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GCSE · Ratio, Proportion and Rates of Change

5.5D Speed and Density (Compound Measures): Fluency and Exam Drill

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A car travels 150 km in 3 hours, travelling at a constant average speed.
Calculate the average speed of the car in km/h.

(Total for Question 1 is 2 marks)

A block of wood has a mass of 45 g and a volume of 60 cm^3.
Calculate the density of the wood, in g/cm^3.

(Total for Question 2 is 2 marks)

A cyclist travels at an average speed of 12 m/s for 25 seconds.
Calculate the distance travelled.

(Total for Question 3 is 2 marks)

A metal cube has a volume of 8 cm³ and a density of 7.9 g/cm^3.
Calculate the mass of the cube.

(Total for Question 4 is 2 marks)

Convert a speed of 20 m/s into km/h.

(Total for Question 5 is 2 marks)

A train travels 234 km in 2 hours 30 minutes, at a constant average speed.
Work out the average speed of the train in km/h.

(Total for Question 6 is 3 marks)

A liquid has a density of 0.92 g/cm^3.
Calculate the mass, in kg, of 5 litres of the liquid. (1 litre = 1000 cm³)

(Total for Question 7 is 3 marks)

Sara walks 3.6 km in 45 minutes, at a constant speed.
Calculate her average speed in m/s. Give your answer to 3 significant figures.

(Total for Question 8 is 4 marks)

A gold bar is a cuboid with dimensions 20 cm by 8 cm by 4.5 cm. The density of the gold is 19.3 g/cm^3.

(a)Work out the volume of the gold bar.(2)

(b)Work out the mass of the gold bar in kg.(2)

(Total for Question 9 is 4 marks)

Aisha and Priya both run a 400 m race.
Aisha finishes the race in 58 seconds.
Priya runs at an average speed of 6.7 m/s.
Work out who is faster, and find the difference between their times for the race, giving your answer to the nearest second.

(Total for Question 10 is 5 marks)

A small sculpture is made from a mixture of two metals, bronze and silver, fused together with no gaps.
The whole sculpture has a total mass of 940 g and a total volume of 110 cm^3.
The bronze part has a volume of 60 cm³ and a density of 8.9 g/cm^3.
The rest of the sculpture is silver.
Calculate the density of the silver part, giving your answer to 3 significant figures.

(Total for Question 11 is 5 marks)

The density of sea water is 1.03 g/cm^3.
A cube-shaped fish tank has a side length of 45 cm.
Work out the mass of sea water, in kg, needed to fill the tank completely. Give your answer to 3 significant figures.

(Total for Question 12 is 4 marks)

A car travels the first 50 km of a journey at an average speed of 40 km/h, then the next 100 km at an average speed of 55 km/h.
Work out the average speed for the whole 150 km journey. Give your answer correct to 1 decimal place.

(Total for Question 13 is 4 marks)

A metal alloy is made by mixing 300 g of metal A, which has a density of 8.4 g/cm³, with 500 g of metal B, which has a density of 7.2 g/cm^3.
Assuming there is no change in total volume when the metals are mixed, calculate the density of the alloy. Give your answer to 3 significant figures.

(Total for Question 14 is 5 marks)

Water flows through a pipe into a cylindrical tank of radius 0.5 m and height 2 m, filling the tank completely in 40 minutes.
The density of water is 1000 kg/m^3.
(Volume of a cylinder = π r² h)

(a)Calculate the volume of the tank, in m³, giving your answer to 3 significant figures.(2)

(b)Work out the mass flow rate of the water, in kg per second, giving your answer to 3 significant figures.(3)

(Total for Question 15 is 5 marks)

The density of a liquid is given by the formula d = m / V, where m is mass and V is volume.

(a)Rearrange the formula to make V the subject.(1)

(b)A liquid has a density of 1.25 g/cm^3. Find the volume of 250 g of the liquid.(2)

(Total for Question 16 is 3 marks)

A cuboid is made of a material with density 2.7 g/cm^3. The cuboid has a mass of 486 g.
The cuboid has a square cross-section of side length x cm and a length of 3x cm.

