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Well the trick here is that any repeating decimal can be represented as a fraction by taking the Nr as the repeated part and the Dr as the number of 9s as the number of digits that gets repeated.

So say for instance .4545454545.... = 45/99

.405405405.......... = 405/999

So in our case we have .1405405405...

So multiply by 10 and divide by 10.

So we have 1.405405405.../10 = (1 + .405405405405)/10

You know that both Nr and Dr are divisible by 9 (as the sum of the digits in each number is divisble by 9).Now you will get 45/111 now again you know this is divisble by 3.

Anybody knows a good link to a site explaining these division rules.

Express the repeating decimal .1405405405405405.... as a fraction in lowest terms.

=26/185

Alright guys, here is the process:
suppose x=.1405405405405405, which is actually is not a repeating decimal. lets make it as repeting decimal by multiplying 10 (because with this multiplication 1 comes before the decimal and remaining decimal will be the repeating decimal) as under:
(i) 10x=1.405405405405
(ii) multiply this eq by 10^3 (because there are 3 repeating numbers: 4, 0, and 5) => 1,000(10x)=1405.405405405405405405405405405405
(iii) now, substract 10x from both side (because we are eliminating the decimal)

9,990x=1405.405405405-1.405405405

x=1404/9990=26/185

i believe, this method can be applied in any fractions. for example
x=0.2222222222222222222222222222222
(i) x=0.222222222222222
(ii) 10x= 2.2222222222222222
(iii) 9x=2
x =2/9

You know that both Nr and Dr are divisible by 9 (as the sum of the digits in each number is divisble by 9).Now you will get 45/111 now again you know this is divisble by 3.

thank you 700Plus for the tip!!, now I remember reading this on the Kaplan Math workbook....

Re: Express the repeating decimal .1405405405405405.... as a [#permalink]

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22 Dec 2014, 11:02

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