Radioactivity: Edexcel International GCSE Physics questions

Every Edexcel International GCSE Physics exam question on Radioactivity (specification points 7.2–7.16), 2019–2025, with its mark scheme and examiner’s report.

All 150 questions on Radioactivity, as a list

June 2025, Paper 1, Q4(d)(i) (1 mark): State what is meant by the term ionising radiation. (1)
June 2025, Paper 1, Q4(d)(ii) (1 mark): State what is meant by the term random process. (1)
June 2025, Paper 1, Q9(a) (1 mark): The teacher uses a radiation detector to detect the radiation emitted from the protactinium‑234m. Give the name of a suitable radiation detector. (1)
June 2025, Paper 1, Q9(b) (2 marks): Give two safety precautions to minimise the risk of harm to the teacher from the radiation emitted by the protactinium generator. (2) 1 2
June 2025, Paper 1, Q9(c)(i) (2 marks): The initial count rate is 840 counts per second. Calculate the expected count rate after 140 seconds. (2) count rate = counts per second
June 2025, Paper 1, Q9(c)(ii) (3 marks): The activity of an isotope is the total amount of radiation emitted each second. The teacher estimates that only 5% of the total radiation emitted is detected b
June 2025, Paper 2, Q6(a) (1 mark): Which of these statements is correct for gamma radiation? (1) A gamma radiation has a weaker penetrating ability than alpha radiation B gamma radiation is emitt
June 2025, Paper 2, Q6(b)(i) (1 mark): Give a reason why the student takes repeat readings. (1)
June 2025, Paper 2, Q6(b)(ii) (3 marks): The graph shows the student’s results. 14 000 12 000 10 000 8000 Mean count 6000 4000 2000 0 0 5 10 15 20 25 30 Distance in cm The student claims that the data
June 2025 (R), Paper 1, Q5(a) (2 marks): Describe the structure of an alpha particle. (2)
June 2025 (R), Paper 1, Q5(b) (3 marks): The source used in the cloud chamber contains the isotope radium-226. The incomplete nuclear equation shows the decay of radium-226 into an isotope of radon. Co
June 2025 (R), Paper 1, Q5(c) (2 marks): The tracks in the cloud chamber can be seen through the glass viewing window. Explain why the radioactive source is not a hazard to the person viewing the cloud
June 2025 (R), Paper 1, Q5(d) (3 marks): When radiation is emitted from the source a white line is produced in the cloud chamber. The white line is known as a track. The diagram shows the tracks produc
June 2025 (R), Paper 2, Q2(a) (2 marks): State what is meant by the term isotopes. (2)
June 2025 (R), Paper 2, Q2(b)(i) (1 mark): How many protons are in the nucleus of an atom of uranium‑239? (1) A 92 B 147 C 239 D 331
June 2025 (R), Paper 2, Q2(b)(ii) (1 mark): Give the name of the other particle found in the nucleus of an atom of uranium‑239. (1)
June 2025 (R), Paper 2, Q2(c)(i) (2 marks): A sample of uranium‑239 has an initial mass of 60 g. Calculate the mass of uranium‑239 remaining after 46 minutes. (2) mass = g
June 2025 (R), Paper 2, Q2(c)(ii) (1 mark): When uranium‑239 undergoes beta decay, an isotope of the element neptunium, Np, is produced. Which of these is the correct symbol for the neptunium nucleus prod
November 2024, Paper 1, Q4(a) (1 mark): On the graph, give the unit for activity. (1)
November 2024, Paper 1, Q4(b) (2 marks): Describe what is meant by the term background radiation. (2)
November 2024, Paper 1, Q4(c)(i) (2 marks): Describe what is meant by the term half‑life. (2)
November 2024, Paper 1, Q4(c)(ii) (2 marks): Use the graph to determine the half‑life of this radioactive sample. (2) half‑life = minutes
November 2024, Paper 1, Q4(d) (4 marks): The teacher suggests that this formula shows the relationship between activity and time taken. activity × time taken = constant Use data from the graph to deter
November 2024, Paper 2, Q1(a) (1 mark): Which of these is the same as a beta particle? (1) A electromagnetic wave B electron C helium nucleus D neutron
November 2024, Paper 2, Q1(b) (1 mark): Which of these is the same as an alpha particle? (1) A electromagnetic wave B helium nucleus C neutron D proton
