The clock keeps an old habit of counting in sixties, the sexagesimal system, which comes from Mesopotamia and survives in time and angles. If what you need is to convert a fraction of an hour, use fraction of an hour to minutes, which gives the result in minutes and seconds. Counting divisors shows that 60 splits into more whole pieces than 100 does.
Worked example: one third of an hour and one third of 100
You want to split an hour into three equal parts. How many minutes does each last? Compare with an imaginary clock that has 100 minutes per hour.
- Multiply by 60. (1/3) × 60 = 60/3.
- Divide. 60 ÷ 3 = 20, with no remainder.
- Repeat with 100. (1/3) × 100 = 100/3 = 33 with 1 left over.
- Write the decimal. 100/3 = 33.3̅, a repeating decimal (the 3 repeats forever).
Result: 20 minutes with 60, and 33.3̅ minutes with 100.
Check: three parts of 20 minutes add up to 60. Three parts of 33.33 minutes (≈ 33.3̅) add up to 99.99, which is 0.01 minutes short; however many decimals you add, you never reach exactly 100 with a finite number of digits.
The difference is that 3 divides 60 but does not divide 100. The table of divisors and the table of fractions of an hour below both follow from that observation.
The divisors of 60 and 100
A divisor of a number is a whole number that divides it with no remainder. Counting divisors is the most direct way to measure how many simple fractions of a unit give whole-number results.
| Number | Divisors | How many |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 6 |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 8 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | 12 |
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 | 9 |
| 360 | 24 divisors, among them 8, 9, 40 and 45 | 24 |
100 lacks six of the divisors of 60, among them 3, 6, 12 and 15. Those gaps are what leave remainders in the most common fractions of an hour. 60 is also the smallest number that can be divided by each of 1, 2, 3, 4, 5 and 6, which is why it comes up again and again in explanations of the origin.
To count divisors without listing them, break the number into prime factors, add one to each exponent and multiply. Since 60 = 2² × 3 × 5, you get (2+1)(1+1)(1+1) = 12. Since 100 = 2² × 5², you get (2+1)(2+1) = 9. With 360 = 2³ × 3² × 5 you get 4 × 3 × 2 = 24. If you would rather not do it by hand, the divisors of a number tool lists them, and the prime factorization tool shows the breakdown.
Which fractions of an hour are whole minutes
A unit fraction 1/n of an hour gives a whole number of minutes when n divides 60. The other fractions with that denominator come from adding: 2/3 of an hour is 40 minutes and 3/4 is 45.
| Fraction | With 60 minutes | With 100 minutes |
|---|---|---|
| 1/2 | 30 | 50 |
| 1/4 | 15 | 25 |
| 1/5 | 12 | 20 |
| 1/6 | 10 | 16.6̅ |
| 1/12 | 5 | 8.3̅ |
| 1/15 | 4 | 6.6̅ |
With 100, fourths and fifths still come out whole because 4 and 5 divide 100. With 60 you also get sixths, twelfths and fifteenths. The decimal wins for prices and measurements in tens, and 60 wins when the split is in thirds. To convert any fraction or decimal of an hour, use fraction of an hour to minutes; to take a part of any quantity, use fraction of a quantity.
What is known about the origin
The story of the 60 minutes is a chain of inheritances with gaps in it, and it helps to separate what the sources say from what is only supposed.
- Supported attribution. According to the history of mathematics, the sexagesimal system comes from the Sumerians, and the Babylonians inherited it. MacTutor, from the University of St Andrews, considers that Sumerian origin fairly certain.
- Hypotheses about why 60. Theon of Alexandria argued that 60 is the smallest number divisible by 1, 2, 3, 4 and 5, so the number of divisors is maximized. MacTutor calls this too scholarly a reason and notes that if divisors were the goal, 12 would be the more likely base. Among the other ideas are counting the finger joints of one hand with the fingers of the other, and the merging of two peoples with different bases, 12 and 5. MacTutor presents the last as the most widely accepted theory, though it remains a hypothesis. MacTutor also reports Moritz Cantor’s idea that base 60 came from a year of 360 days, along with the objection that the Sumerians knew for certain the year was longer than 360 days.
- What is not known. MacTutor (in its author’s opinion) does not think anyone chose the base, and points instead to how counting arose; a separate theory, from Neugebauer, ties it to measures of weight and length divided into thirds. Wikipedia notes that the many divisors make calculating with fractions easier, but none of the sources consulted shows that this was the reason for adopting it; Theon and Neugebauer put it forward as hypotheses.
Ancient astronomy wrote fractions in sexagesimal. In that notation Ptolemy writes π as 3;8,30, that is, 3 + 8/60 + 30/3600. In the measurement of time, traces of the sexagesimal system remain, with 60 minutes in an hour and 60 seconds in a minute. Units smaller than the second use the decimal system.
Why a circle has 360 degrees
The Babylonian tradition divided the circle into 360 equal arcs, which are the degrees. What can be checked is the arithmetic: 360 has 24 divisors, and since 8 divides 360, 1/8 of a turn is 360 ÷ 8 = 45 degrees, a whole number. The guide on fractions of a turn and angles has a table of turns to degrees, with cases such as 1/8 of a turn equal to 45°, and the article on leap years shows where a fraction of a day (≈ 1/4 of a day per year) decides when a day is added to the calendar.
The attempt at a 100-minute hour
The French Revolution tried decimal time. A decree of October 5, 1793 divided the day into ten parts, and each part into ten more. On November 24, 1793 those parts were named hours, with decimal minutes of one hundredth of an hour and decimal seconds of one hundredth of a minute. Official use began on September 22, 1794, and the requirement was suspended on April 7, 1795.
The reform did not take hold. The sources consulted do not name divisibility as the cause of its abandonment. What the episode does leave is a comparison: a ten-hour day splits into halves and fifths, but not into thirds. To see whether a unit fraction of a day is exact in hours, just ask whether the denominator divides the number of hours.
Check three fractions of an hour
Work out each case before opening the answers.
Three checks
- How many minutes is 5/6 of an hour?
- How many minutes is 2/5 of an hour?
- How many minutes is 3/10 of an hour?
See the answers
- 50 minutes, because (5/6) × 60 = 300/6 = 50.
- 24 minutes, because 1/5 of an hour is 12 minutes and 12 × 2 = 24.
- 18 minutes, because 60 ÷ 10 = 6 and 6 × 3 = 18.
Sources
- Wikipedia, “Sexagesimal”: Sumerian origin, the twelve divisors of 60, 60 minutes and 60 seconds, 360 in a circle, and Ptolemy’s π as 3;8,30 (accessed October 17, 2026).
- MacTutor, “Babylonian numerals”: Sumerian origin, Theon of Alexandria’s argument about divisors, and alternative hypotheses about 60 (accessed October 17, 2026).
- Wikipedia, “Decimal time”: decree of October 5, 1793, names of decimal hours and minutes of November 24, 1793, official use from September 22, 1794, and suspension on April 7, 1795 (accessed October 17, 2026).