Find the fraction that equals 0.7266666666 (repeating 6's)

Not sure how to do this one, thanks!

AnonymousConfusedGuy
Jan 15, 2018

#1**+1 **

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

ElectricPavlov
Jan 16, 2018

#2**+1 **

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.

Guest Jan 16, 2018

#3**+1 **

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

CPhill
Jan 16, 2018