What is #0.583333333...# (repeating 3) as a fraction?

4 Answers
Nov 7, 2015

As others have said #0.583bar3 =7/12#. Here is another way to see this.

Explanation:

Let #x=0.5833bar3#

Multiply by powers of 10 to get repeated 3's only, on the right for two different multiples of #x#.

#1000x=583.3bar3#
#100x =58.3bar3#

Subtract to get
#900x=525#

So #x=525/900 = (25xx21)/(25xx36) = 21/36=7/12#

Nov 7, 2015

#7/12#

Explanation:

Here's yet another way:

Knowing that #0.bar(3) = 1/3#, try subtracting that first:

#0.58bar(3) - 0.bar(3) = 0.58bar(3)-0.33bar(3) = 0.25 = 1/4#

So #0.58bar(3) = 1/3+1/4 = 4/12+3/12 = 7/12#

May 9, 2017

#525/900 =7/12#

Explanation:

In addition to all the methods above which explain what to do and why it works, there is also just a simple rule which you can learn and apply quickly to get the fraction. It is based on one of the methods already described.

There are two cases:

  • If ALL the digits recur: write a fraction made up as follows:

#"the recurring digits"/"a 9 for each recurring digit"#

Then simplify if possible.

Eg: #0.432432432.... = 432/999 =16/37#

Eg: #0.62626262.... = 62/99#

  • If only SOME of the digits recur: write a fraction made up as:

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

Then simplify if possible.

We have:

#0color(blue)(.58)color(red)(3)33333... = (583-58)/(color(red)(9)color(blue)(00)) = 525/900#

#525/900 =7/12#

Another example

#7.4561616161... = 7 (4561-45)/9900 = 7 4516/9900#

#=7 1129/2475#

Oct 17, 2017

#0.58bar(3) = 7/12#

Explanation:

If you have access to a pocket calculator, then another way of finding fractions from decimal expansions uses continued fractions.

  • Note down any whole number part of the given number.

  • Subtract the whole number part.

  • Take the reciprocal.

Repeat these steps until there is no remainder, rounding to eliminate rounding errors.

Use the numbers you have noted down to write a terminating continued fraction, then simplify it.

In our example, given #0.58bar(3)#, type as many digits of the expansion as will fit into your calculator, say:

#0.583333333#

This has no whole number part so just take the reciprocal to get:

#1.714285715#

Note down the whole number part #color(red)(1)# and subtract it to get:

#0.714285715#

Take the reciprocal to get:

#1.399999998#

There's an obvious rounding error here, so round to:

#1.4#

Note down the whole number part #color(red)(1)# and subtract it to get:

#0.4#

Take the reciprocal to get:

#2.5#

Note down the whole number part #color(red)(2)# and subtract it to get:

#0.5#

Take the reciprocal to get:

#2#

Since this is a whole number, note it, #color(red)(2)#, and stop.

Taking the numbers we found, we can deduce that:

#0.58bar(3) = 1/(color(red)(1)+1/(color(red)(1)+1/(color(red)(2)+1/color(red)(2)))) = 1/(1+1/(1+2/5)) = 1/(1+5/7) = 7/12#