1/6 as a Decimal

Long division, one digit at a time

1/6 means 1 divided by 6, and long division shows why the answer never stops. Six does not fit into 1, so the quotient starts at 0 and a decimal point follows. Bring down a zero: 10 divided by 6 is 1 with a remainder of 4. Bring down another zero: 40 divided by 6 is 6 with a remainder of 4. The remainder is 4 again, so the next digit is 6 again, and the loop repeats without end.

Written out, 1/6 is 0.1666 with the 6 repeating. A bar drawn over the repeating part, a 6 under a bar after 0.1, records the pattern in a single symbol. The repeating block is one digit here, which makes one sixth one of the easier repeating decimals to read and to explain.

The reason it repeats sits in the denominator. A fraction terminates only when its bottom number, in lowest terms, is built from 2s and 5s, because those are the factors of 10. Six is 2 times 3, and the 3 leaves a remainder at every step of the division. One eighth terminates at 0.125, one fifth terminates at 0.2, and one sixth cannot. Khan Academy's fractions and decimals unit (accessed 2026) works the same long division for the sevenths and other denominators that repeat.

As a percent the same number is 16.6666 percent, which rounds to 16.67 percent at two places. The calculator on this page prints the decimal to ten places, marks the repeating block, and shows the percent at four places, so the size of the rounding stays visible instead of hidden.

Sixths in a table

The sixths family shares one repeating digit, and the table keeps every form of it side by side. Halving a sixth gives 1/12 at 0.0833, and pushing past the whole gives 7/6 at 1.1666, which is more than 1 because the fraction is bigger than the unit.

Three of the eight rows are exact. 3/6 reduces to 1/2, which is 0.5, and 6/6 is 1. Every other row carries the same tail of repeating sixes or threes, because 1/6, 2/6, 4/6 and 5/6 all reduce to fractions with a 3 left in the bottom, and a 3 in the bottom always repeats.

The rounded percent column is what a price tag or a grade sheet would print. One sixth is 16.67 percent for display, while the full percent 16.6666 keeps going. Two sixths is 33.33 percent for display, and five sixths is 83.33 percent. The pattern climbs by 16.6666 points a rung, with the two exact rows breaking the rhythm.

Each pair adds to the whole. 1/6 and 5/6 sum to 1, and their percents sum to 100 once the rounding evens out: 16.67 plus 83.33 is exactly 100. That symmetry is a handy check on any table built from sixths.

What rounding does to a sixth

0.1666 rounded to one place is 0.2, to two places is 0.17, to three places is 0.167, and to four places is 0.1667. Each step moves the value up, because the digit being dropped is 6, which is above 5. NIST Special Publication 811, Appendix B (2008 edition) sets the rounding rule for converted values: round once, at the end, and keep enough digits that the error stays below the tolerance of the job.

Money makes the error countable. A sixth of $100 is $16.6666, which the register rounds to $16.67. Pay that six times and the total is $100.02, two cents more than the bill. A sixth of $300 is exactly $50.00, while 16.67 percent of $300 is $50.01, so the rounded percent lands a cent high on that base.

The fix is to hold the fraction through the arithmetic. Split $300 six ways by dividing 300 by 6 rather than by multiplying 300 by 0.1667, and the answer is $50.00 with nothing left over. The rule of thumb is to round only the number shown to a person, never the number carried through a calculation.

The same caution applies in the other direction. A base of $60 divides by 6 into $10.00 with no remainder, so no rounding is needed anywhere. Bases built from multiples of 6 are friendly; bases that are not will always leave a remainder for the rounding rule to place.

The ladder below puts every rounding step next to what it costs when the value is multiplied back by 6. The last row is the fraction itself, which is the only row that lands exactly on 1.

Each extra place divides the leftover error by 10, which is why the four place value is already good enough for payroll while the two place value drifts by two percent of a dollar every time it moves. The widget prints ten places for this reason.

A sixth of an hour and a sixth of an inch

Sixty minutes divided by six is 10, so one sixth of an hour is 10 minutes, and a time card that records decimals shows it as 0.1667 hours. Five sixths of an hour is 50 minutes, or 0.8333 hours. A shift of 7.5 hours is 450 minutes, and a sixth of it is 75 minutes.

Lengths behave the same way. An inch is 25.4 millimeters, so one sixth of an inch is 4.2333 millimeters and five sixths is 21.1667 millimeters. A plan that calls for a 1/6 inch hole is asking for about 4.2 millimeters, and the nearest bit in a metric set will be 4.2 or 4.5 millimeters, since no common bit is cut to a sixth.

Percent of an inch is a share rather than a length. One sixth of an inch is 16.67 percent of an inch, the same 16.67 that one sixth of an hour or one sixth of a dollar produces. The unit changes, the fraction does not.

The tenth of an hour shows why the repeating tail matters at work. Ten minutes written as 0.17 hours at two places reads as 10.2 minutes when someone multiplies back by 60, and 0.1667 hours reads as 10.002 minutes. The four place version is close enough for payroll, and the two place version drifts.

