Decimal Hours Calculator

Why this conversion exists

Payroll multiplies hours by a rate, so it needs a decimal number. Clocks and time sheets are base 60. The two notations describe the same length of time, and confusing them is the most expensive silent error on a time sheet:

• 7:45 means seven hours and forty-five minutes.
• 7.75 is the same amount in decimal hours.
• 7.45 is neither — it is 7 hours 27 minutes, and 18 minutes short.

The error survives every check downstream, because 7.45 is a perfectly valid number. Nobody catches it until the paycheck, and often not then.

The formula

Minutes to decimal:

decimal hours = hours + (minutes ÷ 60)

Decimal back to minutes:

minutes = decimal fraction × 60

So 0.25 × 60 = 15 minutes, and 0.4 × 60 = 24 minutes. The quarter hours are worth memorising — 0.25, 0.5, 0.75 — because they cover most shifts.

Which minutes have an exact decimal

There is a rule for this and it is one line long: a whole number of minutes has an exact decimal precisely when it divides by three.

• 3 minutes is 0.05, 9 minutes is 0.15.
• 21 minutes is 0.35, 33 minutes is 0.55.
• 57 minutes is 0.95 — the last exact one before the hour.

All twenty of them land on a hundredth. Not one needs a third decimal place, which is the part worth knowing: two places is not a compromise for these minutes, it is exact.

Every other minute repeats forever. One minute is 0.01666…, ten minutes is 0.1666…, twenty minutes is 0.333…. The reason is that an hour is 60 minutes and 60 is 2 × 2 × 3 × 5: dividing by it terminates only when the 3 cancels, and the 3 cancels only for a multiple of three. No number of decimal places repairs the rest — three and four places exist on this page for those minutes alone, and all they change is how far down the remainder is pushed.

20 minutes is the one everybody meets. It is 0.33 at two places, 0.333 at three, and never quite right at any. That is not a flaw in the conversion; it is what base 10 does to base 60. Over a 40-hour week the difference between 0.33 and 0.333 is under two minutes, which is why two places is the payroll standard.

Going back the other way rounds to the nearest whole minute, because a time sheet cannot record half a minute. Entering 0.333 gives 0:20, and 0:20 gives 0.33 — the round trip is stable even though the decimal is not exact.

Tenths of an hour, and why the unit is six minutes

Law firms, agencies and a good many contractors do not bill in hours and minutes at all. They bill in tenths, and a tenth of an hour is six minutes: 0.1 is 6 minutes, 0.2 is 12, 0.3 is 18, and so on to 1.0. There is no table to memorise — the minutes are the tenth times six — and every one of them is exact, because six divides by three.

Both conventions you meet are built the same way. A quarter hour is 15 minutes, also a multiple of three, also exact at two places. They are the two increments the federal rounding regulation names by name: 29 CFR 785.48(b) accepts recording punches to the nearest five minutes, tenth of an hour or quarter of an hour, on the condition that the practice does not, over time, leave people unpaid for time they worked.

The size of the increment is the size of the error it can hide. Rounding to the nearest unit moves a figure by at most half of it, so a quarter hour can move a single punch by 7 minutes 30 seconds, a tenth by 3 minutes, five minutes by 2 minutes 30. That is per punch, and a day usually has two.

Billing in tenths has a second edge, and it is not rounding. A tenth is normally a minimum, so ten separate two-minute phone calls are ten tenths — a billed hour for twenty minutes of work. The arithmetic is correct and the invoice is defensible; it is still the thing to look for on one.

One minute this page used to lose

Converting a decimal back to minutes rounds to the nearest whole minute, half up. 1.025 hours is exactly 61 minutes and 30 seconds, so the rule says 62. This page said 61.

The cause is not the rule but the multiplication. A browser holds 1.025 as the nearest available binary fraction, which is a hair under 1.025, and evaluates 1.025 × 60 as 61.49999999999999. Rounding that is not rounding a half at all — it is rounding something below a half, and it goes down. Counting every decimal typed to three places below 200 hours, 68 of them land on that edge, each losing exactly one minute. 4.225, 8.075, 8.325 and 16.025 are the next four.

The fix is one line: the product is trimmed to twelve significant figures before it is rounded. Twelve figures is more precision than any time sheet can carry, and still coarse enough to sweep away an error that lives in the fourteenth. A test now pins all five of the values named above, so it cannot come back quietly.

One minute, on a three-place decimal, is not what separates a right paycheck from a wrong one, and it would be dishonest to dress it up as one. It is written down here for a different reason: nothing about 61 looks wrong. A conversion that is off by a minute produces a number of exactly the right shape, on a page whose entire subject is a number of the right shape being the wrong one.

Where the numbers come from

If you need to work out the hours before converting them, the Work Hours Calculator turns two clock times into a duration, and the Time Card Calculator totals a whole week — both of which already show decimal hours alongside the clock figure.

Common uses

• Filling in a payroll system that only accepts decimal hours.
• Checking a time sheet someone else converted.
• Turning a decimal figure on a payslip back into hours and minutes.
• Reading an invoice billed in tenths, or writing one.