November 2024, Paper 2, Q1(c) (3 marks): Complete the nuclear equation for an alpha decay. (3) α 95Am → 237 241 Np +
June 2024, Paper 1, Q2(a) (1 mark): The nucleus of an atom of carbon has 6 protons and 8 neutrons. Which row of the table shows the nucleus of an atom that is a different isotope of carbon? (1) Nu
June 2024, Paper 1, Q2(b) (1 mark): Which type of radiation is a high-energy electron? (1) A alpha B beta C gamma D neutron
June 2024, Paper 1, Q2(c) (1 mark): A nucleus emits radiation. This causes the mass number to decrease by one. The atomic number stays the same. Which type of radiation does the nucleus emit? (1)
June 2024, Paper 1, Q2(d) (1 mark): The nucleus of an isotope of uranium can be represented using this symbol. 92U 238 The nucleus forms part of a positively charged ion. How many electrons could
June 2024, Paper 1, Q2(e) (1 mark): A radioactive isotope has an initial activity of 400 Bq. The half-life of the isotope is 8 hours. What is the activity of the isotope after 16 hours? (1) A 25 B
June 2024 (R), Paper 1, Q8(a)(i) (1 mark): How many neutrons are in the nucleus of an atom of iodine-131? (1) A 53 B 78 C 131 D 184
June 2024 (R), Paper 1, Q8(a)(ii) (2 marks): Iodine-131 is radioactive and decays with a half-life of 8 days. State what is meant by the term half-life. (2)
June 2024 (R), Paper 1, Q8(a)(iii) (3 marks): The cross (×) on the graph shows the initial number of atoms in a sample of iodine-131. Draw three more crosses (×) on the graph to show how the number of atoms
June 2024 (R), Paper 1, Q8(a)(iv) (2 marks): Use a curve of best fit on the graph to estimate the time taken for the number of atoms in the sample to decrease to 5000. (2) time taken = days
June 2024 (R), Paper 1, Q8(b)(i) (1 mark): State the name of a piece of equipment that can detect the radiation emitted by iodine-131. (1)
June 2024 (R), Paper 1, Q8(b)(ii) (1 mark): Give a reason why gamma radiation is more likely to be detected outside the body than beta radiation. (1)
June 2024 (R), Paper 1, Q8(c) (3 marks): Technetium-99m is another radioactive isotope. Iodine-131 and technetium-99m are both used as medical tracers. Medical tracers use radiation detected outside th
November 2023, Paper 1, Q7(a)(i) (1 mark): State the name of particle X. (1)
November 2023, Paper 1, Q7(a)(ii) (2 marks): Carbon-14 decays by beta decay. State what happens to the number of protons and the number of neutrons in a carbon-14 nucleus when it decays. (2)
November 2023, Paper 1, Q7(b)(i) (3 marks): Plot the scientist’s data on the grid. (3)
November 2023, Paper 1, Q7(b)(ii) (1 mark): Draw the curve of best fit. (1)
November 2023, Paper 1, Q7(b)(iii) (2 marks): Use the graph to determine the age of a sample of wood that has 36% of its carbon-14 remaining. (2) age = years
November 2023, Paper 1, Q7(c)(i) (1 mark): A tree absorbs carbon-14 during its lifetime. A student suggests that trees grown after 1950 are contaminated with carbon-14. Give a reason why the student’s su
November 2023, Paper 1, Q7(c)(ii) (2 marks): Explain how nuclear weapons testing affects the determination of the age of a sample of wood produced after 1950. (2)
June 2023, Paper 1, Q1(a) (1 mark): Which of these is the unit of activity? (1) A becquerel B kilogram C newton D pascal
June 2023, Paper 1, Q1(b) (1 mark): Which of these is the correct description of the term half-life? (1) A time taken for the activity of a substance to halve B half of the time taken for the mass
June 2023, Paper 1, Q1(c)(i) (2 marks): The box gives the names of different pieces of equipment. ruler stopwatch balance newton meter protractor GM tube voltmeter ammeter Complete the sentences using
June 2023, Paper 1, Q1(c)(ii) (1 mark): The graph shows the teacher’s results. 50 40 30 Count rate in counts / s 20 10 0 0 10 20 30 40 50 Time in s Draw a circle around the anomalous result. (1)
June 2023, Paper 1, Q1(c)(iii) (2 marks): Use the graph to determine the half-life of the radioactive sample. (2) half-life = s
June 2023, Paper 1, Q1(c)(iv) (1 mark): Give a reason why the teacher should not expect the data points to lie exactly on the curve of best fit. (1)
