One quantity per another
Density, concentration, pressure, speed ratio and mechanical advantage look like five formulas to memorize. They are one idea: divide one measurement by another, so the answer means the same thing whatever the size of the sample. This page is about reading that idea, and what each one explains.
Why divide one quantity by another?
Because the raw numbers mislead. A big log is far heavier than a pebble, yet the log floats and the pebble sinks. A person in boots sinks into deep snow, and the same person on snowshoes does not, though their weight has not changed. To compare fairly, you need to know how much there is for each unit: each millilitre, each square metre, each turn.
That is all “per” means: for each one. Dividing by a volume, an area or a number of turns takes the size of the sample out of the answer and leaves a property you can compare between any two things. And that property is usually the one that decides what happens: floating, dissolving, sinking into snow, turning faster or pushing harder.
Where it turns up
- Weather reports — Environment and Climate Change Canada gives air pressure in kilopascals, about 101 kPa at sea level, and a falling reading often means stormy weather is on its way
- Fuel ratings — new vehicles in Canada are rated in litres of fuel per 100 kilometres, so a small car and a pickup truck can be compared on the same scale; the lower number uses less
- Maple syrup — roughly 40 litres of sap boil down to 1 litre of syrup, and producers check the syrup's density with a floating hydrometer to know when enough water has boiled off
- Bicycle gears — a mountain bike's lowest gear pairs a small front gear with the biggest rear one, so a turn of the pedals turns the back wheel less than once: slow, but easy up a steep hill
- 1. The words, first
- 2. The unit is the formula
- 3. Density: why oil floats
- 4. Concentration: comparing drinks fairly
- 5. Pressure: why snowshoes work, and how a jack lifts a car
- 6. Speed ratio and mechanical advantage: turns per turn, force per force
The words, first
The idea: “Per” means “for each”, and a slash in a unit means “per”: g/mL is grams for each millilitre.
| Word | What it means |
|---|---|
| Per | For each one. 60 km per hour is 60 km for each hour. |
| Rate | One quantity divided by another of a different kind, such as grams per millilitre. It has a unit. |
| Ratio | One quantity divided by another of the same kind, such as teeth ÷ teeth. The units cancel, so it has none. |
| Density | Mass per unit of volume, in g/mL or g/cm³. |
| Concentration | Grams of solute per 100 mL of solution. |
| Pressure | Force per unit of area: newtons per square metre (N/m², called pascals) or newtons per square centimetre (N/cm²). |
| Speed ratio | Turns of the driving gear for each turn of the driven gear. |
| Mechanical advantage | Output force ÷ input force: newtons out for each newton put in. |
The unit is the formula
The idea: Whatever comes before the slash goes on top; whatever comes after goes underneath. Read the unit and you cannot divide the wrong way.
| Quantity | Unit | So you divide |
|---|---|---|
| Density | g/mL or g/cm³ | grams ÷ millilitres |
| Concentration | g/100 mL | grams ÷ millilitres, then × 100 |
| Pressure | N/m² (Pa) or N/cm² | newtons ÷ area |
| Speed ratio | none | driven teeth ÷ driving teeth |
| Mechanical advantage | none | output force ÷ input force |
Worked: forgetting the formula. A stone has a mass of 60 g and a volume of 24 cm³. What is its density? The unit, g/cm³, says grams ÷ cubic centimetres: 60 ÷ 24 = 2.5 g/cm³. Divide the other way, 24 ÷ 60 = 0.4, and the unit would be cm³/g, which is not a density at all.
Why a ratio has no unit. The same unit sits above and below the line and cancels: 36 teeth ÷ 12 teeth = 3. That is why a speed ratio can be written 3:1, and a mechanical advantage as a plain 4.
Density: why oil floats
The idea: Density does not depend on how much you have. That is what makes it useful: it describes the material, not the sample.
