
Let me show you. 1/20 converts to a terminating decimal because the prime factorisation of 20 is 2x2x5 (so only 2s and 5s).
1/9 converts to a recurring decimal because the prime factorisation of 9 is 3x3. (3s are not allowed).
Use this rule to see if you can find some more fractions that convert to terminating decimals."
Sudeep, Twitter
Monday, May 6, 2024
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