How do you convert the recurring decimal 0.bar(32) to a fraction?

2 Answers
Apr 24, 2018

x = 32/99

Explanation:

x = 0.bar(32)

2 digits are recurring :
100x = 100xx0.bar(32)
100x = 32.bar(32)

=> x = 0.bar(32) and 100x = 32.bar(32):

100x - x = 32.bar(32) - 0.bar(32)
99x = 32
x = 32/99

Apr 24, 2018

0.bar(32) = 32/99

Explanation:

There is a nifty short cut method to change recurring decimals into fractions:

If all the digits recur

Write a fraction as :

("the recurring digit(s)")/( 9" for each recurring digit")

Then simplify if possible to get simplest form.

0.55555..... = 0.bar5 = 5/9

0.272727... = 0.bar(27)= 27/99 = 3/11

0.bar(32) = 32/99

3.bar(732) = 3 732/999= 3 244/333

If only some digits recur

Write a fraction as:

("all the digits - non-recurring digits")/(9 " for each recurring " and 0 " for each non-recurring digit")

0.654444... = 0.65bar4 = (654-65)/900 = 589/900

0.85bar(271) = (85271-85)/99900 = 85186/99900 = 42593/49950

4.167bar(4) = 4 (1673-167)/9000 = 4 1506/9000= 4 251/1500