1. GCSE Compound Measures & Rates of Flow
Dr J Frost
@DrFrostMaths
Last modified: 21st March 2018

2. Starter :: Speed-Distance-Time
A car travels at 7 m/s. How much time does it take to travel 35m?
If 7m travelled each second, then 35m travelled in 5 seconds.
𝟑𝟓 𝟕 =𝟓 seconds ?
Jessica Ennis ran her 0.8km Olympic heptathlon race in 2012 in a time of 2min 8.65s. What was her average speed in metres per second?
(Hint: convert quantities to metres and seconds first)
0.8 km = 800 m
2min 8.65s = 𝟐×𝟔𝟎 +𝟖.𝟔𝟓=𝟏𝟐𝟖.𝟔𝟓 s
𝒔𝒑𝒆𝒆𝒅= 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒕𝒊𝒎𝒆 = 𝟖𝟎𝟎 𝟏𝟐𝟖.𝟔𝟓 =𝟔.𝟐𝟏 m/s ? 𝑠 𝑑 𝑡
𝑠= 𝑑 𝑡 𝑠 𝑑 𝑡
𝑡= 𝑑 𝑠
In Physics you may have used a “sdt” triangle (letters from left to right) which helps us relate speed 𝑠, distance 𝑑 and time 𝑡: 𝑑 𝑠 𝑡 𝑠 𝑑 𝑡
𝑑=𝑠×𝑡

3. Quickfire Questions 𝑑 𝑠 𝑡 ? ? ? ? ? 𝑘𝑚/ℎ Speed Distance Time 2m/s 10m
0.2km/s
50km
250s ?
5km/h
10km
7200s ?
36 km/h
60km
1 hr 40 min
Fro Tip: Notice how the unit gives us the formula for speed:
𝑘𝑚/ℎ
The same applies with formulae for density and pressure.
Method 1:
1 hr 40 min = 100 mins
𝑠= =0.6 km/min
= 36 km/hr
Method 2:
1 hr 40 min = hr
𝑠= = 36 km/hr
distance
over
time
You go 60 times further in an hour than in a minute.
This time we kept the time in hours.

4. Further Examples
Jeremy Clarkson travels 250 miles at a constant speed of 100mph. How much time has elapsed, in hours and minutes?
𝒕= 𝟐𝟓𝟎 𝟏𝟎𝟎 =𝟐.𝟓 hours
= 2 hr 30 minutes ?
To convert from hours to hours and minutes, multiply the decimal part by 60.
0.5×60=30 mins.
On a calculator, you can use the ° ′ ′′ key.
BEWARE: 3.45 hours wouldn’t mean 3hrs 45mins
Getting your units right!
Hans Solo and Chewbacca activate their warp drive and travel at the speed of light. Light travels at a speed of km/s. What distance have they travelled after a year?
Choose 
6,460,000,000,000,000m 
Choose
9,460,000,000,000,000m 
Choose
12,460,000,000,000,000m 
Choose
15,460,000,000,000,000m

5. Test Your Understanding1 2 ?
1 hr 20 mins = 80 mins
𝒔= 𝟏𝟖 𝟖𝟎 =𝟎.𝟐𝟐𝟓 km/min
=𝟏𝟑.𝟓 km/h ?
45.5 ?
46.0

6. Density
Density gives us a measure of how tightly packed matter is within a space.
The density of a rock is 2.3 g/cm3.
Work out the mass of a piece of this rock with a volume of 20 cm3.
This is a rock. ?
Intuitively, if there is 2.3 g for each cm3, then for 20cm3, mass must be
𝟐.𝟑×𝟐𝟎=𝟒𝟔 𝐠
Test Your Understanding: 𝑚
𝑑𝑒𝑛𝑠𝑖𝑡𝑦= 𝑚𝑎𝑠𝑠 𝑣𝑜𝑙𝑢𝑚𝑒 𝑑 𝑣 ?
19.3
Again, the unit of g/cm3 allows you to work out the formula for density if you forget.

