Rate of pay
earnings = rate × time
$18.70per hour
×
15.5 h=
$289.85If the time is given in hours and minutes, convert it to hours first.
Chapter 12
Rates
Understand common measures of rate, solve average-speed problems accurately, and apply compound measures such as pressure, density and population density with confidence.
By the end of this chapter, you should be able to
- interpret a rate as “one quantity per another quantity”;
- use common measures of rate such as rate of pay and rate of flow;
- calculate average speed and rearrange the speed formula for distance or time;
- convert between units such as seconds, minutes and hours or metres and kilometres;
- apply the given formulas for pressure, density and population density correctly;
- avoid unit mistakes and round sensible answers clearly.
1
Subtopic 1
Common measures of rate
A rate compares two different quantities. It often uses the word per, meaning “for each” or “for every”. Examples include dollars per hour, litres per minute and kilograms per second.
Rate of flow
rate of flow = volume ÷ time
25 L/min
For example, 25 litres per minute means 25 litres flow every minute.
Core relationships
For ordinary rates, you usually use:
- \[\text{rate}=\frac{\text{quantity}}{\text{time}}\]
- \[\text{quantity}=\text{rate}\times\text{time}\]
- \[\text{time}=\frac{\text{quantity}}{\text{rate}}\]
Worked example · Easy
Rate of pay
A student earns \(\$7.80\) per hour. How much does she earn for 12 hours of work?
\[7.80\times12=\boxed{\$93.60}\]
Worked example · Medium
Hours and minutes
A worker is paid \(\$18.70\) per hour and works for 15 hours 30 minutes. Find the total pay.
Convert 30 minutes to 0.5 hours.
\[15\text{ h }30\text{ min}=15.5\text{ h}\]\[18.70\times15.5=\boxed{\$289.85}\]Worked example · Medium
Find a rate of flow
A tank holds 1000 litres and takes 30 minutes to fill. Find the average rate of flow in litres per hour.
30 minutes = 0.5 hours.
\[\text{Rate}=\frac{1000}{0.5}=\boxed{2000\text{ L/h}}\]Worked example · Exam-style
Find the time needed
Water flows into a container at 25 litres per minute. How long will it take to fill a 500-litre bath?
\[\text{time}=\frac{500}{25}=20\text{ minutes}\]
Answer: \(\boxed{20\text{ minutes}}\).
Common mistake
Do not mix hours with minutes. If the rate is in litres per minute, the time must be in minutes.
Additional exam-focused worked example
Worked example · Exam-style
A tank fills at 24 litres/minute. It already contains 80 litres and has capacity 500 litres. How long until full?
Step-by-step solution
Remaining volume \(500-80=420\) L. \[t=\frac{420}{24}=\boxed{17.5\text{ min}}\] That is 17 min 30 s.
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
A pump moves 180 litres in 6 minutes. Find its flow rate.
Show Answer
Steps: \[180\div6=\boxed{30\text{ L/min}}\]
2Easy practice
A worker earns $14 per hour for 8 hours. Calculate pay.
Show Answer
Steps: \[14\times8=\boxed{\$112}\]
3Medium practice
A tap flows at 2.5 L/min. How many litres in 18 minutes?
Show Answer
Steps: \[2.5\times18=\boxed{45\text{ L}}\]
4Medium practice
A pipe delivers 600 litres at 20 litres per minute. Find the time.
Show Answer
Steps: \[t=600\div20=\boxed{30\text{ minutes}}\]
5Exam-style practice
A worker is paid $16.80 per hour for 7 h 45 min. Find total pay.
Show Answer
Steps: Convert \(45\text{ min}=0.75\text{ h}\), so time 7.75 h. \(16.80\times7.75=\boxed{\$130.20}\).
2
Subtopic 2
Speed
Average speed links distance and time. The units always look like “distance per time”, such as km/h or m/s.
