Write each of these numbers as an ordinary number.
(a)3.6 x 105(1)
(b)4.02 x 10-3(1)
(Total for Question 3 is 2 marks)
4
720000 is to be written in standard form. Put a ring around the number that is correctly written in standard form.
A) 72 x 104
B) 7.2 x 105
C) 0.72 x 106
D) 7.2 x 104
(Total for Question 4 is 1 mark)
5
Here are four numbers written in standard form.
3.2 x 103 5.6 x 102 2.1 x 104 9.9 x 103
Put the numbers in order of size, starting with the smallest.
(Total for Question 5 is 2 marks)
6
Work out (2 x 105) x (3 x 103). Give your answer in standard form.
(Total for Question 6 is 2 marks)
7
Work out (8 x 106) / (2 x 102). Give your answer in standard form.
(Total for Question 7 is 2 marks)
8
Work out (4.5 x 103) + (2.3 x 102). Give your answer in standard form.
(Total for Question 8 is 3 marks)
9
Work out (6.4 x 10-2) - (3.2 x 10-3). Give your answer in standard form.
(Total for Question 9 is 3 marks)
10
Light travels at a speed of 3 x 108 metres per second. The distance from the Sun to Neptune is 4.5 x 1012 metres.
(a)Work out how long it takes light to travel from the Sun to Neptune. Give your answer, in seconds, in standard form.(3)
(b)Given that 1 hour = 3600 seconds, work out the time taken in hours, to the nearest hour.(1)
(Total for Question 10 is 4 marks)
11
The mass of a hydrogen atom is 1.67 x 10-27 kg. The mass of a carbon atom is 1.99 x 10-26 kg. Work out how many times heavier a carbon atom is than a hydrogen atom. Give your answer correct to 3 significant figures.
(Total for Question 11 is 3 marks)
12
In 2024 the population of the United Kingdom was 6.8 x 107. The area of the United Kingdom is 2.4 x 105 km2.
(a)Work out the population density of the United Kingdom, in people per km2, correct to 3 significant figures.(3)
(b)State one limitation of using this calculation to describe how crowded the United Kingdom is.(1)
(Total for Question 12 is 4 marks)
13
Work out (3 x 104)2. Give your answer in standard form.
(Total for Question 13 is 3 marks)
14
Show that (5 x 10-3) x (4 x 106) = 2 x 104
(Total for Question 14 is 3 marks)
15
A single red blood cell has a volume of 9.0 x 10-11 cm3. An adult has approximately 2.5 x 1013 red blood cells in their body. Work out the total volume of all the red blood cells in the adult's body. Give your answer, in cm3, in standard form, correct to 2 significant figures.
(Total for Question 15 is 4 marks)
16
a = 3 x 105 b = 4 x 10-2
(a)Work out a x b. Give your answer in standard form.(2)
(b)Work out a / b. Give your answer in standard form.(2)
(Total for Question 16 is 4 marks)
17
The Andromeda Galaxy is 2.4 x 1019 km from Earth. A spacecraft travels at a constant speed of 4.0 x 104 km/h.
(a)Work out how long the journey would take. Give your answer, in hours, in standard form, correct to 3 significant figures.(3)
(b)There are approximately 8.76 x 103 hours in a year. Use your answer to part (a) to work out how many years the journey would take, correct to 3 significant figures.(2)
(Total for Question 17 is 5 marks)
18
√5.8 x 109
(a)Work out √5.8 x 109. Give your answer as an ordinary number correct to the nearest whole number.(2)
(b)Write your answer to part (a) in standard form, correct to 3 significant figures.(2)
(Total for Question 18 is 4 marks)
19
A company produces microchips. Each microchip has a mass of 2.5 x 10-4 g. The company packs the microchips into boxes. Each box has a maximum mass of 1.5 kg.
(a)Work out the maximum number of microchips that can be packed into one box.(3)
(b)The company has an order for 3.6 x 107 microchips. Work out the minimum number of boxes needed to pack this order.(2)
(Total for Question 19 is 5 marks)
20
A cuboid has a volume of 4.8 x 108 mm3. The length of the cuboid is 1.2 x 104 mm and the width of the cuboid is 8 x 102 mm.
Diagram: a cuboid labelled with length 1.2 x 104 mm, width 8 x 102 mm and an unknown height h.
