Explanation:
To convert a recurring decimal into a fraction algebraically, let x equal the recurring decimal written out with several repeating digits. Multiply x by powers of 10 (10x, 100x, 1000x) so that two equations have the exact same recurring tail after the decimal point.
Subtract the smaller equation from the larger equation so that all the recurring digits after the decimal point cancel out to zero, then divide and simplify the fraction.
Example 1: Prove algebraically that 0.̇4̇5 = 511.
Solution:
Let x = 0.454545...
Multiply by 100 (2 repeating digits): 100x = 45.454545...
Subtract (100x − x): 99x = 45
x = 4599 = 511 (divide top and bottom by 9)
Answer: 511
Example 2: Convert 0.2̇7̇3 (where 73 recurs) into a fraction in its simplest form.
Solution:
Let x = 0.2737373... → 10x = 2.737373...
Multiply by 1000: 1000x = 273.737373...
Subtract (1000x − 10x): 990x = 271
Divide by 990: x = 271990
Answer: 271990