1/6 as a Percent

Divide first, then multiply by 100

A percent is a fraction of one hundred, so the route from 1/6 to a percent has two steps. Divide the top by the bottom: 1 / 6 = 0.1666..., where the six carries on without end. Then multiply by 100: 0.1666... x 100 = 16.666... percent. Rounded to two decimal places, that is 16.67%.

The order of the steps does not matter. Multiplying first gives 100 / 6 = 16.666..., the same value. Dividing first is easier to check on a phone, because the decimal is a familiar number that sits between 0.16 and 0.17.

One sixth is a little under one fifth. A fifth is 0.2, and a sixth is 0.1666..., so the sixth is smaller by 0.0333..., which is 3.33 percentage points. That gap is worth remembering when a price cut is advertised as a sixth off rather than 20 percent off.

The long division that never finishes

Long division shows why the six repeats. Write 1.000 and divide by 6. Six does not fit into 1, so the whole part is 0. Bring down a zero to make 10, and six goes in once with 4 left over. Bring down another zero to make 40, and six goes in six times, 6 x 6 = 36, with 4 left over again.

The remainder is 4 at every pass, so every pass produces the same digit. The quotient is 0.1666..., and the block that repeats is the single digit 6. Long division never reaches a remainder of zero, which is the signal that the decimal is infinite.

The denominator explains it in one line. Six factors as 2 x 3. A reduced fraction ends only when the denominator is built from 2s and 5s, and the 3 in six blocks that. Compare 1/8 = 0.125, which stops after three digits, because 8 = 2 x 2 x 2.

Four passes are enough to see the whole story, and the table records what each pass leaves behind. From the second pass on, nothing changes: same division, same digit, same remainder of 4.

The calculator on this page compresses that table into one line. For 1 and 6 it prints the percent 16.67%, the decimal 0.166667, the reduced fraction 1/6, and a repeating pattern line that names a block of one digit: 6. Nothing in that line is rounded except the percent, which is the form people quote.

Rounding 16.666 to a percent you can say out loud

Nobody quotes 16.666... percent in a sentence, so the value gets rounded. Two decimal places is the common choice for money and prices, which gives 16.67%. One decimal place gives 16.7%, and a whole number gives 17%.

Each extra digit cuts the rounding error by a factor of ten. The table shows the written form and how far it sits from the exact value, measured in percentage points.

The exact value is 16.666666... percent, so rounding at the end of a calculation is safer than rounding at the start. The NIST guide to the SI, SP 811 (National Institute of Standards and Technology, 2008) makes the same point about converted values: carry the digits through the work and round the final answer, because early rounding stacks up.

Every sixth in one table

Sixths come in three families. The halves and whole numbers land on clean values, the thirds repeat a single digit, and the remaining sixths repeat the same digit pattern as the thirds. The table gives the fraction in lowest terms, the decimal to six places, and the percent to two places.

The pairs are worth noting. One sixth and five sixths add to 1, and their percents add to 100. Two sixths and four sixths also add to 1, and their percents add to 100. That symmetry is a quick way to catch a wrong row in a table like this one.

A pizza cut into six equal wedges makes the fraction physical. One wedge is 16.67% of the pie, two wedges are 33.33%, and three wedges are half. A six pack of cans splits the same way, and so does a tray of six muffins: one muffin is 16.67% of the tray. Halving each wedge gives twelfths, and 1/12 works out to 8.33%, exactly half of a sixth.

One sixth of a number you can check by hand

Dividing by six is the direct route. One sixth of 240 is 40, because 240 / 6 = 40. One sixth of 90 is 15. One sixth of 60 is 10, which is why a sixth of an hour is 10 minutes.

Using the rounded percent instead of the fraction adds a small drift. On $1,200, the exact sixth is $200.00, while 16.67% of $1,200 is $200.04, four cents more. On $12,000 the drift grows to 40 cents, so the exact fraction is the better tool for large amounts and the rounded percent is fine for a quick estimate.

Work backward to check the answer. If one sixth of a number is 40, the number is 40 x 6 = 240. If 16.67% of a number is 40, dividing 40 by 0.1667 returns 239.95, close enough to confirm the estimate but not exact enough for a ledger.