June 2023 (R), Paper 1, Q7(a)(i) (1 mark): Give a suitable unit for activity. (1)
June 2023 (R), Paper 1, Q7(a)(ii) (3 marks): On the axes below, sketch a graph for the decay of the sample of protactinium‑234 during its first three half‑lives. (3) 800 600 Activity 400 in 200 0 0 5 10 15
June 2023 (R), Paper 1, Q7(a)(iii) (2 marks): When protactinium‑234 undergoes beta (β–) decay it becomes uranium‑234. The incomplete nuclear equation shows this process. 234Pa 234 β 92U + Complete the nucle
June 2023 (R), Paper 1, Q7(b)(i) (1 mark): Which of these is the dependent variable in the investigation? (1) A count measured by the detector B distance between source and detector C material between so
June 2023 (R), Paper 1, Q7(b)(ii) (3 marks): The student’s method does not allow for background radiation. Describe how the student’s method should be modified to allow for background radiation. (3)
June 2023 (R), Paper 1, Q7(b)(iii) (2 marks): Describe how the student’s method could be modified to improve the reliability of the results. (2)
June 2023 (R), Paper 1, Q7(b)(iv) (3 marks): Evaluate the data from the experiment to conclude the type of radiation emitted by protactinium‑231. (3)
January 2023, Paper 1, Q4(a)(i) (1 mark): State one source of background radiation. (1)
January 2023, Paper 1, Q4(a)(ii) (4 marks): Describe how the teacher could measure the count rate from a radioactive source and correct the count rate for background radiation. (4)
January 2023, Paper 1, Q4(b)(i) (2 marks): Calculate the mean count rate when the thickness of lead is 6.0 mm. (2) mean count rate = Bq
January 2023, Paper 1, Q4(b)(ii) (3 marks): Plot a graph of mean count rate against thickness of lead. (3)
January 2023, Paper 1, Q4(b)(iii) (1 mark): Draw the curve of best fit. (1)
January 2023, Paper 1, Q4(b)(iv) (2 marks): When there is not a sheet of lead between the radioactive source and the radiation detector, the mean count rate is 484 Bq. Use the graph to determine the thick
January 2023, Paper 1, Q4(c) (2 marks): The radioactive source emits only one type of radiation. Explain which type of radiation this radioactive source emits. (2)
January 2023 (R), Paper 1, Q5(c) (3 marks): The half-life of a radioactive isotope is 12 years. The activity of a sample of this isotope is 120 kBq. Calculate the activity of this sample after 48 years. (
January 2023 (R), Paper 1, Q9(a)(i) (1 mark): State a precaution the teacher should take to make sure they are working safely with the radioactive sources. (1)
January 2023 (R), Paper 1, Q9(a)(ii) (1 mark): State the name of a detector the teacher could use to detect the radiation from each source. (1)
January 2023 (R), Paper 1, Q9(b) (3 marks): Draw crosses (× ) in the table to show which type of radiation cannot penetrate each material in the table. (3) Material Type of radiation 10 cm of air 2 cm of
June 2022, Paper 1, Q2(a) (1 mark): Which statement describes the term atomic number for a nucleus? (1) A number of electrons in the nucleus B number of neutrons in the nucleus C number of protons
June 2022, Paper 1, Q2(b) (1 mark): Which statement describes the term mass number for a nucleus? (1) A number of electrons in the nucleus B number of neutrons in the nucleus C number of protons i
June 2022, Paper 1, Q2(c) (1 mark): Which statement describes the term isotopes? (1) A atoms with the same number of electrons but a different number of protons B atoms with the same number of n
June 2022, Paper 1, Q2(d) (1 mark): An atom contains 8 protons, 8 neutrons and 8 electrons. Which of these would result in a negatively charged ion of the same element? (1) A adding an electron B
June 2022, Paper 1, Q2(e) (1 mark): It is not possible to predict exactly when a radioactive nucleus will decay. Which feature of radioactive decay best explains this observation? (1) A radioactiv
June 2022, Paper 1, Q2(f) (1 mark): Which of these is the correct unit for measuring the activity of a radioactive sample? (1) A becquerel (Bq) B coulomb (C) C joule (J) D watt (W)
