how do you write this rational number in simplest form for each decimal expansion??? help

Question

         ----

1). 0.15

           ----

2). 1.290

Answer 1

let n=0.151515...

two digits repeat in the repetend

multiply both sides of the equation by 100 (10²)

100n=15.151515...

now subtract the first equation from this new one

 

100n=15.151515...

-

      n=0.151515...

_________________

    99n=15

       n=15/99

       n=5/33

 

n=1.2909090...

two digits repeat so multiply both sides of the equation by 100

100n=129.090909...

-

     n=    1.290909...

_________________

   99n=127.8

       n=127.8/99

       n=1278/990

       n=142/110 (divide both terms by 9)

       n=71/55 (divide both terms by 2)

       n=1 16/55  (if you divide 16 by 55 you get 0.2909090...)

 

Here's another solution for 1.2909090...

multiply (and divide) the number by that power of ten that separates the lag from the repetend

that number is 10

10(1.2909090...)/10

12.909090.../10

there is a shortcut for expressing a repeating decimal as a fraction if there is no lag

examples:

0.333...=3/9=1/3

0.454545...=45/99=5/11

0.123123123...=123/999=41/333

do you see the pattern ??

back to the problem...

12.909090.../10=(12 90/99)/10=(12 10/11)/10

(12 10/11)/10=(142/11)/10=142/110=71/55=1 16/55 again