Many of the amounts people split by six divide without a remainder, and every row below comes out even.

The exact share line in the calculator reports $200.00 for the last row, and the rounded share line reports $200.04, because 16.67% of 1,200 overshoots the exact sixth by four cents. A crew of six splitting a $1,200 job pays $200 each with nothing left over. When the amount refuses to divide evenly, only the leftover changes: split $245 six ways and each share is $40 with $5 left over, since 40 x 6 = 240 and 245 - 240 = 5.

Percent values that get mistaken for a sixth

The first mix-up is 16 percent against 16.67 percent. On a $600 bill, 16% is $96.00, 16.67% is $100.02, and the exact sixth is $100.00. The two cent gap between the last two comes from rounding the percent, not from the fraction.

The second mix-up is the decimal. One sixth is 0.1666..., which is not 0.6, not 0.06, and not 6. A sixth is smaller than a fifth, so any decimal above 0.2 cannot be one sixth.

The third mix-up is direction. A number that grows by a sixth is multiplied by 7/6, which is 1.1666..., so it rises by 16.67%. A number that shrinks by a sixth is multiplied by 5/6, which is 0.8333..., so it falls by 16.67%. The same percent moves in two different directions, and the base changes with it: a $600 bill plus a sixth is $700, while $700 minus a sixth is $583.33.

A fourth mix-up hides in the word sixth itself. One sixth of a number and a sixth off a number are different operations, and the percent 16.67% belongs to both of them. On a $600 bill, adding a sixth gives $700, while taking a sixth away gives $500. The first move multiplies by 7 and divides by 6; the second multiplies by 5 and divides by 6. Writing the direction down before doing the arithmetic is the cheapest safeguard on this page.

Where a sixth turns up in daily life

Time is the most common place. An hour holds six blocks of 10 minutes, so 10 minutes is 16.67% of an hour and 20 minutes is 33.33%. A workday of 7.5 hours holds 450 minutes, and one sixth of that is 75 minutes, or 1 hour 15 minutes.

Chance is the second place. A single face on a fair six sided die has a 1 in 6 chance, which is 16.67%. Two faces cover 33.33%, three faces cover 50%, and the whole die covers 100%.

Scores are the third. Sixty questions with ten wrong leaves 50 right, which is 83.33%, the same as 5/6. Statistical agencies report percent changes the same way, as a rounded figure computed from unrounded index values, which is why the BLS Handbook of Methods for the Consumer Price Index (US Bureau of Labor Statistics) describes the calculation before it describes the published table.

A useful habit: when a share is described as a sixth, convert it to 16.67% once and then keep the fraction for the arithmetic. The fraction divides cleanly and the percent is only a label.

The clock makes the table concrete. Sixty minutes divides by six into blocks of ten, and the percent for each block follows the sixths table from the section above.

Payroll systems that round to the nearest tenth of an hour work on a six minute grid, because 0.1 of 60 minutes is 6. A quarter hour uses a different grid: 15 minutes is 25% of the hour, against 16.67% for the ten minute block. Keeping the two grids apart clears up most timesheet confusion, and the clock chart on this page draws the six block split with one block shaded.

Two more places reward the same division. A playlist of 72 songs gives 12 songs per sixth, enough for a short commute. A $54,000 bonus pool split six ways is $9,000 a share. A 90 minute film spends 15 minutes on each sixth, so the opening sixth ends at the quarter hour mark and the closing sixth starts at 75 minutes.

Sixths in music, where a bar divides the same way

Music notation divides time the same way a clock does. In 6/8 time a bar holds six eighth notes, so one eighth note is one sixth of the bar and takes 16.67% of its length. Three eighth notes fill half the bar, which is the 50% row of the sixths table above.

A sextuplet packs six notes into the space of four, so each stroke lasts two thirds of a beat while the group as a whole covers a single beat. That is a different question from a sixth of a bar, and players keep the two apart by counting the group against the pulse rather than inside it.

A slow bar of 6/8 lasts three seconds, so each eighth note lasts half a second, and a passage of 24 bars holds 144 eighth notes. Sheet music publishers set the time signature at the head of the staff for exactly this reason: the signature states the denominator that the beat divides into.