13/16 as a Decimal

Thirteen divided by sixteen, step by step

Thirteen sixteenths is a division problem before it is anything else. Sixteen does not fit into 13, so the decimal starts at 0.8, and the four passes below finish the job with nothing left over.

The digits read 8, 1, 2, 5, so 13/16 = 0.8125. The last remainder is zero, which is what separates a decimal that stops from one that repeats. Compare 1/9, where the remainder returns to 1 on every pass and the digits never end.

A second route confirms it. Sixteen times 625 is 10,000, so 13/16 equals 8,125/10,000, and that fraction is 0.8125 written in four places. Both routes land on the same number, which is the check to run whenever a hand calculation feels off.

Why sixteenths always stop

A fraction in lowest terms stops when its denominator has no prime factors other than 2 and 5. Sixteen is 2 x 2 x 2 x 2, so every sixteenth lands on an exact decimal. One eighth stops at three places, one sixteenth at four, and one thirty-second at five.

The rule also explains the neighbors that repeat. A third, a sixth, a seventh and a ninth all carry a prime other than 2 or 5 in the denominator, so their decimals run on. Thirteen is prime, but the denominator decides, not the top number.

Because the denominator is a power of two, 0.8125 is exact in binary floating point as well. A computer stores it as 13 multiplied by 2 to the power of minus 4, with nothing left over. One tenth has no such form, which is why 0.1 plus 0.2 prints 0.30000000000000004 in some languages while 0.8125 prints clean.

The sixteenths table, decimals and percents

Every rung of the sixteenths ladder adds 0.0625 to the decimal and 6.25 to the percent. The table lists the whole family, with the reduced form where one exists.

Four sixteenths is 0.25 and 25 percent, eight is 0.5 and 50 percent, twelve is 0.75 and 75 percent. Those quarter marks are the ones most people already know, and the other rows fill the gaps between them.

Thirteen sixteenths sits one mark below fourteen: 0.75 plus 0.0625 is 0.8125, and 75 percent plus 6.25 is 81.25 percent. The complement is 3/16 at 0.1875 and 18.75 percent, and the two halves of the pair add to 1 and to 100 percent.

Thirteen sixteenths of an inch

An inch is 25.4 millimeters exactly, a factor fixed by the 1959 agreement on the international yard and pound and listed in NIST Special Publication 811. Multiply 0.8125 by 25.4 and the answer is 20.6375 millimeters.

That is a real size in a workshop. A 13/16 inch wrench opening measures 20.6375 millimeters across, and the nearest metric size, 21 millimeters, is 0.3625 millimeters larger. On a soft fastener the metric wrench may still turn it, and on a tight one it will round the corners instead.

Sixteenths also divide cleanly on a tape measure, since each small mark is 1/16 inch. Thirteen marks past the inch line is 13/16, and the same length in millimeters is 20.6375, so a plan drawn in inches converts without a repeating decimal anywhere in it.

A cup, an hour, a dollar and a mile

The fraction works the same way on any unit that splits into sixteen. A US cup holds 16 tablespoons, a fact the USDA Agricultural Research Service lists in its measurement conversion tables, so 13/16 cup is 13 tablespoons. An hour holds 60 minutes, and 0.8125 times 60 is 48.75, which is 48 minutes 45 seconds.

Money is the row people check first. Thirteen sixteenths of $1,000 is $812.50, and of $2,400 it is $1,950.00. The same share of a $40 bill is $32.50, which you can verify by halving twice rather than multiplying decimals.

Distances follow the same multiplication. Thirteen sixteenths of a mile is 4,290 feet, and of a yard is 29.25 inches. Each row is one multiplication by 0.8125, and each answer can be checked by dividing the result back by 0.8125 to recover the whole.

Cents give the smallest row of all. Thirteen sixteenths of a dollar is 81.25 cents, which is the percent read as money, and a register that rounds a 13/16 share to the nearest cent can be off by at most half a cent because the decimal stops. Nothing in a sixteenth repeats, so no share drifts no matter how many times the total gets split.

Reading 0.8125 as a percent, and where rounding bites

Multiplying 0.8125 by 100 gives 81.25 percent, an exact figure with two decimal places. That is unusual for a percent: most fractions produce a repeating percent, and 81.25 stops because the decimal underneath it stops.

At two places the shortfall is 0.0025, which is 25 units out of 10,000. At three places the rounding runs the other way, because the fourth digit is 5 and the value rounds up to 0.813. Four places are exact, which is why machine drawings and schedules that involve sixteenths often carry four.

Statistics agencies publish percents to one decimal as a rule, and the Bureau of Labor Statistics reports its price changes that way, so 81.25 would appear as 81.3 in a printed series. Keep the exact fraction for the calculation and round only the sentence that reports it, and the two never disagree by more than the rounding you chose.

Two quick checks that confirm 0.8125

The fastest check is to multiply back. Take 0.8125 and multiply by 16: you get 13, which is the numerator you started with, so the conversion is right. The second check uses the complement. The gap from 13/16 to a full whole is 3/16, and 3 divided by 16 is 0.1875. Add the pair together and 0.8125 plus 0.1875 equals exactly 1, with no rounding anywhere. Both checks work because sixteen is a power of two, so every sixteenth terminates in at most four decimal places. You can also double your way around: half of 13/16 is 13/32, or 0.40625, and twice 13/16 is 13/8, or 1.625. Each of those still stops after a finite number of digits, which is exactly the behavior you never get with thirds, sixths, or ninths.