what is the procedure for turning .583333... into 7/12

Question

I have 46.58333.... my answer is supposed to be 46 and 7/12 but I do not understand exactly how that was determined. If I put 7/12 into the calculator I get .58333 but how do I figure this out on my own. How does .583333.....become 7/12.

 

Answer 1

Bee, maybe I can help a little.

the decimal is 0.583333...

look at the repetend only, not the lag

the repetend is made up of the 3s and the lag is 58

the lag is what precedes the repetend

how many different digits repeat ? only one, the 3

let x=0.58333...

multiply both sides by 10 to the first power; why ? because one digit repeats, the 3

10x=5.83333...

 - x=0.58333...

_____________

  9x=5.25000...

  9x=5.25

    x=5.25/9

    x=525/900

simplify by dividing both terms by 25

   x=21/36

simplify again by dividing by 3

   x=7/12

here is another example...

0.6818181...

let x=0.681818181...

the lag is 6 and the repetend is 818181...

how many digits repeat ? two, the 8 and the 1

multiply both sides of the equation by 10 to the second power(100) because two digits repeat

100x=68.18181818...

-      x=0.68181818...

__________________

  99x=67.50000000...

      x=67.5/99

      x=675/990

simplify by dividing by 5

      x=135/198

simplify again by dividing by 9

      x=15/22

look at your original problem 0.58333...

here is another way to solve the problem

multiply the decimal by the power of ten that puts the decimal point between the lag and the repetend

after that divide by the same power of ten

(100)(0.58333...)/(100)

58.333.../100

(58 1/3)/100      0.333...=3/9=1/3, 0.555...=5/9, and so on

also 0.727272...=72/99=8/11, 0.272727...=27/99=3/11 and so on

also 0.123123123...=123/999=41/333, 0.351351351...=351/999=117/333=39/111=13/37 and so on

getting back to the problem...

(58 1/3)/100=(175/3)/100=175/300=7/12

look at 0.6818181...

(10)(0.6818181...)/10

6.818181.../10

(6 81/99)/10

(6 9/11)/10=(75/11)/10=75/110=15/22

if you have any more questions let me know

      

 

 

 

Answer 2

Hi Bee,

 

Lets proceed as follows:

 

Let x=0.58333....

 

Then, 100x=58.333.... And 1000x=583.333....

 

Consider 

 

1000x-100x=900x

                 =583.333...-58.333...

                 =525

 

Solving 900x=525 leaves us with 

 

x=525/900

  =105/180

  =21/36

  =7/12