The speed triangle
d
s
t
- Cover d: speed × time
- Cover s: distance ÷ time
- Cover t: distance ÷ speed
Useful conversions
1 hour= 60 minutes
1 minute= 60 seconds
1 km= 1000 m
Convert the units before using the formula if necessary.
Speed formulas
\[\text{speed}=\frac{\text{distance}}{\text{time}}\qquad \text{distance}=\text{speed}\times\text{time}\qquad \text{time}=\frac{\text{distance}}{\text{speed}}\]Worked example · Easy
Average speed
A car travels 75 km in 1.5 hours. Find its average speed.
\[\text{speed}=\frac{75}{1.5}=\boxed{50\text{ km/h}}\]
Worked example · Medium
Find time
How many minutes does it take to walk 2 km at a speed of 5 km/h?
\[\text{time}=\frac{2}{5}\text{ h}=0.4\text{ h}\]\[0.4\times60=\boxed{24\text{ minutes}}\]
Worked example · Medium
Convert m/s to km/h
A bus travels at an average speed of 5 m/s. Convert this speed to km/h.
Use \(1\text{ m/s}=3.6\text{ km/h}\).
\[5\times3.6=\boxed{18\text{ km/h}}\]Worked example · Exam-style
Mixed units
A cyclist rides 0.6 km in 3 minutes. Calculate the average speed in km/h.
3 minutes = \(\frac{3}{60}=0.05\) hours.
\[\text{speed}=\frac{0.6}{0.05}=\boxed{12\text{ km/h}}\]Worked example · Exam-style
Rearranging for distance
A train travels for 2 h 24 min at 80 km/h. How far does it travel?
\(2\text{ h }24\text{ min}=2.4\text{ h}\).
\[\text{distance}=80\times2.4=\boxed{192\text{ km}}\]Additional exam-focused worked example
Worked example · Exam-style
A bus covers 32 km in 40 min, stops for 20 min, then covers 48 km in 60 min. Find its overall average speed including the stop.
Step-by-step solution
Total distance \(32+48=80\) km. Total time \(40+20+60=120\) min = 2 h. \[\text{average speed}=80\div2=\boxed{40\text{ km/h}}\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
A cyclist rides 42 km in 2 hours. Find speed.
Show Answer
Steps: \[s=42\div2=\boxed{21\text{ km/h}}\]
2Easy practice
A train moves at 75 km/h for 3 hours. Find the distance.
Show Answer
Steps: \[d=75\times3=\boxed{225\text{ km}}\]
3Medium practice
Find the time to travel 36 km at 8 km/h.
Show Answer
Steps: \[t=36\div8=\boxed{4.5\text{ h}}\]
4Medium practice
Convert 12 m/s to km/h.
Show Answer
Steps: \[12\times3.6=\boxed{43.2\text{ km/h}}\]
5Exam-style practice
A runner covers 800 m in 3 min 20 s. Calculate average speed in m/s.
Show Answer
Steps: Convert time: \(3\times60+20=200\) s. \[s=\frac{800}{200}=\boxed{4\text{ m/s}}\]
3
Subtopic 3
Applying other measures of rate
Some other rates combine two quantities in a formula. In exam questions, these formulas are usually given to you. You must still use them correctly and keep units consistent.
Pressure
force
pressurearea
\(\text{pressure}=\frac{\text{force}}{\text{area}}\)
Typical units: N/m\(^2\)
Density
mass
densityvolume
\(\text{density}=\frac{\text{mass}}{\text{volume}}\)
Typical units: g/cm\(^3\)
Population density
population
densityarea
\(\text{population density}=\frac{\text{number of people}}{\text{area}}\)
Typical units: people/km\(^2\)
Worked example · Easy
Pressure
A force of 200 N acts on an area of 2.5 m\(^2\). Calculate the pressure.
\[\text{pressure}=\frac{200}{2.5}=\boxed{80\text{ N/m}^2}\]
Worked example · Medium
Pressure with unit conversion
A heavy box exerts a force of 120 N on a base measuring 50 cm by 40 cm. Find the pressure in N/m\(^2\).
