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how to write 0.877777777 as a fraction

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6 Answers

here are two other ways to solve the problem
1)
 
0.877777...
let x equal the decimal
 
x=0.877777...
multiply both sides by 10 because only one digit repeats
10x=8.7777...
subtract the first equation from the second new equation
 
10x=8.77777...
-  x=0.87777...
  9x=7.90000...
x=7.9/9
x=79/90
2)
 
 
0.877777...
multiply the decimal by 10 and then divide by 10
(10*0.877777...)/10     
(8.77777...)/10
now you should realize that 0.77777....is equal to 7/9
(8  7/9)/10=(79/9)/10=79/90 again  (9*8+7=79)
note: when one digit repeats, the fraction is that digit over 9
0.88888...=8/9, 0.444444...=4/9
when two digits repeat(without a lag), the fraction is the two digits over 99
0.36363636...=36/99=4/11
0.7272727272...=72/99=8/11
look for patterns in repeating decimals; they make your work easier
 
n = 0.87 where underline means repeat forever.
. .10n = 8.77
. . -n = -0.87
10n-n = 7.9
9n = 7.9
n = 7.9/9 = 79/90

Comments

This method is shown in textbooks and is used by Tom. I wanted to show the connection of this problem with 
geometric series which is helpful in more complex situations. Going back to the method you and Tom just used, it can be modified this way.
Given n = 0.8[7] (as Tom used).
Let's put it this way. The repeated part can be written as
 
            7/100 + 7/1000 + ........ = (7/100)[1+ (1/10) +(1/102) + (1/103)......]
 
What we have in square brakets is a sum of geometric series which is equal to
 
                             1/(1-1/10) = 10/9
Thus your number is
 
         0.87777... = (8/10) +(7/100)(10/9) = (8/10) + (7/90) =79/90
Let x=.8777777.... (equation 1)
 
Multiply both the sides by 10
 
10x=8.777777..... (equation 2)
 
subtract the equation 1 from equation 2
 
10x=8.777777.....
   x=  .877777.....
-
—————————
 9x=7.900000...
 
 9x=7.9
 
divide both the sides by 9
 
9x/9=7.9/9
 
x=7.9/9
 
x=79/90
87 =
99
 
 
write   8 and repeated pattern 7 on top ( numerator) , and 9's equal to number of digits at the bottom
denominator.
 
  Like   : 1/ 6 = 0. 1666666
                
           1/6 = 16 /99
             16 * 6 = 69
 
 
             
Let .8[7] be defined as .8777.... where SQUARE brackets are used to denote repeating quantities
 
.8[7]= .8 + [.07] = .8 + [.7]/10
 
Let x= [.7]
 
10x= 7 + x    <----think about that for awhile (it's a recursion relation)
 
x=7/9
 
Now we have all we need
 
.8[7]=8/10+(7/9)/10 = (8+ 7/9)/10 = 79/90

Comments

I am just trying to finish my comment to Steve's solution suddenly interrupted.
Multiply n = 0.8[7] one time by 100 and then by 10, and subtract from each other.
We will obtain
                       90 n =  79   or n = 79/90
Yes.  More than one way to skin a dog.  I like to see a variety of solutions in these forums.  Variety benefits the student.  I liked your geometric series approach too.
 
Cheers!