7. Pressure ? 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒= 𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎
Pressure is how concentrated a force is on a surface.
Suppose I press two different objects against my hand, but with the same force…
The first is going to hurt a lot more because the force is concentrated over a much smaller area. The pressure is therefore greater.
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒= 𝐹𝑜𝑟𝑐𝑒 𝐴𝑟𝑒𝑎
Units: Pascals is Newtons per square metre (N/m2). Pounds per square inch, shortened to psi, is typically used as the unit of pressure underwater. In the Mariana Trench, the deepest point on Earth, the pressure is psi; that’s like a 1000 stone truck placed on each square inch! =4 ?

8. Exercise 1a
Questions on provided sheet.
A silver ring has a volume of 3 cm3 and a mass of 36 grams. Find the density of the silver. 12g/cm3
A force of 108 Newtons is applied across an area of 4.5 m2. Work out the pressure. 24N/m2
Gold plating costs £6 per cm2. How much will it cost to plate a rectangular lid of dimensions 10cm by 24cm? £1440
An arctic hare runs a distance of 216km in 9 hours. How fast does it run? 24 km/h
A penguin flies 4 miles in 15 minutes. What is its speed in mph? 16 mph
Find the distance travelled: (a) 65 mph for 2 hours miles (b) 5 m/s for 1 minute m
In the 2012 Olympics Usain Bolt won the 100m in 9.63 seconds and the 200m in seconds. For what event was he faster? Speeds: 10.38m/s vs 10.35m/s. Faster for 100m.
Find the density of a metal if 100𝑐 𝑚 3 weighs 800 grams. 8 g/cm3
[Edexcel IGCSE June2010-4H Q4a]
Rosetta drives 85 kilometres in 1 hour 15 minutes. Work out her average speed in km/h. 68 km/h
[Edexcel IGCSE May2013-4H Q3]
Yoko flew on a plane from Tokyo to Sydney. The plane flew a distance of 7800 km. The flight time was 9 hours 45 minutes. Work out the average speed of the plane in kilometres per hour. 800 km/h 8 1 ? ? 2 9 ? ? 3 ? 10 ? ? 4
More questions on next slide… ? 5 ? 6 ? 7 ?

9. Exercise 1a ? ? ? ? 11 N1 N2 12 [Edexcel GCSE June2014-1H Q14]
Emily is driving in France. She sees this sign.
Emily is going to drive to Dijon. She plans to drive at an average speed of 50 miles per hour. 
Work out how long it should take Emily to drive to Dijon.
[IMC 2003 Q16] After a year’s training, Minnie Midriffe increased her average speed in the London Marathon by 25%. By what percentage did her time decrease? 20%
[JMC 2011 Q21] Gill leave Lille by train at 09:00. The train travels the first 27km at 96 km/h. It then stops at Lens for 3 minutes before travelling the final 29km to Lillers at 96 km/h. At what time does Gill arrive at Lillers? 9:58 11 N1 ? N2 ?
𝟓𝟎 𝒎𝒑𝒉=𝟖𝟎 𝒌𝒎/𝒉 𝒕= 𝟒𝟖𝟎 𝟖𝟎 =𝟔 hours
[Kangaroo Grey 2015 Q6] A cyclist rides at 5 metres per second. The wheels of his bicycle have a circumference of 125 cm. How many complete turns does each wheel make in 5 seconds? 20 turns ? 12 ?

10. Density involving Volumes
Sometimes we have to work out the volume of the solid ourselves, before subsequently working out density or mass.
Recall that the volume of a prism
= area of cross-section × length ?
𝑉𝑜𝑙𝑢𝑚𝑒= 12×5 2 ×15=𝟒𝟓𝟎 cm3
𝑀𝑎𝑠𝑠=450×6.6=𝟐𝟗𝟕𝟎 𝒈 ?
Fro Tip: Ensure any units are consistent with the others. If the unit of density is g/cm3, then your mass will be in grams.

11. Test Your Understanding?
Cross-section: 2×7 + 5×2 =𝟐𝟒 cm2
Volume =24×200=𝟒𝟖𝟎𝟎 g/cm3
Mass =4800×8=𝟑𝟖𝟒𝟎𝟎 g ? ?
Warning! The units used in the diagram (cm) and the unit of the length of the prism (m) are different. The density is given in g/cm3, so what unit should we use for all lengths?