Convert dimensions to metres: \(50\text{ cm}=0.5\text{ m}\), \(40\text{ cm}=0.4\text{ m}\).
\[\text{area}=0.5\times0.4=0.2\text{ m}^2\]\[\text{pressure}=\frac{120}{0.2}=\boxed{600\text{ N/m}^2}\]Worked example · Easy
Density
The density of gold is 19.3 g/cm\(^3\). Find the mass of a gold bar with volume 30 cm\(^3\).
\[\text{mass}=\text{density}\times\text{volume}=19.3\times30=\boxed{579\text{ g}}\]
Worked example · Medium
Population density
A town has a population of 65 000 and an area of 5.8 km\(^2\). Calculate the population density to the nearest thousand.
\[\text{population density}=\frac{65000}{5.8}=11206.896\ldots\]
Rounded to the nearest thousand:
\[\boxed{11000\text{ people/km}^2}\]Worked example · Exam-style
Find an unknown area
A town has population 200 000 and population density 10 000 people per square mile. Find the area of the town.
\[\text{area}=\frac{\text{population}}{\text{population density}}=\frac{200000}{10000}=\boxed{20\text{ square miles}}\]
Do I need to memorise these?
You are expected to know speed, but for pressure, density and population density the question will usually give you the formula. Your task is to choose the correct formula, substitute carefully and keep the units consistent.
Additional exam-focused worked example
Worked example · Exam-style
A metal block has dimensions 5 cm × 4 cm × 3 cm and mass 462 g. Find its density.
Step-by-step solution
Volume \(5\times4\times3=60\text{ cm}^3\). \[\rho=\frac{462}{60}=\boxed{7.7\text{ g/cm}^3}\]
Practice: from confidence to exam-style
Work each question first, then select Show Answer to reveal the steps.
1Easy practice
Calculate pressure for force 360 N on an area of 6 m².
Show Answer
Steps: \[P=F/A=360\div6=\boxed{60\text{ N/m}^2}\]
2Easy practice
Find density when mass is 350 g and volume 50 cm³.
Show Answer
Steps: \[\rho=m/V=350\div50=\boxed{7\text{ g/cm}^3}\]
3Medium practice
A town has 54 000 people over 18 km². Find population density.
Show Answer
Steps: \[54000\div18=\boxed{3000\text{ people/km}^2}\]
4Medium practice
The pressure is 150 N/m² on 0.8 m². Find the force.
Show Answer
Steps: Rearrange \(P=F/A\): \(F=PA=150\times0.8=\boxed{120\text{ N}}\).
5Exam-style practice
A block has mass 2.4 kg and volume 800 cm³. Express density in g/cm³.
Show Answer
Steps: Convert mass to grams: \(2.4\text{ kg}=2400\text{ g}\). \[\rho=2400\div800=\boxed{3\text{ g/cm}^3}\]
Chapter 12 practice
Begin with straightforward rate calculations, then move on to speed, unit conversion and other compound measures. Keep answers hidden until you have tried each question.
A. Build confidence
1Rate of pay[2 marks]
A student earns \(\$6.50\) per hour. How much does he earn for 9 hours?
Show Answer
\[6.50\times9=\boxed{\$58.50}\]
2Rate of flow[3 marks]
Water flows at 40 litres per minute. How long will it take to fill a 600-litre tank?
Show Answer
\[\text{time}=\frac{600}{40}=\boxed{15\text{ minutes}}\]
3Find a rate[3 marks]
A 20-tonne lorry is loaded in 40 minutes. Find the rate of loading in kilograms per minute.
Show Answer
20 tonnes = 20 000 kg.
\[\text{rate}=\frac{20000}{40}=\boxed{500\text{ kg/min}}\]4Oil usage[3 marks]
An oil tank contains 1500 litres. Oil is used at a rate of 2 litres per hour. How much is left after 7 days?
Show Answer
7 days = \(7\times24=168\) hours.
\[2\times168=336\text{ litres used}\]\[1500-336=\boxed{1164\text{ litres}}\]B. Speed
5Average speed[3 marks]
Find the average speed of a runner who covers 180 m in 40 seconds.
