Rates.
Read “per”, match the units and connect distance, time and speed.
Use pay, exchange, flow and fuel rates · apply supplied formulas · calculate average speed
A rate compares different kinds of quantity
A rate compares quantities with different units. Saying a worker earns $12 per hour links money to time: for every hour, the payment increases by $12 at that constant rate.
The word “per” means “for each one”. Dividing total pay by hours finds pay for one hour; multiplying an hourly rate by hours finds total pay. Always keep the units visible because they tell you which quantity should be multiplied and which should be divided.
“Per” means “for each”. A rate tells you how much of one quantity corresponds to one unit of another. Read the units before choosing the calculation.
| Rate | Calculation | Typical unit |
|---|---|---|
| Hourly pay | earningshours worked | dollars per hour |
| Flow rate | volumetime | litres per minute |
| Fuel use | fuel useddistance travelled × 100 | litres per 100 km |
| Fuel economy | distance travelledfuel used | km per litre |
A worker earns $12 per hour for 712 hours. Find the pay.
For 5 ordinary hours and 2 overtime hours paid at 1.5 times the ordinary rate: 5 × 12 + 2 × 18 = $96.
Water flows at 18 litres per minute. How long to fill a 270-litre tank?
The units cancel: litres divided by litres per minute leaves minutes.
A car uses 36 litres to travel 450 km. Express the consumption in litres per 100 km.
The economy is 450 ÷ 36 = 12.5 km per litre. A lower litres-per-100-km value means less fuel used; a higher km-per-litre value means further travel on each litre.
Pause and explainIf 3 hours cost $24, what is the hourly rate?
$24 ÷ 3 = $8 per hour. We divide to find the amount for one hour.
Speed is distance per unit time
Picture travelling at 60 km each hour. In two hours you would cover twice 60 km, so distance is speed multiplied by time. The familiar formulas are rearrangements of that one relationship.
Speed compares distance with time, so its units must match the inputs. Before using kilometres per hour, write the time in hours. A duration of 30 minutes is half an hour, not 0.30 hours, because an hour contains 60 minutes.
You need to know the speed–distance–time relationship. Make the time unit match the speed unit before substituting.
A cyclist covers 45 km in 3 hours 45 minutes. Find the average speed.
- Convert minutes: 45 minutes = 4560 hour = 0.75 hour.
- Total time = 3 + 0.75 = 3.75 hours.
- Average speed = 453.75 = 12 km/h.
Convert 72 km/h to m/s.
Convert kilometres to metres and hours to seconds together.
Pause and explainHow many hours is 1 hour 30 minutes?
1.5 hours: the extra 30 minutes is 3060 = 12 hour.
Average speed uses the whole journey
A journey may include fast sections, slow sections and stops. Average speed replaces the whole journey with one constant speed that would cover the same total distance in the same total elapsed time.
Add all the relevant distances and all the relevant times, including stops when the whole journey is described. Then divide total distance by total time. A simple average of two speed values is usually wrong because the traveller may spend different times at them.
Add all distances and divide by total elapsed time, including stops if they are part of the stated journey. Do not simply average the speeds unless the time spent at each speed is equal.
A driver travels 60 km at 30 km/h, stops for 30 minutes, then travels 60 km at 60 km/h.
Total distance = 120 km. Total elapsed time = 2 + 0.5 + 1 = 3.5 h.
Without the stop, the moving average is 120 ÷ 3 = 40 km/h, still not (30 + 60) ÷ 2 = 45 km/h.
Pause and explainWhy can a long stop reduce average speed even though the distance is unchanged?
It increases the total elapsed time. The same distance divided by a larger time gives a smaller average speed.
Other rates: use the formula provided
Not every rate concerns movement. Density compares mass with volume, population density compares people with area, and pressure compares force with area. Each describes how much of one quantity corresponds to one unit of another.
Use the relationship provided in the question and identify the units of both inputs. Converting first prevents an answer that mixes square metres with square centimetres or grams with kilograms. After substitution, check that the output unit agrees with the requested rate.
For other measures of rate, use the formula given in the question. Substitute quantities in compatible units and state the resulting unit.
| Measure | Formula | Worked substitution |
|---|---|---|
| Density | massvolume | 540 g ÷ 200 cm³ = 2.7 g/cm³ |
| Pressure | forcearea | 600 N ÷ 0.2 m² = 3000 N/m² |
| Population density | populationland area | 84 000 people ÷ 120 km² = 700 people/km² |
Pause and explainWhat does a population density of 200 people per km² mean?
It is an average of 200 people for each square kilometre of the area, not a claim that every square kilometre has exactly 200 people.
Your turn. Follow the units.
Underline the quantity being found and write its required unit. Then choose the relationship that links it to the given information.
For journey questions, convert clock-style times to one unit before calculating. For any other rate, write the formula from the question in words first and explain which quantity corresponds to “per one unit”.
A worker earns $14 per hour. Find the pay for 6 hours 45 minutes.
Reveal worked answer
A tap delivers 2.4 litres per minute. How much water flows in 1 hour 15 minutes?
Reveal worked answer
A runner travels 1500 m in 5 minutes. Find the average speed in m/s.
Reveal worked answer
A car covers 90 km in 1.5 hours and 60 km in 1 hour, with a 30-minute stop between. Find the average speed for the whole journey.
Reveal worked answer
Using density = massvolume, find the mass of a 60 cm³ object with density 7.8 g/cm³.