Recurring Decimals as Fractions Objectives: Write a recurring decimal as a fraction Terms and Conditions: To the best of the producer's knowledge, the presentation’s academic content is accurate but errors and omissions may be present and Brain-Cells: E.Resources Ltd cannot be held responsible for these or any lack of success experienced by individuals or groups or other parties using this material. The presentation is intended as a support material for GCSE maths and is not a comprehensive pedagogy of all the requirements of the syllabus. The copyright proprietor has licensed the presentation for the purchaser’s personal use as an educational resource and forbids copying or reproduction in part or whole or distribution to other parties or the publication of the material on the internet or other media or the use in any school or college that has not purchased the presentation without the written permission of Brain-Cells: E.Resouces Ltd.

A fraction like 3/8 is another way of writing 3 ÷ 8 Do this division, gives the decimal that is equivalent to 3/8 So 3/8 as a decimal is 3 ÷ 8 = 0.375 To change any fraction into a decimal, we always divide the top (numerator) by the denominator

Change these fractions into decimals 1. 3/5 6. 5/8 2. 1/8 7. 1/4 3. So to change a fraction into a decimal, we divide the numerator (top) by the denominator (bottom) 3/8 as a decimal is 3 ÷ 8 = 0.375 Change these fractions into decimals 1. 3/5 6. 5/8 2. 1/8 7. 1/4 3. 4/5 8. 1/5 4. 3/4 9. 2/5 5. 7/8 10. 1/2

Change these fractions into decimals 1. 3/5 = 0.6 6. 5/8 = 0.625 2. So to change a fraction into a decimal, we divide the numerator (top) by the denominator (bottom) 5/8 as a decimal is 5 ÷ 8 = 0.625 Change these fractions into decimals 1. 3/5 = 0.6 6. 5/8 = 0.625 2. 1/8 = 0.125 7. 1/4 = 0.25 3. 4/5 = 0.8 8. 1/5 = 0.2 4. 3/4 = 0.75 9. 2/5 = 0.4 5. 7/8 = 0.875 10. 1/2 = 0.5

The dot above the three tells us that it’s a recurring 3 Sometimes, the decimal goes on and on. Here is an example: 1 ÷ 3 = 0.33333…. We show that this is a recurring decimal by putting a dot over the number that recurs like this: 1/3 = 0.33333333…  1/3 = 0.3 The dot above the three tells us that it’s a recurring 3

The dot above the three tells us that it’s a recurring 6 Here is another example showing the conversion of 1/6 into a decimal: 1 ÷ 6 = 0.16666666… 1/6 = 0.16 The dot above the three tells us that it’s a recurring 6

Sometimes, the recurring pattern can be several numbers. When, for example, we convert 1/7 into a decimal then 1 ÷ 7 = 0.142857142857142857142857… We show that this is a recurring decimal by putting two dots over the numbers that recur like this: 0.142857142857142857142857… = 0.142857 1/7 = 0.142857

Change these fractions into decimals Change these fractions into decimals. Write down all the figures on your calculator’s display 1. 2/3 6. 5/13 2. 4/9 7. 3/7 3. 4/11 8. 8/9 4. 5/7 9. 9/11 5. 7/12 10. 27/41

Now, give these answers using the notation for recurring decimals 1. 2/3 = 0.666666666 6. 5/13 = 0.384615384 2. 4/9 = 0.444444444 7. 3/7 = 0.428571428 3. 4/11 = 0.363636363 8. 8/9 = 0.888888888 4. 5/7 = 0.714285714 9. 9/11 = 0.818181818 5. 7/12 = 0.583333333 10. 27/41 = 0.658536585

Now, give these answers using the notation for recurring fractions 1. 2/3 = 0.6 6. 5/13 = 0.384615 2. 4/9 = 0.4 7. 3/7 = 0.428571 3. 4/11 = 0.36 8. 8/9 = 0.8 4. 5/7 = 0.714285 9. 9/11 = 0.81 5. 7/12 = 0.583 10. 27/41 = 0.65853

Finding the Fraction for a Given Recurring Decimal We begin by giving four example. Here is the first of these…

What fraction will have the equivalent recurring decimal 0.777777777… ? Let x equal the recurring decimal x = 0.777777777…  10x = 7.777777777… If we multiply both sides of the equation by 10, we get…

Subtract the first equation from the second What fraction will have the equivalent recurring decimal 0.777777777… ? The same decimal x = 0.777777777…  10x = 7.777777777… Subtract the first equation from the second  9x = 7 10x – x = 9x 7.7777… - 7.7777… = 7

We can now find the value of x as a fraction What fraction will have the equivalent recurring decimal 0.777777777… ? x = 0.777777777…  10x = 7.777777777… We can now find the value of x as a fraction  9x = 7  x = 7/9 Check x = 7 ÷ 9 x = 0.77777777…

Here is the second example…

What fraction will have the equivalent recurring decimal 0.2424242424… ? Let x equal the recurring decimal x = 0.24242424…  100x = 24.24242424… If we multiply both sides of the equation by 100, we get…

Subtract the first equation from the second What fraction will have the equivalent recurring decimal 0.2424242424… ? The same decimal x = 0.24242424…  100x = 24.24242424… Subtract the first equation from the second  99x = 24 100x – x = 99x 24.2424… - 0.2424… = 24

We can now find the value of x as a fraction What fraction will have the equivalent recurring decimal 0.2424242424… ? x = 0.24242424…  100x = 24.24242424… We can now find the value of x as a fraction  99x = 24  x = 24/99 Check x = 24 ÷ 99 x = 0.24242424…

Here is the third example…

Let x equal the recurring decimal What fraction will have the equivalent recurring decimal 0.123123123… ? Let x equal the recurring decimal x = 0.123123123…  1000x = 123.123123123… This time, we need to multiply both sides of the equation by 1000. This gives…

So, we subtract the first equation from the second What fraction will have the equivalent recurring decimal 0.123123123… ? The same decimal x = 0.123123123…  1000x = 123.123123123… So, we subtract the first equation from the second  999x = 123 1000x – x = 999x 123.123123… - 0.123123… = 123

Now, we can now find the value of x as a fraction What fraction will have the equivalent recurring decimal 0.123123123… ? x = 0.123123123…  1000x = 123.123123123…  999x = 123 Now, we can now find the value of x as a fraction  x = 123/999 Check x = 123 ÷ 999 x = 0.123123123…

Finally, here is the fourth example…

What fraction is equivalent to the recurring decimal 0.277777… x = 0.277777…  10x = 2.777777…  9x = 2.5  x = 2.5/9 Let x equal the recurring decimal Multiply both sides of the equation by the required power of 10. In this case x 100 Find x as a fraction Subtract first equation from second

What fraction is equivalent to the recurring decimal 0.277777… x = 0.277777…  10x = 2.777777…  9x = 2.5  x = 2.5/9  x = 5/18 We want a fraction with whole numbers so multiply top and bottom by 2 to get… Check x = 5 ÷ 18 x = 0.2777777…

Write these recurring decimals as fractions 1. 0.444444… 6. 0.3131313… 2. 0.1717171… 7. 0.244444… 3. 0.475475… 8. 0.177777… 4. 0.1251251… 9. 0.533333… 5. 0.585858… 10. 0.722222…

Write these recurring decimals as fractions 1. 0.444444… = 4/9 6. 0.3131313… = 31/99 2. 0.1717171… = 17/99 7. 0.244444… = 11/45 3. 0.475475… = 475/999 8. 0.177777… = 8/45 4. 0.1251251… = 125/999 9. 0.533333… = 8/15 5. 0.585858… = 58/99 10. 0.722222… = 13/18