From fraction to decimal is just division
A fraction a/b is a division waiting to happen: the bar means “a divided by b.” So 3/4 = 3 ÷ 4 = 0.75 and 1/3 = 1 ÷ 3 = 0.333…. Both 3/4 and 1/3 are rational numbers, and so are their decimal forms — fraction and decimal are just two faces of the same number.
3/4 as a decimal — long-divide 3.000 by 4:
0.75
┌────────
4 │ 3.00
2 8 (4×7=28)
───
20
20 (4×5=20)
──
0 remainder 0 → it stops
So 3/4 = 0.75 (a terminating decimal)Why some decimals stop and others repeat
When the long division eventually hits a remainder of 0, the decimal stops — that is a terminating decimal, like 0.75 or 0.4. When the remainders start cycling and never reach 0, the same block of digits repeats forever — a repeating decimal, like 1/3 = 0.333… or 1/7 = 0.142857142857…. We write the repeating block with a bar over it. Either way the number is still rational; the deciding factor is just whether the denominator's prime factors are only 2s and 5s (terminates) or include something else (repeats).
Turning a repeating decimal back into a fraction
Because a repeating decimal is rational, you can always recover its fraction with a clean algebra trick: multiply by a power of 10 to shift one full repeat, then subtract to cancel the infinite tail. Always finish by writing the fraction in lowest terms.
Write 0.272727… as a fraction.
Let x = 0.272727…
Two repeating digits → multiply by 100:
100x = 27.272727…
x = 0.272727…
Subtract:
99x = 27
x = 27/99
Reduce by gcd(27,99)=9:
x = 3/11 ✓ check: 3 ÷ 11 = 0.2727…