12. Exercise 1b ? ? 1 2 3 metal bars Wood: 1.11 g/cm3, Water: 1.03 g/cm3
[Edexcel GCSE June2016-2H Q14 Edited]
The diagram shows a metal bar in the shape of a prism. The length of the metal bar is 120 cm. The cross section of the metal bar is shown below. All corners are right angles. The metal bar is made from steel with density 8 g/cm3.
Sean has a trolley. The trolley can carry a maximum mass of 250 kg. How many metal bars can the trolley carry at the same time? 
[Edexcel GCSE(9-1) Mock Set 1 Autumn H Q12 Edited]
The diagram shows a piece of wood in the shape of a cuboid.
The piece of wood is 3 cm by 20 cm by 1.2 m. The mass of the piece of wood is 8 kg.
The piece of wood will float in sea water if the density of the wood is less than the density of the sea water. In a large pool, 1 litre of sea water has a mass of 1030 g.
Will the piece of wood float in this pool? Justify your answer.
[Note: 1 litre = 1000 cm3] 1 2 ?
3 metal bars ?
Wood: 1.11 g/cm3, Water: 1.03 g/cm3
1.11 > 1.03 so wood will sink (i.e. not float)

13. Exercise 1b ? ? 3 4 6kg 302 g [Edexcel GCSE Nov2011-3H Q16]
The diagram shows a solid prism made from metal. The cross-section of the prism is a trapezium. The parallel sides of the trapezium are 8 cm and 12 cm. The height of the trapezium is 6 cm. The length of the prism is 20 cm.
The density of the metal is 5 g/cm3.
Calculate the mass of the prism. Give your answer in kilograms.
[Edexcel GCSE June2008-4H Q13b]
A solid cylinder has a radius of 4 cm and a height of 10 cm. The cylinder is made from wood. The density of the wood is 0.6 grams per cm3.
Work out the mass of the cylinder. Give your answer correct to 3 significant figures. 3 4 ?
6kg ?
302 g

14. Combining parts of a journey
Sometimes a journey may consist of multiple parts. Given the large amount of information to process, you may find it helpful to arrange it in a table.
Use the rows for speed, distance and time, and the columns for each leg of the journey:
F → G
G → H
Speed
40 mph
54 mph
Distance
10 miles
18 miles
Time
1/4 hr
1/3 hrs ? ? ?

15. Considering total time/distance
Pippin the cat runs to the end of my garden at 10mph and back at 20mph. What was her average speed?
Note: We weren’t given the distance! Let’s set the distance arbitrarily for now so that the numbers are easy (e.g. 20 miles). It won’t affect the final answer.
10mph
20mph
Click for Catimation
FROST MANOR
Period 1
Period 2
Overall
Speed
10 mph
20 mph ?
Distance
20 miles
40 miles
Time
2 hours
1 hour
3 hours
Let’s add a 3rd column that represents the whole journey. Note that the ‘overall distance’ is the sum of the distances, and the ‘overall time’ is the sum of the times. The speeds however can’t be added in this way. ? ? ? ? ? ? ?
Speed = 40 3 = mph ?

16. Considering total time/distance
(For more adventurous classes/students!)
We could also do it algebraically to keep the distance general...
10mph
20mph
Period 1
Period 2
Overall
Speed
10 mph
20 mph ?
Distance
𝑑 miles
2𝑑 miles
Time
𝑑 10 hours
𝑑 20 hours
3𝑑 20 hours
Speed = 𝟐𝒅 𝟑𝒅 𝟐𝟎 = 𝟒𝟎𝒅 𝟑𝒅 = 𝟒𝟎 𝟑 mph ? ? ? ? ? ? ? ?

17. Combined Density 𝑚 𝑑 𝑣 ? ? ? ? ? ? Density of B = 1.03 g/cm3 (to 2dp)
The same ‘table’ approach can also be used for a mixture of substances...
Liquid A
Liquid B
Combined (C)
Density
1.42
1.05
Mass
9.94
128.66
138.6
Volume 7
125
132 ? ? ? ? ?
Density of B = 1.03 g/cm3 (to 2dp) ? 𝑚 𝑑 𝑣

18. Test Your Understanding
Papa Smurf walked 10m at 5 m/s from A to B, and then 5m at 10m/s from B to C. What was Papa Smurf’s average speed across the journey? 1
A to B
B to C
Overall
Speed
5m/s
10 m/s
6 m/s
Distance
10m 5m
15m
Time
2 seconds
0.5 seconds
2.5s ? ? ? ? ?
Bob the Builder needs to travel to a building site. He drives the first part of the journey of 10 km at 20 km/h. The total journey time is 1 hour. What speed did he drive for the remainder of the journey if his average speed across the whole journey was 40 km/h? 2
1st
2nd
Overall
Speed
20 km/h
60 km/h
40 km/h
Distance
10 km
30 km
40 km
Time
0.5 hr
1 hr ? ? ?

