Rates & speeds

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A rate is one quantity measured per one of another: kilometres per hour, metres per second, hours per day. It reads with a slash, the way it appears on a road sign or a spec sheet, and Solve treats the whole compound as a single unit you can write, convert and derive.

A unit written with a slash is a rate: a quantity per one of something. km/h, m/s and hours/day are each one unit, not a division, so the compound spelling you would read off a sign is what you can type.

100 km/h // 100.00 km/h
5 m/s // 5.00 m/s
3 hours / day // 3.00 hours/day

A rate converts to another rate, or to any of the single-word speed spellings (mph, kph, mps), by converting the top and the bottom on their own.

100 km/h in mph // 62.14 mph
10 m/s in km/h // 36.00 km/h
60 mph in km/h // 96.56 km/h
100 km/h to m/s // 27.78 m/s

A ship’s or an aircraft’s speed is in knots: one nautical mile an hour, a little faster than a mile an hour. It is written kn, knot or knots. kt is not the knot, because the unit table already reads kt as the kilotonne.

20 knots in km/h // 37.04 km/h
20 kn in mph // 23.02 mph
1 knot in m/s // 0.51 m/s

A knot is a rate, so a speed in knots times a time cancels the hours and leaves the distance covered, in nautical miles.

10 knots * 2 hours // 20.00 nmi

The target can be written the way the rate itself can, with per in place of the slash, and a price with its currency symbol. in miles per hour is the same target as in miles/hour, and in $/day the same as in USD/day, so a speed or a pay rate reads the same on both sides of in.

60 km/h in miles per hour // 37.28 miles/hour
60 km/h to miles per hour // 37.28 miles/hour
$20/hour in $/day // $480.00/day
$20/hour in dollars per day // $480.00/day
$20 per hour in USD per day // $480.00/day
100 Mbps in MB per s // 12.50 MB/s

A target of a bare slash and a unit keeps what the rate counts and changes only what it is per, so a weekly amount reads as a monthly one. A frequency, which is a count per second, converts to a count per any time.

$50/week in /month // $214.29/month
60 mph in /min // 1.00 mi/min
10 Hz in /min // 600.00 /min

A price per hour in another currency per day converts through the exchange rate once one is known, and until then says the rate is missing. in $ on its own is still a currency conversion, and only per and the slash are read in a target: after a conversion, a and each stay prose, so 100 km in miles a day is a distance converted and then made a rate. A target that is not a rate of the same kind is refused, as 5 km in miles per hour is: a distance is not a speed.

$20 in $ // $20.00
100 km in miles a day // 62.14 miles/day
5 km in miles per hour // ERROR: Cannot convert km to miles/hour: they do not measure the same thing

Dividing a distance by a time builds the same rate, and the conversion applies to the whole quotient rather than to the number just before it.

120 km / 2 hours // 60.00 km/hours
120 km / 2 hours in kph // 60.00 kph

Speeds of rotation

An engine, a drill or a record player turns rather than travels, and its speed is counted in revolutions per minute, rpm (or RPM, as a dashboard prints it). One revolution is one full turn, so a turning speed is a frequency: how many times a second something repeats, in hertz (Hz). 3,000 rpm is 50 turns a second. A single revolution is an angle, written revolution or revolutions.

3000 rpm in Hz // 50.00 Hz
50 Hz in rpm // 3,000.00 rpm
1 revolution in deg // 360.00 deg
2 revolutions in rad // 12.57 rad

A turning speed is a frequency, so for a length of time it is a plain count of turns, as any frequency for a time is (see derived units):

3000 rpm * 2 min // 6,000

The boundary: the word turn is not read as a revolution, because it is ordinary English (take turns, turn 3 times). The count is a plain number rather than an angle in revolutions, since a frequency counts repetitions of anything.

Adding and comparing rates

Two rates of the same kind are one quantity written in two units, so they add, subtract and compare the way two lengths do. The right-hand rate is read in the left-hand one’s unit first, by the same conversion in makes, and the answer is in the left-hand unit. A total of rates works the same way.

10 m/s + 36 km/h // 20.00 m/s
10 km/h + 5 mph // 18.05 km/h
10 m/s > 30 km/h // true
total of 10 m/s, 36 km/h // 20.00 m/s

The boundary: two rates of different kinds do not combine, since a speed and a flow of mass per second are not one quantity. Two prices per hour in different currencies are not converted at an exchange rate here either.

10 m/s + 5 kg/s // Cannot combine incompatible units: m/s and kg/s

Cancelling a rate

A rate multiplied by what it is per leaves the other half: a speed for a time is a distance. Dividing a distance by a speed cancels the other way and leaves a time. The single-word speeds cancel as the slash spellings do. Multiplying and dividing units has the full set, prices and coverage rates included.

60 km/h * 2 h // 120.00 km
120 km / 60 km/h // 2.00 h
60 mph * 30 min // 30.00 mi

Drive time

A distance at a speed is a duration: how long the journey takes. The answer comes back in the largest sensible time unit.

250 miles at 60 mph // 4.17 h
100 km at 60 kph // 1.67 h

For a different unit, convert the whole thing: (250 miles at 60 mph) in minutes. A quantity that is not a distance, so does not match the speed, is reported as an error rather than a wrong number.