Fraction to Decimal

What this tool is used for

• Converting a fractional measurement into a decimal for a calculation.
• Turning a mixed number into a decimal for a spreadsheet.
• Checking whether a fraction terminates or repeats.
• Converting an imperial fraction for a metric conversion.
• Confirming a decimal you were given matches a fraction.

Frequently Asked Questions

Which fractions give a terminating decimal?

Only those whose reduced denominator has no prime factors other than 2 and 5, because ten is 2 × 5. So 1/8 terminates at 0.125, while 1/3 and 1/7 repeat forever.

How long can the repeating part be?

Up to one less than the denominator. 1/7 repeats with a six-digit cycle (0.142857…), which is the maximum for 7. Such denominators are called full reptend primes, and 7, 17 and 19 are examples.

How is a repeating decimal written?

With a bar (vinculum) over the repeating digits, or in plain text with the cycle in brackets: 0.1[6] for 1/6. Writing 0.1666 without a marker is an approximation and loses the information that it recurs.

Why does 1/3 + 1/3 + 1/3 not equal 1 in decimal?

Because 0.333 truncated is not 1/3. Summing the rounded values gives 0.999, short by the truncation. The exact fraction sums to 1 — which is one reason financial systems store fractions or integers rather than decimals.

Why can computers not store 0.1 exactly?

Because binary fractions use negative powers of two, and 1/10 has a factor of 5 in the denominator — so it repeats forever in binary just as 1/3 does in decimal. That is the true cause of 0.1 + 0.2 ≠ 0.3.

How do I recover the fraction from a repeating decimal?

Put the repeating digits over the same number of nines: 0.overline{27} is 27/99, which reduces to 3/11. A non-repeating lead-in adds a power of ten to the denominator. Every repeating decimal is rational, and this construction proves it.

Is 0.999… really equal to 1?

Yes, exactly — not approximately. By the rule above it is 9/9, and the difference between it and 1 would have to be a positive number smaller than every positive number. They are two spellings of the same value.

Common errors and gotchas

• Rounding a repeating decimal and treating it as exact, which then fails to convert back.
• Losing the whole-number part of a mixed number.
• Assuming every fraction terminates, when only those whose denominator has factors of 2 and 5 do.
• Reading a slash in a date or a ratio as a fraction.
• Converting to decimal and then working in a trade that uses fractions, which needs converting back.