19. Exercise 2
[IMC 2009 Q19] Driving to Birmingham airport, Mary cruised at 55 miles per hour for the first two hours and then flew along at 70 miles per hour for the remainder of the journey. Her average speed for the entire journey was 60 miles per hour. How long did Mary’s journey to Birmingham Airport take? A 6 hours B hours C 4 hours D hours E 3 hours
[Edexcel GCSE(9-1) Mock Set 2 Spring 2017 1H Q9]
1 litre of a liquid 𝑷 has a mass of 𝑝 grams. 1 litre of a liquid 𝑸 has a mass of 𝑞 grams.
A liquid 𝑹 is made by mixing a volume of liquid 𝑷 with a volume of liquid 𝑸 in the ratio 3:7
Find an expression, in terms of 𝑝 and 𝑞, for the mass of 50 litres of liquid 𝑹.
𝟏𝟓𝒑+𝟑𝟓𝒒
[IMC 2002 Q17] I walk to the bike shop at 3 miles per hour and cycle back along the same route at 12 miles per hour. What is my average speed, in miles per hour, for the time I am actually travelling on the route?
4.8 mph
[Edexcel GCSE Jun2015-2H Q16]
Liquid A has a density of 0.7 g/cm3. Liquid B has a density of 1.6 g/cm3.
140 g of liquid A and 128 g of liquid B are mixed to make liquid C. Work out the density of liquid C.
0.957 g/cm3
Catniss runs the first lap of a race at a speed of 10m/s. The second lap she changes speed, and takes 40 seconds in this lap. Determine the speed she needs to go in the second lap in order to achieve an overall speed of 20 m/s with an overall time of 60 seconds.
25 m/s
[Edexcel GCSE(9-1) Mock Set 2 Spring H Q9]
The densities of three metal alloys, 𝐴, 𝐵 and 𝐶, are in the ratio
         13:15:21
1 m3 of alloy 𝐵 has a mass of 8600 kg.
Work out the difference between the mass of 5 m3 of alloy A and 3 m3 of alloy C. Give your answer correct to 3 significant figures.
1146kg 4 1 ? 2 5 ? 3 ? 6 ? ?

20. Exercise 2
[Cayley 2015 Q1] A train travelling at constant speed takes five seconds to pass completely through a tunnel which is 85m long, and eight seconds to pass completely through a second tunnel which is 160m long. What is the speed of the train?
Note that the train enters the tunnel when the front enters, but exits the tunnel when the back of the train leave. Let the length if the train be 𝒙 metres.
Speed in first tunnel: 𝟖𝟓+𝒙 𝟓
Speed in second tunnel: 𝟏𝟔𝟎+𝒙 𝟖
But the train is going at a constant speed:
𝟖𝟓+𝒙 𝟓 = 𝟏𝟔𝟎+𝒙 𝟖 𝟖 𝟖𝟓+𝒙 =𝟓 𝟏𝟔𝟎+𝒙 𝒙=𝟒𝟎
So speed of train = 𝟖𝟓+𝟒𝟎 𝟓 =𝟐𝟓 m/s N1
[JMO 2009 B3] Tom left a motorway service station and travelled towards Glasgow at a steady speed of 60mph. Tim left the same service station 10 minutes after Tom and travelled in the same direction at a steady speed, overtaking Tom after a further 1 hour 40 minutes. At what speed did Tim travel? When Tim catches Tom, Tom has travelled for 1 hour 50 minutes.
Distance = 𝟔𝟎×𝟏 𝟓 𝟔 =𝟏𝟏𝟎 miles
Thus for Tim: 𝒕=𝟏 𝟐 𝟑 , 𝒅=𝟏𝟏𝟎, 𝒔= 𝟏𝟏𝟎 𝟏 𝟐 𝟑 =𝟔𝟔 mph N2 ? ?

