Changing Recurring Decimals into Fractions 0.833… = 0.151515… = 0.1566566566… = © T Madas

A recurring decimal can always be written as a fraction. A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern. = 0.333… 1.1666… 0.8333… 2.58333… 0.0913913913… 0.285714285714285714… 1 3 7 6 5 31 12 913 9990 2 A recurring decimal can always be written as a fraction. © T Madas

These recurring decimals are worth remembering. A recurring decimal is a never ending decimal, whose decimal part repeats with a pattern. = 0.333… 0.666… 0.1666… 0.8333… 1 3 2 6 5 These recurring decimals are worth remembering. © T Madas

A terminating decimal comes to an end after a number of decimal places. = 0.25 0.4 0.52 0.3875 2.2975 1.4755859375 1 4 2 5 13 25 31 80 919 400 1511 1024 © T Madas

THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM It turns out that: The number we are dividing (i.e. the numerator) plays no role into whether the decimal will be terminating or recurring. The divisor (i.e. the denominator) is important: If the denominator can be broken into prime factors of 2 and/or 5 only then the decimal will be terminating. If the denominator contains any other prime factors then the decimal will be recurring. THIS ONLY HOLDS TRUE PROVIDED THE FRACTION IS IN ITS SIMPLEST FORM © T Madas

Converting Recurring Decimals into Fractions © T Madas

Convert 0.444… into a fraction Let x = 0.444… Since the recurring decimal has a one-digit pattern we multiply this expression by 10 10x = 4.444… x = 0.444… 9x = 4. ... 4 9 x = 4 9 0.444… = © T Madas

Convert 0.363636… into a fraction Let x = 0.363636… Since the recurring decimal has a two-digit pattern we multiply this expression by 100 100x = 36.3636… x = 0.3636… 99x = 36. ... 36 99 4 11 x = = 4 11 0.363636… = © T Madas

Convert 0.411411411… into a fraction Let x = 0.411411411… Since the recurring decimal has a three-digit pattern we multiply this expression by 1000 1000x = 411.411411… x = 0.411411… 999x = 411. ... 411 999 137 333 x = = 137 333 0.411411411… = © T Madas

Convert 0.3777… into a fraction Let x = 0.3777… Since the recurring decimal has a one-digit pattern we multiply this expression by 10 10x = 3.777… x = 0.377… 9x = 3. 4 ... 3.4 9 34 90 17 45 x = = = 17 45 0.3777… = © T Madas

Convert 1.01454545… into a fraction Let x = 1.01454545… Since the recurring decimal has a two-digit pattern we multiply this expression by 100 100x = 101.454545… x = 1.014545… 99x = 100. 4 4 ... 100.44 99 10044 9900 2511 2475 x = = = 279 275 0.01454545… = © T Madas

Convert 2.9135135135… into a fraction Let x = 2.9135135135… Since the recurring decimal has a three-digit pattern we multiply this expression by 1000 1000x = 2913.5135135… x = 2.9135135… 999x = 2910. 6 ... 2910.6 999 29106 9990 539 185 x = = = [HCF:54] 539 185 2.9135135135… = © T Madas

Convert 0.153846153846153846… into a fraction Let x = 0.153846153846153846… Since the recurring decimal has a six-digit pattern we multiply this expression by 1000000 1000000x = 153846.153846153846… x = 0.153846153846… 999999x = 153846. ... 153846 999999 17094 111111 5698 37037 518 3367 2 13 x = = = = = ÷9 ÷3 ÷11 ÷259 2 13 0.153846153846153846… = © T Madas

It is worth noting a pattern in some recurring decimals: 4 9 31 99 107 999 0.444… = 0.313131… = 0.107107… = 7 9 8 99 23 999 0.777… = 0.080808… = 0.023023… = 5 9 37 99 163 999 2.555… = 2 1.373737… = 3.163163… = 1 3 This might save a bit of work when converting: “write as a decimal” 5 11 x9 5 11 45 99 0.454545… = = x9 © T Madas

© T Madas

by recognising patterns: 17 33 Write as a recurring decimal Method A 5 1 5 1 17 33 by division: 3 3 1 7 = 0.515151… 5 17 5 17 Method B by recognising patterns: x3 17 33 51 99 0.515151… = = x3 © T Madas

© T Madas

by immediate recognition: Calculate the mean of 0.6 and 0.16 giving your final answer as a recurring decimal by immediate recognition: Mean add them divide by 2 2 3 0.6 0.666… = = 1 6 0.16 0.1666… = = Method 1 Method 2 x 2 2 3 1 6 4 6 1 6 5 6 0.6666… + + x 2 5 12 0.1666… = = = + 2 2 2 0.8333… 1 4 1 6 6 4 1 6 6 1 2 5 2 2 8 3 3 3 8 8 8 1 1 1 5 12 5 12 = 0.416 = 0.416 © T Madas