Recurring decimals

How to start… Write the following in full to 9 decimal places… a) 0.6 b) 0.53 c) 0.72 d) 0.479 e) 0.328 ● ● ● ● ● ● ● ●

How to start… Write the following in full to 9 decimal places… a) 0.6 b) 0.53 c) 0.72 d) 0.479 e) 0.328 ● ● ● ● ● ● ● ● → 0.666666667 → 0.533333333 → 0.727272727 → 0.479479479 → 0.328282828

Learning Objective How do we convert recurring decimals to fractions?

Learning Outcomes To convert fractions to decimals To convert a recurring decimal to a fraction

Fraction to decimal How do we change a fraction to a decimal?

Fraction to decimal We divide the numerator by the denominator.

Fraction to decimal e.g. 8 1

Fraction to decimal e.g. 8 1. 0 0 0 0.1 2 5

Fraction to decimal Sometimes the decimal will be a recurring decimal 3 1. 0 0 0 e.g.

Fraction to decimal Sometimes the decimal will be a recurring decimal 3 1. 0 0 0 0.3 3 3 e.g. Because the same division is repeating over and over we can say it is recurring. The digits which recur are marked with a dot above them.

Fraction to decimal

Fraction to decimal 11 3. 0 0 0 0 0.2 7 2 7

Fraction to decimal Change into a decimal.

Fraction to decimal 6 5. 0 0 0 0 0. 8 3 3 3 Change into a decimal. Notice only the 3 has a dot over it as only the 3 recurs.

Fraction to decimal Change these fractions into decimals.

Learning Outcomes To convert fractions to decimals To convert a recurring decimal to a fraction

The algebra of recurring decimals What is a recurring decimal?

The algebra of recurring decimals A decimal where the digit(s) repeat.

The algebra of recurring decimals

The algebra of recurring decimals Check this on a calculator by dividing 4 by 9.

Let’s change 0.7 into a fraction The algebra of recurring decimals

Let x= 0.7 so . . Let’s change 0.7 into a fraction The algebra of recurring decimals

Let x= 0.7 so 10x = 7.7 . . Let’s change 0.7 into a fraction Multiply! The algebra of recurring decimals

Let x= 0.7 so 10x = 7.7 . . 10x= 7.7777777777777777... x= 0.7777777777777777... Let’s change 0.7 into a fraction Multiply! Subtract! The algebra of recurring decimals

Let x= 0.7 so 10x = 7.7 . . 10x= 7.7777777777777777... x= 0.7777777777777777... 9x= 7 Let’s change 0.7 into a fraction Multiply! Subtract! The algebra of recurring decimals

Let x= 0.7 so 10x = 7.7 . . 10x= 7.7777777777777777... x= 0.7777777777777777... 9x= x= 7 9 Let’s change 0.7 into a fraction Multiply! Subtract! Divide! The algebra of recurring decimals 7

Change 0.47 into a fraction . . The algebra of recurring decimals

Change 0.47 into a fraction . . Let x= 0.47 . . The algebra of recurring decimals

Change 0.47 into a fraction . . Let x= 0.47 . . Multiply! The algebra of recurring decimals With what this time?

Change 0.47 into a fraction . . Let x= 0.47 . . Multiply! The algebra of recurring decimals With 100

Change 0.47 into a fraction . . Let x= 0.47 . . so 100x = 47.47 . . Multiply! The algebra of recurring decimals

Change 0.47 into a fraction . . Let x= 0.47 100x= 47.4747474747474747... x= 0.4747474747474747... . . so 100x = 47.47 . . Multiply! Subtract! The algebra of recurring decimals

Change 0.47 into a fraction . . Let x= 0.47 100x= 47.4747474747474747... x= 0.4747474747474747... 99x= 47 . . so 100x = 47.47 . . Multiply! Subtract! Divide! The algebra of recurring decimals

Change 0.47 into a fraction . . Let x= 0.47 100x= 47.4747474747474747... x= 0.4747474747474747... 99x= x= 47 99 . . so 100x = 47.47 . . Multiply! Subtract! Divide! The algebra of recurring decimals 47

Steps: 1. Write the recurring decimal out to nine decimal places 2. Multiply the decimal by powers of 10 until the part of the decimal that repeats is in full at the front 3. Write the amount that you multiplied by ‘ ’ = the result from above 4. Write ‘ the original decimal 5. Subtract the two 6. Divide to get The algebra of recurring decimals

Recurring Decimals

Change 0.39 into a fraction . . Let x= 0.39 100x= 39.3939393939393939... x= 0.3939393939393939... 99x= x= 39 99 . . so 100x = 39.39 . . Multiply! Subtract! Divide! 39 Recurring Decimals

Change 0.39 into a fraction . . Let x= 0.39 100x= 39.3939393939393939... x= 0.3939393939393939... 99x= x= 39 99 . . so 100x = 39.39 . . Multiply! Subtract! Divide! 39 = 13 as required 33 Recurring Decimals

Challenge 0.45 .

Challenge 0.45 . Let x= 0.45 .

Challenge 0.45 . Let x= 0.45 . so 100x = 45.5 .

100x= 45.555555555555555… x= 0.455555555555555… Challenge 0.45 . Let x= 0.45 . so 100x = 45.5 .

100x= 45.555555555555555… x= 0.455555555555555… Challenge 0.45 . Let x= 0.45 . 99x= 45.1 so 100x = 45.5 .

100x= 45.555555555555555… x= 0.455555555555555… Challenge 0.45 . Let x= 0.45 . 99x= 45.1 990x= 451 so 100x = 45.5 .

100x= 45.555555555555555… x= 0.455555555555555… Challenge 0.45 . Let x= 0.45 . 99x= 45.1 990x= 451 x= 451 990 = 41 90 so 100x = 45.5 .

Change 0.213 into a fraction . . Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 . . so 1000x = 213.213 . . Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 . . so 1000x = 213.213 . . Multiply! Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 1000x= 213.213213213213213... x= 0.213213213213213... . . so 1000x = 213.213 . . Multiply! Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 1000x= 213.213213213213213... x= 0.213213213213213... . . so 1000x = 213.213 . . Multiply! Subtract! Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 1000x= 213.213213213213213... x= 0.213213213213213... 999x= . . so 1000x = 213.213 . . Multiply! Subtract! 213 Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 1000x= 213.213213213213213... x= 0.213213213213213... 999x= . . so 1000x = 213.213 . . Multiply! Subtract! Divide! 213 Recurring Decimals

Change 0.213 into a fraction . . Let x= 0.213 1000x= 213.213213213213213... x= 0.213213213213213... 999x= x= 213 999 . . so 1000x = 213.213 . . Multiply! Subtract! Divide! 213 Recurring Decimals

Change these recurring decimals into fractions. Recurring Decimals (This is different from the previous example.) (= 0.235235235….)

Change these recurring decimals into fractions. Recurring Decimals

Learning Outcomes To convert fractions to decimals and vice versa To convert a recurring decimal to a fraction

Extension Convert the recurring decimal to a mixed number. Give your answer in its simplest form.

Extension Convert the recurring decimal to a mixed number. Give your answer in its simplest form. x= 1000x=2136.363636… 1000x=2136.36363636… 10x= 21.36363636… 990x= 2115 x=

Extension Convert the recurring decimal to a mixed number. Give your answer in its simplest form.

Extension Convert the recurring decimal to a mixed number. Give your answer in its simplest form. x= 100x=206.66666666… 100x=206.66666666… 10x= 20.66666666… 90x=186 x=

Learning Objective How do we convert recurring decimals to fractions?

Recurring decimals

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