21. Super Challenging Extension Question
We can use this table approach for exceedingly difficult Olympiad problems…
Monday
Tuesday
Wednesday
Walk
Run
Speed 𝑠 2𝑠
Distance
2𝑠𝑡
𝑠𝑡−2𝑠
4𝑠𝑡−8𝑠
Time 2𝑡 𝑡
𝑡−2
2𝑡−4 ?1 ?2 ?4 ?5 ?7 ?8 ?3 ?6
Using the fact the total distance must be the same on Monday and Tuesday, we can work out 𝑡:
𝟐𝒔𝒕+𝟐𝒔𝒕=𝒔𝒕−𝟐𝒔+𝟒𝒔𝒕−𝟖𝒔 𝒕=𝟏𝟎
And therefore the total distance:
𝟒𝒔𝒕=𝟒𝒔×𝟏𝟎=𝟒𝟎𝒔
And therefore the time on Wednesday:
𝟒𝟎𝒔 𝒔 =𝟒𝟎 minutes
If time on Monday was 3𝑡, time on Tuesday was 3𝑡−6 (using information given). Splitting this in the ratio 1:2, we get the times as indicated. ?9
?10
?11

22. Rates of Flow Questions
ROUND 1:
Mr T fills a cuboid container with the dimensions shown. If he fills it at a constant rate of 25 cm3 per second, how long will it take him to completely fill the container?
50cm
Volume:
𝟖×𝟓×𝟓𝟎=𝟐𝟎𝟎𝟎 cm3
Time:
𝟐𝟎𝟎𝟎 𝟐𝟓 =𝟖𝟎 seconds
5cm ?
8cm ?

23. Rates of Flow Questions
ROUND 2: Mr T builds a T shaped container using two cuboids, with dimensions as pictured. Mr T pours in water at a constant rate. In 9 hours the water level reaches 45cm from the bottom of the container, as shown.
Determine the time to completely fill the tank.
The key to solving these questions is:
Work out the rate (cm3/hr or cm3/s) at which the container fills, using the time given and the volume of liquid poured in in this time.
Use this rate in relation to the total volume. This might be working out the total volume given the total time, or vice versa.
50cm
15cm
40cm
40cm
Volume filled in 9 hours:
𝟑𝟎×𝟏𝟎×𝟏𝟎 + 𝟒𝟎×𝟒𝟎×𝟏𝟓 =𝟐𝟕𝟎𝟎𝟎 cm3
Rate at which container fills:
𝟐𝟕𝟎𝟎𝟎 𝟗 =𝟑𝟎𝟎𝟎 cm3/hr
Total volume:
𝟑𝟎×𝟏𝟎×𝟏𝟎 + 𝟒𝟎×𝟒𝟎×𝟓𝟎 =𝟖𝟑𝟎𝟎𝟎 cm3
Therefore time to fill:
𝟖𝟑𝟎𝟎𝟎 𝟑𝟎𝟎𝟎 =𝟐𝟕 𝟐 𝟑 hrs ?
30cm ? ?
10cm
10cm ?

24. Rates of Flow Questions
And what if we were given the time and needed to find one of the lengths?
ROUND 3: Mr T builds another T shaped container using two cuboids, with dimensions as pictured. Mr T pours in slime at a constant rate. In 12 hours the water level reaches 25cm from the bottom of the container, as shown.
If it takes 21.6 hours to completely fill the tank, determine the height ℎ of the upper cuboid. ℎ
5cm
10cm
10cm
Volume filled in 12 hours:
𝟐𝟎×𝟓×𝟓 +(𝟏𝟎×𝟏𝟎×𝟓)=𝟏𝟓𝟎𝟎 cm3
Rate at which container fills:
𝟏𝟓𝟎𝟎 𝟏𝟐 =𝟏𝟐𝟓 cm3/hr
Total volume:
𝟐𝟎×𝟓×𝟓 + 𝟏𝟎×𝟏𝟎×𝒉 =𝟓𝟎𝟎+𝟏𝟎𝟎𝒉
But also:
𝟐𝟏.𝟔×𝟏𝟐𝟓=𝟐𝟕𝟎𝟎 cm3
∴𝟓𝟎𝟎+𝟏𝟎𝟎𝒉=𝟐𝟕𝟎𝟎 𝟐𝟐𝟎𝟎=𝟏𝟎𝟎𝒉 𝒉=𝟐𝟐 cm ? ?
20cm ?
5cm
5cm ? ?

