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Mathematics · pattern A, Numeric fields

Decimal to a fraction in lowest terms.

Enter a decimal to get the equivalent fraction reduced by its greatest common divisor, with the power of ten the conversion started from.

Inputs

The formula used

write the decimal over a power of ten, then divide by the greatest common divisor

The number of decimal places sets the power of ten. Recurring decimals need the algebraic method — multiply by a power of ten and subtract.

Fraction

3/8

Numerator before reducing
375
Denominator before reducing
1,000
Greatest common divisor
125
As a percentage
37.500000 %
Reciprocal
2.66666667
Check: back to a decimal
0.37500000

Recurring decimals need algebra, not counting.

For 0.333… let x be the value, multiply by ten to get 10x = 3.333…, then subtract: 9x = 3, so x = 1/3. That subtraction removes the infinite tail, and the same method works for any recurring block using the matching power of ten.

Questions about decimal to fraction

How do I convert 0.142857 recurring?
Multiply by 10⁶ because the block is six digits, subtract the original and solve. The answer is 1/7.
Why reduce by the GCD?
Because lowest terms is the conventional form and the smallest numbers to work with. 375/1000 and 3/8 are the same number.
Can every decimal become a fraction?
Every terminating or recurring one. Irrational numbers cannot, by definition.

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Arithmetic runs in double-precision floating point, so results beyond about fifteen significant figures are not exact. Where a question wants an exact fraction or surd, keep the exact form rather than a decimal.