(a)Work out the volume of the cuboid.(2)

(b)Form an equation in x using the volume of the cuboid, and solve it to find the value of x. Give your answer to 3 significant figures.(4)

(Total for Question 17 is 6 marks)

A solid metal sphere has radius 4.5 cm and density 11.3 g/cm^3.
(Volume of a sphere = (4/3) π r³)
Calculate the mass of the sphere, in kg. Give your answer to 3 significant figures.

(Total for Question 18 is 5 marks)

A ship travels from Port A to Port B, a distance of 84 nautical miles, at a constant average speed of 14 knots (nautical miles per hour).
The ship leaves Port A at 09:15. On the way, it stops at a checkpoint for 45 minutes.
The ship must reach Port B by 15:30 the same day.

(a)Work out the clock time at which the ship arrives at Port B.(3)

(b)Determine, giving a reason, whether the ship reaches Port B on time.(1)

(Total for Question 19 is 4 marks)

A composite solid is made by joining a cylinder and a cone of the same radius, made from the same material throughout.
The cylinder has radius 3 cm and height 10 cm. The cone has radius 3 cm and height 8 cm.
The material has density 2.4 g/cm^3.
(Volume of a cylinder = π r² h. Volume of a cone = (1/3) π r² h)

(a)Calculate the total volume of the solid, giving your answer to 3 significant figures.(2)

(b)The solid is fully submerged in water, and displaces its own volume of water. The density of water is 1 g/cm^3.
Calculate the mass of water displaced, in kg, giving your answer to 3 significant figures.(2)

(c)State, with a reason, whether the solid would float or sink in water.(2)

(Total for Question 20 is 6 marks)

Mark scheme · 5.5D Speed and Density (Compound Measures): Fluency and Exam Drill

Question 1

M1 speed = distance / time i.e. 150 / 3 oe
A1 50 (km/h) cao
Answer: 50 km/h

Question 2

M1 density = mass / volume i.e. 45 / 60 oe
A1 0.75 (g/cm³) cao
Answer: 0.75 g/cm³

Question 3

M1 distance = speed x time i.e. 12 x 25 oe
A1 300 (m) cao
Answer: 300 m

Question 4

M1 mass = density x volume i.e. 7.9 x 8 oe
A1 63.2 (g) cao
Answer: 63.2 g

Question 5

M1 20 x 3.6 oe, e.g. 20 x 3600 / 1000
A1 72 (km/h) cao
Answer: 72 km/h

Question 6

M1 2 hours 30 minutes = 2.5 hours oe, e.g. 5/2 hours
M1 234 / 2.5 oe (ft their time in hours)
A1 93.6 (km/h) cao
Answer: 93.6 km/h

Question 7

M1 5 litres = 5000 cm³ oe
M1 0.92 x 5000 oe (ft their volume in cm³)
A1 4.6 (kg) cao (allow 4600 g if not converted to kg for this mark)
Answer: 4.6 kg

Question 8

M1 3.6 km = 3600 m oe
M1 45 minutes = 2700 seconds oe
M1 3600 / 2700 oe (ft their distance in m and time in s)
A1 1.33 (m/s) awrt
Answer: 1.33 m/s (3 s.f.)

Question 9

(a) M1 20 x 8 x 4.5 oe
(a) A1 720 (cm³) cao
(a) Answer: 720 cm³
(b) M1 19.3 x their (a), i.e. 19.3 x 720 (ft)
(b) A1 13.896 (kg) cao, accept awrt 13.9
(b) Answer: 13.896 kg (accept awrt 13.9 kg)

Question 10

M1 Priya's time = 400 / 6.7 oe
A1 awrt 60 (s), from 59.7... seconds rounded to the nearest second
M1 compares Aisha's time (58 s) with Priya's time (ft their rounded time)
A1 Aisha is faster (oe), correct conclusion with correct reasoning
B1 difference = 2 (seconds) cao
Answer: Aisha is faster; difference = 2 seconds

Question 11

M1 mass of bronze = 8.9 x 60 oe
A1 534 (g) cao
M1 mass of silver = 940 - their 534 (ft)
M1 volume of silver = 110 - 60 = 50 oe; density of silver = their mass of silver / 50 (ft)
A1 8.12 (g/cm³) awrt
Answer: 8.12 g/cm³

Question 12

M1 volume = 45³ oe
A1 91125 (cm³) cao
M1 mass = 1.03 x their 91125 (ft), converted to kg
A1 93.9 (kg) awrt
Answer: 93.9 kg (3 s.f.)

