0.16 repeating as a fraction
0.(16) = 16/99.
Said aloud, this is the same phrase as a different number. Here the repeating block is 6 in one reading and 16 in the other, so the answers differ: 0.(16) is 16/99, while 0.1(6) is 1/6. Which one you meant depends on where the bar sits over the digits.
Checked by test against every decimal in this family · How we check
How it gets there
- Call the number x, so x = 0.(16). It cannot go over a power of ten the way a terminating decimal can, because the digits never stop.
- Multiply by 100, so that it and the original have the same endless tail: 100x = 16.16… against x = 0.16…
- Subtract. The tails are identical, so everything after the point cancels exactly: 99x = 16, which is 16/99.
- Nothing divides both, so that is the answer: 16/99.
Questions people actually ask
- What is half of 0.(16)?
- 8/99 — a repeating decimal, so the fraction is the only exact way to write it. Halving a fraction doubles the denominator rather than halving the top, unless the numerator is even and cancels. Doubled instead, 16/99 is 32/99.
- What is 0.(16) as a fraction?
- 16/99, in lowest terms. 16/99 is the same value before reducing, and dividing both by 1 gets there.
- Why can't 0.(16) be written over a power of ten?
- Because the digits never stop, and a power of ten only clears a decimal that terminates. The algebra above multiplies by two different powers of ten so the endless tails line up, then subtracts them away — which is the standard proof that every repeating decimal is a rational number.
- Is 0.(16) a rational number?
- Yes. A number is rational exactly when it can be written as one whole number over another, and 16/99 is that. Every repeating decimal is rational and every rational number's decimal either terminates or repeats — the two statements are the same fact from opposite ends.
- Is 16/99 in simplest form?
- Yes. The numerator and denominator share no factor above 1, which is what simplest form means. 16/99 is the same number written before that reduction.
- What is 0.(16) on a tape measure?
- It falls between marks. The nearest sixteenth is 3/16, which is off by about 0.66 mm on an inch — close enough for carpentry and not for machining.
- What is 16/99 as a decimal?
- 0.(16). This page runs the conversion in the direction people ask for it, and dividing 16 by 99 runs it back the other way.
The same number, other ways
- Fraction, lowest terms
- 16/99
- Before reducing
- 16/99
- As a percentage
- 16.161…%
- As a ratio
- 16:99
- Equivalent fractions
- 32/198, 48/297, 64/396, 80/495
- Nearest sixteenth on a tape
- 3/16 (nearest)
A tape measure only has sixteenths, so 0.(16) does not land on a mark. The nearest one is 3/16, which is 0.414 of a sixteenth away — about 0.66 mm on an inch.
Work it out for any decimal
The two calculators this page is built from take any numbers you like: reduce a fraction shows the divisor it used, and fraction to decimal goes the other way, to the nearest 16th, 32nd and 64th.
Decimals either side
Cite this page
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- Chicago
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