Show Answer
\[\text{speed}=\frac{180}{40}=\boxed{4.5\text{ m/s}}\]
6Find time[3 marks]
A runner’s average speed in an 80 m race is 7 m/s. Find the time to the nearest 0.1 s.
Show Answer
\[\text{time}=\frac{80}{7}=11.42857\ldots\text{ s}\]
To the nearest 0.1 s, \(\boxed{11.4\text{ s}}\).
7Convert units[3 marks]
A bus travels at 72 km/h. Convert this speed to m/s.
Show Answer
Either divide by 3.6 or convert units directly.
\[72\div3.6=\boxed{20\text{ m/s}}\]8Distance travelled[4 marks]
A car travels at an average speed of 85 km/h for 1 hour 36 minutes. Find the distance travelled.
Show Answer
1 h 36 min = \(1+\frac{36}{60}=1.6\) h.
\[\text{distance}=85\times1.6=\boxed{136\text{ km}}\]9Cyclist[4 marks]
A cyclist rides 0.6 km in 3 minutes. Calculate the average speed in km/h.
Show Answer
3 minutes = 0.05 h.
\[\text{speed}=\frac{0.6}{0.05}=\boxed{12\text{ km/h}}\]C. Other compound measures
10Pressure[3 marks]
Calculate the pressure exerted by a force of 150 N on an area of 0.5 m\(^2\).
Show Answer
\[\text{pressure}=\frac{150}{0.5}=\boxed{300\text{ N/m}^2}\]
11Density[3 marks]
A stone has mass 350 g and volume 40 cm\(^3\). Find its density.
Show Answer
\[\text{density}=\frac{350}{40}=\boxed{8.75\text{ g/cm}^3}\]
12Population density[3 marks]
A region has population 60 000 and area 8 km\(^2\). Calculate the population density.
Show Answer
\[\frac{60000}{8}=\boxed{7500\text{ people/km}^2}\]
13Find a mass[3 marks]
A foam plastic ball has volume 20 cm\(^3\) and density 0.3 g/cm\(^3\). What is its mass?
Show Answer
\[\text{mass}=0.3\times20=\boxed{6\text{ g}}\]
D. Exam-style challenge
14Pay and overtime[5 marks]
A worker earns \(\$5.70\) per hour. One week she works 42 hours. How much does she earn?
Show Answer
\[5.70\times42=\boxed{\$239.40}\]
15Journey time[5 marks]
A journey of 180 km is made at an average speed of 72 km/h. Find the time taken in hours and minutes.
Show Answer
\[\text{time}=\frac{180}{72}=2.5\text{ h}\]
0.5 h = 30 min, so the journey takes \(\boxed{2\text{ h }30\text{ min}}\).
16Density and volume[5 marks]
A rubber ball has volume 28.3 cm\(^3\) and density 0.7 g/cm\(^3\). Find its mass.
Show Answer
\[\text{mass}=0.7\times28.3=\boxed{19.81\text{ g}}\]
17Town area[5 marks]
A town has population 276 300 and population density 9800 people per square mile. Find the area of the town.
Show Answer
\[\text{area}=\frac{276300}{9800}=\boxed{28.193877\ldots\text{ square miles}}\]
Approximately \(\boxed{28.2\text{ square miles}}\).
Chapter 12 high-grade checklist
Read the units
I check whether the question wants an answer in minutes, hours, m/s, km/h, litres per minute or another unit.
Use the right formula
I know speed and can use the given formulas for pressure, density and population density correctly.
Rearrange carefully
If the question asks for time, distance, mass or area, I rearrange the formula logically rather than guessing.
Convert before substituting
I change minutes to hours or centimetres to metres before using the formula if needed.
Reasonableness
I check whether a higher speed should mean a shorter time and whether a larger density should mean more mass for the same volume.
Final accuracy
I round only at the end unless the question tells me otherwise.