Question

write each repeating decimal as a fraction in simplest form.
(a)0.16666666666666666666666666666666......66666666.....
(b)o.4166666666666666666666666666666.....666666666666.....

Answer 1

let's do the first one and you can do the same with the other...

Answer 2

okay

Answer 3

let
x = 0.16666...

now,
10x = 1.6666.... and
100x = 16.6666..

notice that 100x - 10x = 90x
and 16.6666.... - 1.6666... = 15

so we have 90x = 15 or x=15/90... reduce that fraction....

Answer 4

do you need anything clarified?

Answer 5

um, I kind of get thegist of it, but I don't understand... you don't need to do the infinite sum of a geometric series?

Answer 6

This one of those things that you can do mentally:\[ 0.4166666666 = \frac{416-41}{900} = \frac 5 {12} \]

Answer 7

so what if it asks for u to solve it wth the infiinite sum of a geometric series way?

Answer 8

That's how @dpaInc solved it.

Answer 9

so this method IS the sum of an Infinite series way?

Answer 10

0.1 + 6/100 + 6/1000 + 6/10000 +...

so

1/10 + sum of infinite geom. series

where an = 6*(1/10)^(n+1))

a = 6/100 and r = 1/10

sum = a/(1-r)
= 6/100 / (1 - 1/10)
= 6/100 / 9/10
= 6/100 * 10/9
=6/90

NOT done... add this to 1/10

1/10 + 6/90 = 15/90 now reduce....

Here is a link: http://en.wikipedia.org/wiki/Geometric_series

More than one way to do this.

Answer 11

thank you johnrr!