How do you write 27/99 as a decimal?

1 Answer
Jun 21, 2018

#0.27272727...->0.27bar(27)#

Explanation:

A fraction written as a decimal is either terminating (has a fixed number of decimal places) or has a cycle of digits that repeat for ever. As we have 99 in the denominator I suspect that the decimal has an infinitely repeating set of digits.

#color(blue)("Using a calculator I get "0.272727....)#

The repeating part can be indicated by putting a bar over the appropriate digits. So I would chose to write this as:

#0.27bar(27)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("What if you do not have a calculator?")#

We are dividing 99 into a number that is less (do NOT use the word 'smaller')

However I can do a sort of cheat. 27 is the same as #270xx1/10#

This idea can be repeated as many times as you wish as long as you apply the #xx1/10xx1/10xx# however many # 1/10# you end up with. This will be clearer when I use it.

#color(white)()#
#color(white)()#

#27 ->color(white)("ddd")270 color(magenta)(xx1/10)#

#color(red)(2)xx99-> ul(198larr" Subtract")#
#color(white)("ddddddddd")72#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
But #72<99" so write it as "720xx1/10#

#color(white)("ddddddddd")720color(magenta)(xx1/10)#

#color(red)(7)xx99->color(white)("d")ul( 693 larr" Subtract")#
#color(white)("dddddddddd")27#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(purple)("And so the cycle goes on and on for ever.")#

Thus so far we have:

#color(red)(27)color(magenta)(xx1/10xx1/10) = 0.27#

But the repeats give: #0.27272727...... ->0.27bar27#