Sixths of an hour and sixths of an inch share one table, because both come from dividing a fixed unit by 6. The minute column steps in tens; the millimeter column multiplies the same share of 25.4.

The bottom row is the definition of the unit, which is the fastest check on the rest of the column. Halfway down, the half hour and the half inch sit on 12.7 millimeters and 30 minutes, and every other row is a whole number of steps away from them.

Turning 0.1666 back into a fraction

The repeating decimal can be converted back to a fraction without guessing. Call the value x, so x = 0.1666. Multiply by 10 and 10x = 1.6666. Subtract the first line from the second: 10x minus x is 9x, and 1.6666 minus 0.1666 is 1.5. So 9x = 1.5, which makes x = 1.5/9, and multiplying top and bottom by 2 gives 3/18, which reduces to 1/6.

The same trick handles any single repeating digit. 0.3333 gives 9x = 3 and x = 1/3. 0.6666 gives 9x = 6 and x = 2/3. 0.08333 gives 9x = 0.75 and x = 1/12. The multiplier of 9 comes from the length of the repeating block, which is why a one digit repeat needs one 9.

A shorter route exists when the digits are known, but it has a trap. 0.1666 looks like 16/100, yet 16/100 reduces to 4/25, which is 0.16 and stops. The repeating tail carries the difference between 0.16 and 0.1666, and only the algebra above recovers the exact fraction. That is why a decimal rounded to two places cannot be turned back into a fraction with confidence.

The check runs forward again. Multiply 1/6 by 6 and the product is exactly 1, which is the definition of one sixth. Multiply a rounded 0.1667 by 6 and the product is 1.0002, close but not exact, and the gap is the fingerprint of the rounding.

Where a sixth shows up

Splitting a bill six ways is the everyday version. Six people at a $100 table owe $16.67 each when the cents round up, and the sixth person pays $16.65 so the total lands on $100.00. A $1,000 loan repaid in six equal installments takes $166.67 a month, and the last payment absorbs the two cents of rounding.

Recipes scale by fractions the same way. A recipe written for six servings feeds two at one third of every quantity, since 2/6 reduces to 1/3. A cup of stock for six becomes 0.1667 cups for one, which is 2 tablespoons and 2 teaspoons, an amount a measuring spoon set handles without a fuss.

Grades and weighted scores use it too. A final exam worth one sixth of a course carries 16.67 percent of the grade, so a score of 90 on that exam contributes 15 points to a 100 point total, and 90 times 0.1667 would give 15.003. The remaining five sixths of the course carry 83.33 percent between them.

Six equal parts show up in schedules as well. A six month warranty is one sixth of a year of coverage, and a budget spread across six categories gives each category 16.67 percent of the total. In every case the decimal is 0.1666, and the percent quoted is the rounded two place version of it.

Splitting money six ways collects the rounding in one place, usually the last share. The table shows four totals, the printed per head figure, and who absorbs the leftover cent or two.

The two exact rows are the multiples of 6, and the pattern generalizes: any total built from sixths divides cleanly, and any other total leaves a remainder of a few cents for the last payer. Restaurants that print per head figures usually follow the second column and let the final bill absorb the difference.

A fair die carries six faces, so the chance of landing on the face named in advance is 1/6, which a scorekeeper would report as 16.67 percent. Each throw is a separate event, so the same one sixth applies every time the die leaves the hand, and six throws give each face a fair expectation of one hit. Games that roll for initiative or for loot are running this decimal every session without printing it anywhere.

Reading a calculator display

A calculator asked for 1/6 shows 0.16666667 on an eight digit display. The final 7 is not the next digit of the quotient, it is the result of rounding the string of sixes up at the edge of the screen. A ten digit display shows 0.1666666667, and a percent key shows 16.666666667.

Truncation and rounding differ at that last digit. Cutting the display at eight places gives 0.16666666, while rounding gives 0.16666667, a difference of one hundred millionth. Neither display is wrong, but only the rounded one is closer to the true value, and only the true value is exactly one sixth.

The habit that keeps a spreadsheet honest is to store the fraction or the full decimal and format only for display. Multiply a rounded 0.17 by 6 and the product is 1.02, not 1. Multiply 0.1667 by 6 and the product is 1.0002. Multiply 1/6 by 6 and the product is exactly 1. The closer the stored value sits to the fraction, the closer the total lands.

Long division can settle any doubt about a digit. At the eighth step of the division the remainder is 4 again, so the eighth digit is 6, and every later step repeats the pair. No digit after the first 1 is anything other than 6, which is the whole content of the repeating bar.

Displays of different widths show the same number with different endings, and the table lines them up against what the widget prints.

Only the width of the screen changes between the rows; the quotient behind them is one number. That is the reason a digit that differs between two calculators is a display artifact rather than a disagreement about the math.