Multiplying and dividing units

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When two quantities are multiplied, their units are multiplied too. A room 5 metres long and 3 metres wide covers 15 square metres: a metre times a metre is a new unit, the metre squared, which measures area. Dividing runs the other way, so 15 square metres along a 3 metre wall leaves 5 metres. And a unit that appears on both sides of a division cancels, the way a number divided by itself leaves 1: a price in dollars per kilowatt-hour, multiplied by a number of kilowatt-hours, leaves dollars.

Solve follows these rules, so an answer comes out in the unit it really has. Where the units combine into something the engine cannot show, such as a kilogram times a kilogram, the answer is an error that says so rather than a number in the wrong unit.

Areas and volumes

A length times a length is an area, and three lengths multiplied are a volume. The left operand’s unit sets the answer’s, the same rule addition follows. An area or volume the engine works out is printed with a superscript: m² is square metres and m³ cubic metres.

5 m * 3 m // 15.00 m²
2 m * 3 m * 4 m // 24.00 m³
5 m * 3 ft // 4.57 m²
5 m2 * 3 m // 15.00 m³

A square or cubic unit can be typed with the superscript, with a plain digit (m2), or in words (square metres, sq ft, cubic feet). Every spelling of one unit is the same unit, so they convert, compare and add together.

15 m² in ft² // 161.46 ft²
3 m * 2 m in square feet // 64.58 square feet
15 sq ft in m² // 1.39 m²
15 m² + 10 sq ft // 15.93 m²
2 cubic metres in litres // 2,000.00 litres

A power can also be written on the unit: m^2 is square metres. The power belongs to the unit it is written on, so 5 m^2 is five square metres, the way a physics book reads it, and not five metres squared, which would be 25 square metres. To square a whole quantity, put it in brackets.

5 m^2 // 5.00 m²
10 m^3 in litres // 10,000.00 litres
(3 m)^2 // 9.00 m²

After a slash, the power belongs to the unit after it, the same way. A density is a mass per volume, so water at 1,000 kilograms per cubic metre is written 1000 kg/m^3, and it converts to grams per cubic centimetre or per millilitre, which are the same size. A time squared after a slash is the time of an acceleration (see named derived units).

1000 kg/m^3 // 1,000.00 kg/m³
1 g/cm^3 in kg/m^3 // 1,000.00 kg/m³
1 g/cm³ in g/mL // 1.00 g/mL
9.81 m/s^2 in ft/s^2 // 32.19 ft/s²

A check compares two such units whenever one converts into the other, so a density in grams per millilitre can be checked against one in grams per cubic centimetre.

check 1 g/mL == 1 g/cm^3 // ✓
check 1 kg/m^3 < 1 g/cm^3 // ✓

Dividing back down

Dividing an area by a length gives a length: the answer to “how long is a 15 square metre room that is 3 metres wide?”. A volume divided by an area is a length too, and a volume divided by a length is an area. The answer is measured in the length the area or volume is written in, or in the divisor’s when the area has a name of its own, such as a hectare.

15 m² / 3 m // 5.00 m
24 m³ / 6 m² // 4.00 m
24 m³ / 6 m // 4.00 m²
1 ha / 50 m // 200.00 m

A square root takes an area back to the length it is the square of, and a cube root does the same for a volume. A power of a half, or of a third, is the same root written another way.

sqrt(16 m²) // 4.00 m
cbrt(27 m³) // 3.00 m
(9 m²)^0.5 // 3.00 m
sqrt(1 ha) // 100.00 m

Rates that cancel

A rate is one quantity per one of another: a price per kilogram, a speed in miles per hour, a paint’s coverage in square metres per litre. Multiplying a rate by the thing it is per cancels that unit and leaves the other, which is how a unit price and a quantity make a bill.

6 kWh * $0.30/kWh // $1.80
3 kg * $5/kg // $15.00
2 kW * 3 h * $0.30/kWh // $1.80
60 mph * 2 hours // 120.00 mi

A price written straight after an amount belongs to it, so $5/kg is one rate wherever it stands in a line: 3 kg * $5/kg is fifteen dollars, not fifteen dollars per kilogram. The single-word rates, such as mph, mpg and Mbps, cancel in the same way as the ones written with a slash. A price per unit is money, so the bill is exact to the cent: 12.3 kWh * $0.15/kWh is exactly $1.845, shown as $1.85, the same as $0.15 * 12.3 (see money precision).

Dividing a quantity by a rate for it cancels the other half, and answers how many of the thing the rate is per: how many litres of paint a wall needs, how long a journey takes, how much a sum of money buys.

20 m² / (5 m²/l) // 4.00 l
120 miles / 60 mph // 2.00 h
100 miles / 30 mpg // 3.33 gal
$100 / ($5/kg) // 20.00 kg

Two rates that share a unit cancel it between them: an hourly rate times hours a day is a daily rate, and a speed divided by a fuel use per hour is a distance per litre.

$30/hour * 8 hours/day // $240.00/day
(100 km/h) / (10 l/h) // 10.00 km/l

A price per kilowatt-hour, in reading order

An electricity cost is usually said in the order it is worked out: the price, then the power, then how long. A kilowatt-hour is one kilowatt for one hour, so a price per kilowatt-hour times a power is a price per hour, and that price per hour times the time is the bill. The line reads left to right in that order, and the bill is the one the energy-first order gives.

