Quick question

Let x = .72666...

100 x then equals  72.666...

1000x = 726.666.....

1000x - 100x = 900x     substitute

726.666.... - 72.666.... = 900x

654 = 900x      sove for 'x'

654/900  = x =

218/300=

109/150

0.7266666666.............etc. Take the reciprocal of the fraction

1.376146788990825688073394508038 - 1      Do the same......

2.6585365853658536585365852766211 - 2    Do the same......

1.5185185185185185185185187242798 - 1    Do the same.......

1.9285714285714285714285706632653 -1     Do the same.......

1.076923076923076923076923964497 - 1      Do the same.......

13

Now, start with 13 and multiply it by the number above it, and by the number above that.......and so on until the the 2nd last number.

13 x 1.0769...... x 1.928571..... x 1.51851......x 2.658536...... =109 - This is the numerator of the fraction.

Then multiply 109 x 1.3761467........ =150 - this is your denominator. So, your fraction is:

109/150 - This is called "continued fraction method"

Note: EP's method above is much simpler than this, but I thought you should know about this method because "continued fractions" are very important in Math.

Here's another way

.726...... [ 6 repeating  ]

Forget the decimal

Subtract  the   non-repeating part from the whole  = 726 - 72  = 654    (1)

Write a number that  has the same number of 9's  as the number of digits in the repeating part and append to this  the same number  of 0's as the number of digits in the non-repeating part  = 900   (2)

Put   (1)  over (2)   =   654/900  = 109/150

Thank you all so much!  This was super helpful!