How do you convert 2.136 (36 repeating) as a fraction?

Question

Eli Assouline · Accepted Answer

#2.1bar(36) = 47/22#

Repeating decimals can be indicated with a notation that positions a bar above the repeating pattern.

The original decimal representation is written using that notation:

#2.1bar(36)#

To make this into an integer, multiply by #10(100-1) = 1000-10#. The factor #10# here is to shift the repeating section to just after the decimal point. the factor #(100-1)# shifts the number by an additional #2# places (the length of the repeating pattern) and subtracts the original number to cancel out the repeating part:

#(1000-10)*2.1bar(36) = 2136.bar(36) - 21.bar(36) = 2115#

Then divide both sides by #(1000-10)# and simplify to find:

#2.1bar(36) = 2115/(1000-10) = 2115/990 = (color(red)(cancel(color(black)(3*3*5)))*47)/(color(red)(cancel(color(black)(3*3*5)))*22) = 47/22#

Julia Adams · Answer

undefined To convert 2.136 (36 repeating) as a fraction:

1. Let x = 2.136 (36 repeating).
2. Multiply x by 100 to eliminate the decimal: 100x = 213.636363...
3. Subtract x from 100x to eliminate the repeating decimal: 100x - x = 213.636363... - 2.136363... = 211.5
4. Subtract the original x from the result: 100x - x = 211.5 - 2.136 = 209.364
5. Now, write down the equation: 100x - x = 209.364
6. Solve for x: 99x = 209.364
7. Divide both sides by 99: x = 209.364 / 99
8. Simplify the fraction: x = 209.364 / 99 = 209364 / 9900
9. Reduce the fraction to its simplest form: x = 2327 / 1100