$0.30/kWh * 2 kW // $0.60/h
$0.30/kWh * 2 kW * 3 h // $1.80
$0.30/kWh * 3 h * 2 kW // $1.80
$0.30/kWh * 2 kW * 180 min // $1.80
$300/MWh * 2000 W * 3 h // $1.80
2 kW * 3 h * $0.30/kWh // $1.80

The time can come before the power: a price per kilowatt-hour for three hours is a price per kilowatt, which the power then cancels. The answer stays exact to the cent, as a price times an energy is, so half a cent a kilowatt-hour rounds the way a till rounds it.

$0.30/kWh * 3 h // $0.90/kW
$0.305/kWh * 2 kW * 3 h // $1.83

This works for any rate whose unit is a product of the next quantity and a time: a price per kilowatt-hour meets a power or a time. A price per kilowatt-hour times a mass makes nothing, and is refused by name.

$0.30/kWh * 2 kg // ERROR: Cannot multiply a rate per kWh by a quantity in kg: they measure different things, and together they make no unit.

One per something

A plain number divided by a quantity is its reciprocal: how many of something fit in one of the unit, or how often a thing happens per unit. One divided by two metres is half of one per metre, written /m, the same per-unit rate the engine uses anywhere a count is per something. It cancels like any other rate, so multiplying it by a length gives back a plain number. A rate turns over, so the reciprocal of a speed is the time a kilometre takes, and the reciprocal of a frequency is its period in seconds.

1 / (2 m) // 0.50 /m
10 / (5 s) // 2.00 /s
1 / (2 m) * 4 m // 2
1 / (50 Hz) // 0.02 s
1 / (2/week) // 0.50 weeks

The brackets matter. A fraction written in front of a unit is that much of the unit, the way a recipe or a timesheet reads it, so 1/2 hour is half an hour and 3/4 cup three quarters of a cup. Bracket the quantity to ask for the reciprocal instead.

1/2 hour // 0.50 hours
3 / 4 cup // 0.75 cup
1 / (2 hour) // 0.50 /hour

A temperature has no reciprocal: it is measured from a zero point of its own, so there is no “per degree” to show, and it is refused by name.

1 / (20 C) // ERROR: A number divided by a temperature in C has no unit: a temperature is measured from a zero point of its own, so there is no "per degree" to show it in.

Worked through

Paint for one wall, two coats, from a tin that covers 12 square metres a litre:

wall = 4 m * 2.5 m // 10.00 m²
coats = 2 // 2
paint = wall * coats / (12 m²/l) // 1.67 l

What a 3 kW kettle costs to run for twenty minutes at 28p a kilowatt-hour. A power for a time is an energy, named here in kilowatt-hours (see named derived units):

kettle = 3 kW // 3.00 kW
used = kettle * 20 min // 1.00 kWh
cost = used * £0.28/kWh // £0.28

What has no unit

Only lengths square into a new unit, and only up to a volume. A mass times a mass, a time times a time, or money times money has no unit the engine can show; neither has an area times an area, which would be a metre to the fourth power. A price per hour divided by a time is a rate of a rate. Each is refused by name. A speed divided by a time is the exception, since that is an acceleration (see named derived units). A like product used to keep the left operand’s unit, so 2 kg * 3 kg was reported as 6 kg, a confident answer in the wrong unit.

2 kg * 3 kg // ERROR: A quantity in kg times one in kg has no unit: mass times mass is not a unit. Lengths multiply into an area or a volume, and no other quantity squares into one.
$5 * $3 // ERROR: A quantity in USD times one in USD has no unit: money times money is not a unit. Lengths multiply into an area or a volume, and no other quantity squares into one.
5 m2 * 3 m2 // ERROR: A quantity in m2 times one in m2 has no unit: lengths multiply into an area or a volume, and a product of more than three lengths is not a unit.
$100/h / 2 h // ERROR: A quantity in USD/h divided by one in h has no unit: nothing cancels, and USD/h per h is a rate of a rate, which is not a unit.
sqrt(5 m) // ERROR: sqrt: a quantity in m has no square root with a unit; only an area has a length as its root.

Money times a count is the exception that stays. Thirty dollars times four days is the cost of four days at thirty dollars a day, because one side is money and the other only says how many.

$30 * 4 days // $120.00

The boundary

This covers the units the engine already holds: lengths, areas and volumes, rates of one unit per another, and the named physical units on the derived units page. It is not a general algebra of units, so a product such as a kilogram-metre, or a metre to the fourth power, is refused rather than shown.

• A reciprocal is a per-unit rate (/s), not a named unit, so ten per second is 10.00 /s rather than ten hertz. The two convert into each other with in (10 Hz in /s), but a quotient is not renamed in hertz on its own.
• A rate meets a quantity of another measure only when the two make up what the rate is per with a time or a named unit: a price per kilowatt-hour meets a power or a time. A rate per something that no such pair makes (a price per hour times a mass) is refused.
• A single capital letter after a slash, as in $0.50/W, is read as a variable called W, since N and W are common names for a count. Write the unit as a word, $0.50 per watt.
• There is no unit for an amount of substance (mol), so gas-law formulas such as PV = nRT are not yet available.