Diagram NOT accurately drawn
(a)Work out the height of the cuboid, in mm. Give your answer in standard form.(3)
(b)Given that 1 m = 1000 mm, work out the height of the cuboid in metres. Give your answer in standard form.(3)
(Total for Question 20 is 6 marks)
Mark scheme · 5.4 Standard Form
Question 1
B1 4.5 x 104 oe
Answer: 4.5 x 104
Question 2
B1 3.2 x 10-3 oe
Answer: 3.2 x 10-3
Question 3
(a) B1 360000 cao
(a) Answer: 360000
(b) B1 0.00402 cao
(b) Answer: 0.00402
Question 4
B1 B
Answer: B (7.2 x 105)
Question 5
M1 converts numbers to a comparable form, e.g. as ordinary numbers 3200, 560, 21000, 9900 oe
A1 5.6 x 102, 3.2 x 103, 9.9 x 103, 2.1 x 104 cao, all four in correct order
Answer: 5.6 x 102, 3.2 x 103, 9.9 x 103, 2.1 x 104
Question 6
M1 2 x 3 = 6 and 105 x 103 = 108 seen, or 6 x 108 oe unsimplified
A1 6 x 108 cao
Answer: 6 x 108
Question 7
M1 8 / 2 = 4 and 106 / 102 = 104 seen, or 4 x 104 oe unsimplified
A1 4 x 104 cao
Answer: 4 x 104
Question 8
M1 writes both numbers to the same power of 10, e.g. 45 x 102 + 2.3 x 102 oe
M1 dependent, adds correctly to give 47.3 x 102 oe
A1 4.73 x 103 cao
Answer: 4.73 x 103
Question 9
M1 writes both numbers to the same power of 10, e.g. 64 x 10-3 - 3.2 x 10-3 oe
M1 dependent, subtracts correctly to give 60.8 x 10-3 oe
A1 6.08 x 10-2 cao
Answer: 6.08 x 10-2
Question 10
(a) M1 sets up time = distance / speed, e.g. (4.5 x 1012) / (3 x 108)
(a) M1 4.5 / 3 = 1.5 and 1012 / 108 = 104 seen
(a) A1 1.5 x 104 (seconds) cao
(a) Answer: 1.5 x 104 seconds
(b) B1 4 (hours), ft their part (a) answer / 3600
(b) Answer: 4 hours
Question 11
M1 sets up (1.99 x 10-26) / (1.67 x 10-27)
M1 11.9... or 11.916... seen (unrounded or truncated value)
A1 awrt 11.9
Answer: 11.9 (times heavier)
Question 12
(a) M1 sets up (6.8 x 107) / (2.4 x 105)
(a) M1 283.3... or 2.833... x 102 seen (unrounded value)
(a) A1 awrt 283 (people per km2)
(a) Answer: 283 people per km2 (awrt)
(b) B1 any valid limitation, e.g. the calculation assumes the population is spread evenly across the whole area, but in reality people are concentrated much more heavily in cities than in rural areas oe
(b) Answer: The average density does not show that people are actually spread very unevenly, with far higher density in cities than in the countryside.
Question 13
M1 squares the coefficient, 32 = 9
M1 squares the power of 10, (104)2 = 108
A1 9 x 108 cao
Answer: 9 x 108
Question 14
M1 multiplies coefficients, 5 x 4 = 20
M1 multiplies powers of 10, 10-3 x 106 = 103, giving 20 x 103
A1 cso, converts 20 x 103 to 2 x 104, matching the given answer, with no incorrect working seen
Answer: 2 x 104 (shown)
Question 15
M1 multiplies coefficients, 9.0 x 2.5 = 22.5
M1 adds powers of 10, -11 + 13 = 2, giving 22.5 x 102
A1 2.25 x 103 (unrounded standard form value)
B1 ft, awrt 2.3 x 103 (cm3)
Answer: 2.3 x 103 cm3 (awrt, 2 sig figs)
Question 16
(a) M1 3 x 4 = 12 and 105 x 10-2 = 103 seen, or 12 x 103 oe unsimplified
(a) A1 1.2 x 104 cao
(a) Answer: 1.2 x 104
(b) M1 3 / 4 = 0.75 and 105 / 10-2 = 107 seen, or 0.75 x 107 oe unsimplified
(b) A1 7.5 x 106 cao
(b) Answer: 7.5 x 106
Question 17
(a) M1 sets up time = distance / speed, e.g. (2.4 x 1019) / (4.0 x 104)
(a) M1 2.4 / 4.0 = 0.6 and 1019 / 104 = 1015 seen
(a) A1 awrt 6.00 x 1014 (hours)
(a) Answer: 6.00 x 1014 hours
(b) M1 ft, divides their part (a) answer by 8.76 x 103
(b) A1 ft, awrt 6.85 x 1010 (years)
(b) Answer: 6.85 x 1010 years (awrt)
Question 18
(a) M1 correct calculator method, sight of 76157.7... or awrt 76157.7
(a) A1 76158 cao (nearest whole number)
(a) Answer: 76158
(b) M1 ft, converts their part (a) answer to standard form, e.g. 7.6158 x 104
(b) A1 ft, awrt 7.62 x 104
(b) Answer: 7.62 x 104 (awrt)
Question 19
(a) M1 converts 1.5 kg to grams, 1.5 kg = 1.5 x 103 g
(a) M1 sets up (1.5 x 103) / (2.5 x 10-4)
(a) A1 6 x 106 cao (6000000 microchips)
(a) Answer: 6 x 106 microchips
(b) M1 ft, divides 3.6 x 107 by their part (a) answer
(b) A1 6 (boxes) cao, rounded up to a whole number of boxes if not exact
(b) Answer: 6 boxes
Question 20
(a) M1 finds length x width, (1.2 x 104) x (8 x 102) = 9.6 x 106
(a) M1 dependent, divides volume by their length x width, (4.8 x 108) / (9.6 x 106)
(a) A1 5 x 101 cao (50 mm)
(a) Answer: 5 x 101 mm (50 mm)
(b) M1 ft, divides their part (a) answer by 103, e.g. (5 x 101) / (1 x 103)