Decimal to fraction calculator

How do I convert a decimal to a fraction?

Write the digits over 10, 100, 1,000… and simplify: 0.375 = 375/1,000, and dividing both by 125 gives 3/8. 2.5 = 25/10 = 5/2, or 2 1/2.

How do I convert a repeating decimal?

Type the repeating part in parentheses: 0.(3) is 1/3 and 0.1(6) is 1/6. For 0.(3): x = 0.333…, 10x = 3.333…, so 9x = 3 and x = 1/3.

How do I convert a fraction to a decimal?

Divide the top by the bottom: 3/8 = 3 ÷ 8 = 0.375. Some fractions repeat forever: 1/7 = 0.(142857).

What fraction of an inch is 0.4375?

7/16 inch. Tape measures use sixteenths: 0.0625 = 1/16, 0.125 = 1/8, 0.25 = 1/4, 0.5 = 1/2. The feet and inches calculator rounds any length to the nearest 1/16.

What if my decimal is rounded, like 0.333?

Then 0.333 is exactly 333/1,000. The calculator also shows the simplest nearby fraction — here 1/3 — which is usually what a rounded decimal stands for.

Why do some fractions give repeating decimals?

A fraction in lowest terms ends in a finite decimal only when its denominator has no prime factors other than 2 and 5, because those are the factors of 10. So 3/8 ends (0.375), but 1/3 (0.(3)), 1/6 (0.1(6)) and 1/7 (0.(142857)) repeat. For a denominator d, the repeating block is at most d − 1 digits long.

How do I turn 0.1(6) or 0.8333… into a fraction?

Let x be the number, multiply by a power of 10 to line up the repeating part, and subtract. For x = 0.1(6): 100x = 16.666… and 10x = 1.666…, so 90x = 15 and x = 1/6. The calculator does the same algebra: 1/6 for 0.1(6) and 5/6 for 0.8(3).

The method for a terminating decimal

A decimal with k digits after the point equals the digits written as a whole number over 10^k, then reduced by the greatest common divisor. So 0.375 is 375/1,000 = 3/8, 0.0625 is 625/10,000 = 1/16, and 2.5 is 25/10 = 5/2. Because the reduction is exact, no rounding is involved when you type the digits you have.

Worked example: a rounded decimal and a measurement

Typed exactly, 0.333 is 333/1000, which is not 1/3. A rounded decimal usually stands for a simple fraction nearby: with denominators up to 100 the nearest simple fraction to 0.333333 is 1/3, to 0.142857 is 1/7, and to 3.14159 is 22/7 (denominators up to 7) or 355/113 (up to 120). For a shop measurement, 0.4375 in is 7/16 in, and 0.8125 in is 13/16 in, both exact sixteenths.

Common mistakes

Treating a rounded decimal as exact is the main one: 0.67 is 67/100 exactly, not 2/3, and the difference matters if you multiply it up. Another is forgetting to mark the repeating digits; 0.333 and 0.(3) are different numbers. A third is stopping the reduction early (375/1,000 is not in lowest terms until both numbers are divided by 125). For a mixed number, convert only the fractional part, keeping the whole part separate, for example 2.5 = 2 + 1/2.

Sources: OpenStax Prealgebra 2e (fractions and decimals chapters) for converting between decimals and fractions, and OpenStax College Algebra for the geometric-series argument behind repeating decimals; continued fractions (Hardy and Wright, An Introduction to the Theory of Numbers) for the nearest simple fraction. Related: fraction calculator, fraction to percent calculator, feet and inches calculator and LCM and GCF calculator.