A rate compares two different quantities: kilometres per hour, dollars per hour, litres per minute, grams per cubic centimetre. Every rate question is a multiply or a divide, and the units tell you which.
A rate is one quantity per one unit of another. Read the units: km/h means km ÷ h, so distance \(=\) speed \(\times\) time and time \(=\) distance ÷ speed. Average speed \(= \dfrac{\text{total distance}}{\text{total time}}\).
Fig. 1: the speed, distance and time triangle. Distance (D) sits on top, so it is speed (S) times time (T)
Section 01
Speed, distance and time.
Open the lab on Average speed and change the distance and time sliders for each leg. Watch which number the average speed follows: it is always total distance over total time, never the mean of the two speeds.
Speed \(= \dfrac{\text{distance}}{\text{time}}\), so distance \(=\) speed \(\times\) time and time \(= \dfrac{\text{distance}}{\text{speed}}\) (Fig. 1).
Hours and minutes: change minutes to a decimal of an hour before dividing: 45 min \(= \dfrac{45}{60} = 0.75\) h.
km/h to m/s: multiply by 1000 then divide by 3600, which is the same as dividing by 3.6.
Worked example Average speed
A cyclist travels 45 km in 3 hours 45 minutes. What is their average speed?
Change the time to hours: \(45\) min \(= 0.75\) h, so the time is \(3.75\) h.
Speed \(= 45 \div 3.75\).
Answer: 12 km/h
Worked example A journey in two parts
A car travels 60 km at 60 km/h and then another 60 km at 30 km/h. Find the average speed for the whole journey.
Time for part 1: \(60 \div 60 = 1\) h. Time for part 2: \(60 \div 30 = 2\) h.
Total distance \(= 120\) km and total time \(= 3\) h.
Average speed \(= 120 \div 3\).
Answer: 40 km/h (not 45 km/h, the mean of the two speeds)
Your turn
Convert 72 km/h into metres per second.
\(72\) km \(= 72\,000\) m in \(3600\) s, so \(72\,000 \div 3600 = 20\) m/s.
Examiner trap
3 hours 45 minutes is NOT 3.45 hours. 45 minutes is \(\dfrac{45}{60} = 0.75\) of an hour, so the time is 3.75 hours.
Section 02
Everyday rates.
Pay, flow and fuel are all rates. Write the rate with its units and the units guide the calculation.
Worked example Pay with overtime
Kai is paid \(\$14.50\) per hour for a 38-hour week, and time-and-a-half for overtime. One week he works 43 hours. How much is he paid?
Normal pay: \(38 \times 14.50 = \$551\).
Overtime rate: \(1.5 \times 14.50 = \$21.75\) per hour, for \(43 - 38 = 5\) hours.
Overtime pay: \(5 \times 21.75 = \$108.75\).
Total: \(551 + 108.75\).
Answer: \(\$659.75\)
Worked example Fuel consumption
A car uses 6.4 litres of fuel per 100 km. Fuel costs \(\$1.85\) per litre. Find the cost of the fuel for a 350 km journey.