1/9 as a Percent

Divide, then move the point two places

To write 1/9 as a percent, divide 1 by 9 and then multiply by 100. The division gives 0.111..., and multiplying by 100 moves the decimal point two places to the right, so 1/9 is 11.111... percent. Reports and receipts print it as 11.11 percent, which is the same number with the endless tail cut off.

The division runs forever because every pass leaves the same remainder. Ten divided by nine is one with one left over, and the leftover one becomes the next ten. No step ever clears the remainder, so the digit 1 repeats without end, and the percent repeats with it: 11.111111... , one 1 for every digit the division produces.

A percent is a fraction over 100, so 11.11 percent is 11.11/100, or 0.1111 as a decimal. The exact fraction 1/9 is larger than 0.1111 by 0.0000111... per unit. That small gap is the whole reason a rounded percent and an exact ninth disagree on money, and the rest of this page measures it.

The ninths ladder in percent

The ninths family repeats its top number, so the percents form a ladder with a fixed step. Each additional ninth adds 11.11 percentage points, and the eighth rung lands just short of 100. The table carries the ladder across a $1,800 base, where every ninth is a clean $200.

Rows that reduce still follow the ladder. Three ninths is one third at 33.33 percent, and six ninths is two thirds at 66.67 percent. Nine ninths is the whole at 100 percent, which is the only rung where the repeat stops.

Pair the rows and the rounded percents add back to 100.00: 11.11 with 88.89, 22.22 with 77.78, 33.33 with 66.67, and 44.44 with 55.56. That pairing is a fast check on any ninths table, and it fails loudly when a row has been copied wrong. Note that 5/9 rounds up to 55.56 while 4/9 rounds down to 44.44, which is what lets each pair land exactly on 100.00.

11.11 percent of a number

Finding 11.11 percent of a number is one multiplication, and the result sits a few cents below the exact ninth. The table runs the comparison across five bases so the size of the gap is visible.

Take the $45,000 row. One ninth of 45,000 is 5,000, while 45,000 times 0.1111 is 4,999.50, so the rounded percent leaves 50 cents on the table. The same calculation at $270,000 leaves $3.00 behind, and the gap grows in a straight line with the base.

The gap works out to about one dollar for every $90,000 of base. That is small enough to ignore on a receipt and large enough to matter in a budget line that repeats every month. When the base is a multiple of nine, the exact share lands on a whole number: 63,000 divided by 9 is 7,000, while 11.11 percent of 63,000 is 6,999.30, so the rounded label is still 70 cents short on a base that divides cleanly.

Percentage points are not percent change

Two statements about the same move can both be true and still confuse a reader. Going from 11.11 percent to 22.22 percent is a rise of 11.11 percentage points, and it is also a rise of 100 percent, because the share doubled. Percentage points measure the gap between two percents, while percent change measures the gap against the starting value.

The formula behind the last column is (new minus old) divided by old, times 100. From 11.1111 to 12 that is 0.8889 divided by 11.1111, which is 8.0 percent. From 11.1111 down to 5.5556 the same formula gives minus 50 percent, because 5.5556 is half of 11.1111.

Rate tables, fee schedules and score reports mix the two measures constantly. A fee that climbs from 11.11 percent to 12 percent has climbed less than one point, and a headline that calls that a 1 percent increase is wrong in a way that compounds. Say which measure you mean before the numbers go in a report.

Where 11.11 percent shows up

Eleven point one one percent turns up wherever a whole is split nine ways. A nine question quiz with one wrong answer scores 88.89 percent, so 11.11 percent is the price of that single miss. A nine person dinner bill of $180 leaves $20.00 a head, which is 11.11 percent of the total.

Rates use the same figure. A balance of $5,000 at 11.11 percent simple interest for one year earns $555.50, while the exact ninth earns $555.56. A $90 tip at 11.11 percent is $10.00, and the shortcut behind all of these is division by nine.

Dividing by nine is faster than multiplying by 0.1111 and it avoids the rounding gap entirely. Nine people splitting $270 get $30 each, and nine months of a $45,000 contract pay $5,000 a month. The percent is a label for the division, not a replacement for it, and the widget on this page shows both answers side by side for any base you type.

Rounding rules that keep the percent honest

Rounding a repeating percent is a choice about where to stop, and the choice should be made once, at the end of the calculation. Carry more places than you plan to publish, multiply, then round the final figure. Truncating 11.111... to 11.11 always loses a little, while rounding to nearest can go either way depending on the next digit.

A spreadsheet keeps the full value in the cell even when the display shows two decimals, so a total built from those cells stays correct. The trouble starts when the displayed 11.11 is retyped into a new cell: the tail is gone, and every later multiplication uses the shorter number. On a $270,000 line that retyping costs $3.00, which is the gap from the table above.

Published figures follow the same discipline. NIST Special Publication 811 sets out the rounding conventions used for measurement values, and the Bureau of Labor Statistics publishes its price indexes to three decimals for the same reason. Borrowing costs are quoted as percents too, and the Consumer Financial Protection Bureau explains why an APR and an interest rate can differ even when both are written as percentages. The digits after the point carry information, and dropping them early is how a total drifts.

A mental drill that locks the pattern in

Pick a number that divides by nine cleanly and run the percent in your head. Take 45. One ninth of 45 is 5, so 5 is 11.11% of 45, and 11.11% of 45 rounds to 5.00. Take 270. One ninth of 270 is 30, and 11.11% of 270 computes to 29.997, which prints as 30.00 at two decimals. Now stretch the base to 270,000 dollars. The exact ninth is 30,000.00, while 11.11% of the base is 29,997.00, a shortfall of 3.00 dollars. The pattern is always the same shape: the gap equals one ninth of one percent of the base, so every 900 dollars of base costs you about 1.00 dollar of rounding. Once you can recite that ratio, you can estimate the damage of a rounded ninth on any amount without a calculator.