How would you convert a repeating decimal like 1.27 or 0.6 into a fraction?

2 Answers
Apr 7, 2015

The examples in your question are not examples of repeating decimals. A repeating decimal is one in which the same number is repeated infinitely. An example would be #0.666...#, also written as #0.bar6#, or #0.0909...#, also written as #0.bar09#.

#1.27=1 (27)/(100)#

#0.6=6/10=3/5#

Oct 6, 2015

The algorythm is given in the explanation.

Explanation:

The fractions you wrote in the questions are not repeating. You probably meant fractions like #1.(27)# or #0.(6)#.

If you want to sugest that a decimal fraction has a period, you have to put it (the period) in brackets. Other way of indicating a period may be a horizontal bar over the period as suggested in the previous answer. So you can write the repeating fraction as: #1.(27)# or #1.bar(27)#

To show how such decimals can be changed into a fraction let's look closer at the decimals. Let's take #1.(27)#. It can be written as:

#1.(27)=1+0.27+0.0027+0.000027+...#. So you can see, that the fraction can be written as #1# plus the sum of an infinite geometrical sequence, for which: #a_1=0.27#, and #q=0.01#.

Since #|q|<1# the sequence is convergent, so you can calculate the finite sum of all terms.

#S=a_1/(1-q)#

#S=0.27/(1-0.01)=0.27/0.99=27/99=3/11#

So now you can write, that #1.(27)=1+3/11=1 3/11#