Convert Recurring Decimals to Fractions

Let x equal the decimal, multiply so the repeating digits line up, subtract, then solve. For 0.333…: let x = 0.333…, so 10x = 3.333…. Subtracting, 10x − x = 3.333… − 0.333…, so 9x = 3 and x = 3/9 = 1/3. The subtraction removes the never-ending tail, leaving whole numbers.

To shift the digits so the repeating parts line up: 10 when one digit repeats, 100 when two do. For 0.4545…, the repeating part is 45, two digits, so let x = 0.4545… and 100x = 45.4545…. The two tails are identical, so subtracting cancels them exactly: 99x = 45, and x = 45/99 = 5/11.

Make more equations and subtract the pair whose tails match. For 0.1666…: let x = 0.1666…, so 10x = 1.666… and 100x = 16.666…. Then 100x − 10x = 15, so 90x = 15 and x = 1/6. For 0.4777…, 100x − 10x = 47.777… − 4.777… = 43, so 90x = 43 and x = 43/90.

7/9. Let x = 0.777…, so 10x = 7.777…; subtracting gives 9x = 7, so x = 7/9. Any single repeating digit works the same way and ends up over 9: 0.555… = 5/9, and 0.333… = 3/9 = 1/3. Americans say “repeating decimal”; it means the same as recurring.

Yes, every one. Whatever digits repeat, set the decimal equal to x, multiply by 10, 100 or 1,000 to line up the repeating part, subtract to remove it, and solve. That working is also the proof an exam asks for when it says “prove that 0.4545… = 5/11”. A decimal that never repeats, like pi, cannot be written as a fraction.