Question 13

M1 time for first part = 50 / 40 = 1.25 (hours) oe
M1 time for second part = 100 / 55 = 1.818... (hours) oe, awrt 1.82
M1 average speed = 150 / (their total time), using total distance / total time (ft)
A1 48.9 (km/h) awrt
Answer: 48.9 km/h (1 d.p.)

Question 14

M1 volume of A = 300 / 8.4 oe, awrt 35.7
M1 volume of B = 500 / 7.2 oe, awrt 69.4
M1 total volume = their vol A + their vol B (ft), awrt 105
M1 density of alloy = 800 / their total volume (ft)
A1 7.61 (g/cm³) awrt
Answer: 7.61 g/cm³ (3 s.f.)

Question 15

(a) M1 π x 0.5² x 2 oe
(a) A1 1.57 (m³) awrt
(a) Answer: 1.57 m³ (3 s.f.)
(b) M1 mass of water = 1000 x their (a) (ft)
(b) M1 40 minutes = 2400 seconds oe
(b) A1 0.654 (kg/s) awrt
(b) Answer: 0.654 kg/s (3 s.f.)

Question 16

(a) B1 V = m / d oe
(a) Answer: V = m / d
(b) M1 250 / 1.25, using their (a) formula (ft)
(b) A1 200 (cm³) cao
(b) Answer: 200 cm³

Question 17

(a) M1 volume = mass / density = 486 / 2.7 oe
(a) A1 180 (cm³) cao
(a) Answer: 180 cm³
(b) M1 volume of cuboid = x x x x 3x = 3x³ oe
(b) M1 3x³ = their (a) (ft), i.e. 3x³ = 180
(b) M1 x³ = 60 oe (ft)
(b) A1 x = 3.91 (cm) awrt, from the cube root of 60
(b) Answer: x = 3.91 (3 s.f.)

Question 18

M1 (4/3) x π x 4.5³ oe
A1 382 (cm³) awrt, or exact value 121.5pi
M1 mass = 11.3 x their volume (ft)
M1 converts their mass in g to kg, dividing by 1000 oe
A1 4.31 (kg) awrt
Answer: 4.31 kg (3 s.f.)

Question 19

(a) M1 sailing time = 84 / 14 = 6 (hours) oe
(a) M1 total elapsed time = their 6 hours + 45 minutes = 6 hours 45 minutes oe (ft)
(a) A1 16:00 cao (ft their total elapsed time added to 09:15)
(a) Answer: 16:00
(b) B1 no oe, with correct comparison, e.g. 16:00 is later than 15:30, so the ship is 30 minutes late (ft their (a))
(b) Answer: No; the ship arrives at 16:00, which is 30 minutes after the 15:30 deadline.

Question 20

(a) M1 π x 3² x 10 + (1/3) x π x 3² x 8 oe, i.e. 90pi + 24pi
(a) A1 358 (cm³) awrt, or exact value 114pi
(a) Answer: 358 cm³ (3 s.f.)
(b) M1 mass of water displaced = 1 x their (a) (ft), then converted from g to kg
(b) A1 0.358 (kg) awrt
(b) Answer: 0.358 kg (3 s.f.)
(c) M1 compares the density of the solid (2.4 g/cm³) with the density of water (1 g/cm³), e.g. by finding mass of solid = 2.4 x their (a) (ft)
(c) A1 sink oe, because the density of the solid (2.4 g/cm³) is greater than the density of water (1 g/cm³); correct reasoning required
(c) Answer: The solid would sink, because its density (2.4 g/cm³) is greater than the density of water (1 g/cm³).

Speed and Density (Compound Measures): Fluency and Exam Drill - Worksheets, Questions and Revision