What is #0.33%# (repeating) as a fraction?

Question

Julia Adams · Accepted Answer

#0.333...%" "=" "0.00bar3" "=" "1/300#.

Take care when working with percentages—they are not the same as standard decimal numbers! First, we use the following concept to convert the percentage to a decimal number: An acronym for "divide this number by 100" is a percent. For example, #50%# means one-half, right? So #50%# is short for #50 divide 100#, which equals #50/100#, or #5/10#, or #1/2# (one-half). So we now know #0.333...%=0.00333...#, which can be concisely written as #0.00bar3#. But what is this decimal number as a fraction? To find this, we use the following rule. The numerator (top number) of our fraction will be the digits under the bar (in this case, #3#). This series of numbers is used to find the denominator, or bottom number: So, for the number #0.00bar3#, our numerator is 3. For the denominator, we count one digit under the bar, so we write one 9. Since there are two digits between the decimal and the bar, we write this many 0's after the 9. Our denominator is then #900#. (This simplified rule works because the digits prior to the three are all zeros; the fraction requires a little more work when these digits are not zeros.) Alright, so this means that our ultimate figure is: #0.333...%" "=" "0.00bar3" "=" "3/900# But wait—#3/900# can be reduced! Both #3# and #900# are multiples of 3, so we can reduce it by dividing both top and bottom by 3: #3/900=(3divide 3)/(900divide3)=1/300# And there we have it! #" "0.bar3%=1/300#.

Avery Alexander · Answer

#1/300# Write #0.3(3" repeating")%# as #color(white)("ddd")0.3(3" repeating")xx1/100# #color(blue)("Consider just the 0.3(3 repeating) part")# Let #x=0.3333333......# Then #10x=3.3333....# So #10x-x=3.3333....# #ul( color(white)("ddddddddddd") 0.3333....larr" Subtract ")# #color(white)("ddddddddddd") 3.0# #10x-x=3# #x(10-1)=3# #x=3/9 = 1/3# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Putting it all together")# But we have to include the #xx1/100# giving: #1/3xx1/100 = 1/300#

Eva Abramson · Answer

undefined 0.33% (repeating) as a fraction is $ \frac{1}{300} $.