25. Test Your Understanding
Warning: Use consistent units (e.g. metres rather than cm)
Volume of water drained:
𝟐×𝟏×𝟎.𝟐=𝟎.𝟒 m3
Therefore rate per hour:
𝟎.𝟒×𝟐=𝟎.𝟖 m3/hr
Total volume:
𝟏.𝟑+𝟎.𝟓 𝟐 ×𝟐 ×𝟏=𝟏.𝟖 m3
Time to fill:
𝟏.𝟖 𝟎.𝟖 =𝟐.𝟐𝟓 hours
Therefore time extra to wait:
𝟐.𝟐𝟓−𝟎.𝟓=𝟏.𝟕𝟓 hours
or 1 hour, 45 minutes ? ? ? ? ?

26. Exercise 3 ? ? 2 1 65 hours 𝑽=𝟐𝟎×𝟏𝟎=𝟐𝟎𝟎 m3 =𝟐𝟎𝟎 𝟎𝟎𝟎 𝒍
[Edexcel GCSE(9-1) Mock Set 2 Spring 2017 1F Q21, 1H Q4]
The diagram shows a swimming pool. The swimming pool is in the shape of a prism. The swimming pool is filled with water at a rate of 5 litres per second.
Jeremy has 10 hours to fill the swimming pool. 1 m3 = 1000 litres.
Will he completely fill the swimming pool in 10 hours? You must show all your working. 2
[Edexcel GCSE Nov2014-2H Q13a]
The diagram shows a swimming pool in the shape of a prism. The swimming pool is empty. The swimming pool is filled with water at a constant rate of 50 litres per minute.
Work out how long it will take for the swimming pool to be completely full of water. Give your answer in hours. (1 m3 = 1000 litres) 1 ?
𝑽=𝟐𝟎×𝟏𝟎=𝟐𝟎𝟎 m3
=𝟐𝟎𝟎 𝟎𝟎𝟎 𝒍
Volume filled in 10 hours:
𝟏𝟎×𝟑𝟔𝟎𝟎×𝟓=𝟏𝟖𝟎 𝟎𝟎𝟎 𝒍
Therefore he won’t fill the pool. ?
65 hours

27. Exercise 3 ? ? 3 4 𝒉= 𝟏𝟒 𝟑 Volume of water after 37 hours:
[Edexcel GCSE Jun2015-1H Q23]
Useful formulae: Volume of a cone with radius 𝑟 and height ℎ is 1 3 𝜋 𝑟 2 ℎ
The diagram shows a container for grain. The container is a cylinder on top of a cone. The cylinder has a radius of 3 m and a height of ℎ m. The cone has a base radius of 3 m and a vertical height of 4 m. The container is empty.
The container is then filled with grain at a constant rate. After 5 hours the depth of the grain is 6 metres above the vertex of the cone. After 9 hours the container is full of grain.
Work out the value of ℎ. Give your answer as a fraction in its simplest form. You must show all your working.
[Frost] A cone of radius 2cm and height 4cm is gradually filled with water at a constant rate. After 37 hours that water level has risen to 1cm. How long will it take the cone to completely fill with water? 4
4cm
2cm
Volume of water after 37 hours:
𝟏 𝟑 𝝅× 𝟐 𝟐 ×𝟒− 𝟏 𝟑 𝝅× 𝟏.𝟓 𝟐 ×𝟑 = 𝟑𝟕 𝟏𝟐 𝝅
Therefore 𝟏 𝟏𝟐 𝝅 cm3 filled per hour.
Total volume = 𝟏𝟔 𝟑 𝝅
𝟏𝟔 𝟑 𝝅÷ 𝟏 𝟏𝟐 𝝅=𝟔𝟒 hours ? ?
𝒉= 𝟏𝟒 𝟑