Problems we wrote
Unit analysis: let the units cancel
Write every quantity with its unit, and treat units like numbers that can be multiplied and divided. When the unwanted units cancel and the wanted one remains, the calculation is set up correctly.
Rates are fractions
12 litres per minute is 12 L over 1 min. Written as a fraction, it can be multiplied or flipped like any other.
Cancel what you do not want
To find minutes from litres, divide litres by litres per minute. The litres cancel and minutes are left.
Convert by multiplying by one
1,000 m over 1 km equals one, so multiplying by it changes the unit without changing the amount.
Original Years 7 and 8 rate problems, set up with units| Problem | Set-up with units | Answer |
|---|
| A tap fills a tank at 12 litres per minute. How long to fill 300 litres? | 300 L divided by (12 L per min): litres cancel | 25 minutes |
| A car uses 7 litres per 100 km. How many litres for 350 km? | 350 km times (7 L per 100 km): kilometres cancel | 24.5 litres |
| Convert 72 km per hour into metres per second. | 72 km per h times (1,000 m per km) times (1 h per 3,600 s) | 20 metres per second |
| A printer prints 18 pages per minute. How many pages in 2.5 hours? | 2.5 h times (60 min per h) times (18 pages per min) | 2,700 pages |
| Walking at 5 km per hour, how many minutes to walk 1.5 km? | 1.5 km divided by (5 km per h) gives 0.3 h; times 60 min per h | 18 minutes |
| A recipe uses 250 g of rice for 2 serves. How much rice for 7 serves? | 7 serves times (250 g per 2 serves) | 875 grams |
The third problem is the one students most often get wrong by guessing whether to multiply or divide by 3.6. With units written in, there is nothing to guess: kilometres must cancel, so the 1,000 metres per kilometre goes on top; hours must cancel, so the 1 hour per 3,600 seconds goes on top as well. The arithmetic then reduces to 72,000 divided by 3,600, which is 20.
The walking problem shows a second advantage. Dividing 1.5 kilometres by 5 kilometres per hour leaves hours, because kilometres cancel; the answer 0.3 is therefore 0.3 of an hour, not 0.3 of anything else. Students who drop units often write 0.3 minutes or stop at 0.3 altogether. Keeping the unit forces the last conversion, 0.3 hours times 60 minutes per hour, which gives 18 minutes. The same discipline protects every multi-step rate question in Years 7 and 8, from fuel use to flow rates to speed.
Unit analysis also works as a check in a multiple-choice test. If an option is in the wrong unit, or would need the units to multiply into something meaningless such as litres times minutes, it can be ruled out before any calculation.