Worked: a big block and a small block. Two blocks are cut from the same board. The small one has a mass of 12 g and a volume of 20 cm³; the big one, 96 g and 160 cm³.
small: 12 ÷ 20 = 0.6 g/cm³ · big: 96 ÷ 160 = 0.6 g/cm³
Eight times the mass and eight times the volume give the same density. Both float on water, at 1.00 g/cm³, and so would a whole tree trunk of the same wood. The heavier block is no more likely to sink.
Why oil floats on water. Cooking oil is about 0.92 g/mL and water 1.00 g/mL. Each millilitre of oil has less mass than a millilitre of water, so the water settles underneath and the oil rides on top. Spilled oil spreads across a lake's surface for the same reason. An egg that sinks in tap water floats in strong salt water: the dissolved salt raised the water's density above the egg's.
Concentration: comparing drinks fairly
The idea: Scaling both solutions to the same 100 mL is what makes them comparable. Without it, the bigger container always looks stronger.
Worked. A 355 mL can of pop has 39 g of sugar. A 250 mL juice box has 26 g. Which is more concentrated?
can: 39 ÷ 355 × 100 ≈ 11.0 g/100 mL · juice box: 26 ÷ 250 × 100 = 10.4 g/100 mL
The can is slightly more concentrated, and it holds more, so you take in more sugar in total. Both numbers matter: the concentration says how sweet each sip is, and the total says how much sugar you swallow.
The same idea elsewhere. A doctor often works out a child's medicine dose in milligrams per kilogram of body mass, so a smaller child gets a smaller dose. The “per” is what lets one rule fit people of every size.
Pressure: why snowshoes work, and how a jack lifts a car
The idea: The same force on a larger area gives less pressure, and on a smaller area, more. In a hydraulic jack, the same pressure on a larger area gives more force.
Spreading a force out. In Unit A, a 600 N student presses 10 000 Pa into the snow in boots and only 2000 Pa on snowshoes. Caribou do the same with broad hooves, and a tractor with wide tires packs a wet field less.
Concentrating a force. A sharp knife does the opposite. Press with 20 N on an edge touching 0.2 cm² of a tomato, and the pressure is 20 ÷ 0.2 = 100 N/cm²; the flat of the blade, touching 4 cm², gives only 20 ÷ 4 = 5 N/cm², and the tomato just squashes. Needles, thumbtacks and axes all work this way.
Using pressure to multiply force. In a hydraulic jack, a 50 N push on a 2 cm² piston makes a pressure of 50 ÷ 2 = 25 N on every square centimetre of the liquid. The large piston has 40 square centimetres, each pushed up with 25 N: 1000 N in all. The per-area number stays the same; only the area changes. Unit D follows the distances and the work.
Speed ratio and mechanical advantage: turns per turn, force per force
The idea: These two compare like with like, so they have no unit. A machine that gives you more force always gives you less speed or distance, and the two numbers show the trade.
| Speed ratio | Mechanical advantage | |
|---|---|---|
| Compares | Turns of the driving gear for each turn of the driven gear | Newtons of output force for each newton of input force |
| Worked out as | driven teeth ÷ driving teeth | output force ÷ input force |
| Example | 36 ÷ 12 = 3 | 600 N ÷ 150 N = 4 |
| Bigger than 1 means | The driven gear turns more slowly, with more turning force | The machine multiplies your force |
Worked: what friction costs. Unit D's ramp raised a 200 N box 1.5 m by a push of 80 N along 5.0 m. The push travelled 5.0 ÷ 1.5 ≈ 3.3 times as far as the box rose, so with no friction the force would have been multiplied about 3.3 times, to a push of 60 N. The real mechanical advantage was 200 ÷ 80 = 2.5. The gap between 3.3 and 2.5 is friction at work, and it is why the efficiency came out at 75%, not 100%.
Say it in words. Books do not all write a speed ratio the same way up. “The small gear turns 3 times for each turn of the big one” cannot be misread, whichever way the number is written.