June 2022, Paper 1, Q5(a) (1 mark): Granite is formed naturally and contains radioactive isotopes. The granite work surface contributes to the background radiation in the kitchen. Give another nat
June 2022, Paper 1, Q5(b)(i) (2 marks): State two differences between alpha radiation and beta radiation. (2) 1 2
June 2022, Paper 1, Q5(b)(ii) (3 marks): The incomplete nuclear decay equation summarises the decay sequence of thorium-232 into radon‑220. 2 α + 0 β 90Th 220 86Rn + 4 −1 232 Calculate the number of al
June 2022, Paper 1, Q5(c) (3 marks): Thorium‑232 is a solid and remains in the work surface. Radon‑220 is a gas and is emitted from the work surface. Thorium‑232 and radon‑220 both emit alpha radia
June 2022 (R), Paper 1, Q7(a) (2 marks): Describe what is meant by the term background radiation. (2)
June 2022 (R), Paper 1, Q7(b)(i) (2 marks): State what is meant by the term half‑life. (2)
June 2022 (R), Paper 1, Q7(b)(ii) (2 marks): Use the graph to determine the half‑life of radon‑222. (2) half‑life = days
June 2022 (R), Paper 1, Q7(c) (3 marks): Radon‑222 is formed by multiple alpha decays of uranium‑234. Complete the nuclear equation by giving the missing information. (3) 92U → 222Rn + 3 α 234
June 2022 (R), Paper 1, Q7(d) (2 marks): Radon‑222 also emits alpha radiation. Explain the hazard to humans of breathing in air contaminated with radon‑222. (2)
June 2022 (R), Paper 2, Q3(a) (2 marks): Neutral sulfur particles can become positively charged sulfur particles. Describe the difference between a neutral sulfur particle and a positively charged sulf
January 2022, Paper 1, Q2(a) (1 mark): State how the scientist should correct the count rate for background radiation. (1)
January 2022, Paper 1, Q2(b)(i) (2 marks): Explain what is meant by the term half-life. (2)
January 2022, Paper 1, Q2(b)(ii) (2 marks): Use the graph to determine the half-life of this isotope. (2) half-life = minutes
January 2022 (R), Paper 1, Q1(a) (1 mark): Bismuth‑207 is a gamma emitter and is represented by the symbol 207 83Bi How many neutrons are in the nucleus of an atom of bismuth‑207? (1) A 83 B 124 C 207 D
January 2022 (R), Paper 1, Q1(b) (1 mark): Which of these gives the best description of gamma radiation? (1) A a helium nucleus B a high energy electron C a high frequency electromagnetic wave D a subato
January 2022 (R), Paper 1, Q1(c) (3 marks): A technician uses this method to investigate the penetration power of gamma radiation. • place a gamma emitting source at a fixed distance from a radiation dete
November 2021, Paper 1, Q4(a) (1 mark): Give a suitable unit for activity. (1)
November 2021, Paper 1, Q4(b)(i) (1 mark): Label both axes. (1)
November 2021, Paper 1, Q4(b)(ii) (1 mark): Draw the curve of best fit. (1)
November 2021, Paper 1, Q4(b)(iii) (2 marks): Use the graph to determine the half‑life of cobalt‑60. (2) half‑life = years
November 2021, Paper 1, Q4(b)(iv) (2 marks): Estimate the time taken for the activity to decrease to 1 8 of its initial value. (2) time = years
November 2021, Paper 1, Q4(c) (2 marks): Cobalt‑60 is produced when a neutron is absorbed by the nucleus of a stable atom of cobalt‑59. The nuclei of these two isotopes can be represented as 60 59 27Co
November 2021, Paper 1, Q4(d) (2 marks): Cobalt‑60 decays by beta emission. Describe what happens to the nucleus of a cobalt‑60 atom during beta decay. (2)
November 2021, Paper 1, Q4(e) (4 marks): Cobalt‑60 also emits gamma radiation. Cobalt‑60 is produced in a nuclear reactor. Discuss the hazards involved and the precautions taken when disposing of cobal
June 2021, Paper 1, Q7(a) (2 marks): The diagram shows tracks produced by radiation in a device called a cloud chamber. The tracks are formed when particles ionise the material in the cloud chamber
June 2021, Paper 1, Q7(b)(i) (2 marks): Describe what is meant by the term isotope. (2)
June 2021, Paper 1, Q7(b)(ii) (2 marks): Complete the equation for the decay of uranium-235 (2) 2α 92U → 231Th + 235
June 2021, Paper 1, Q7(b)(iii) (5 marks): Uranium-235 decays to thorium-231 with a half-life of 700 million years. When a rock was formed, it contained 6400 million uranium-235 nuclei and no thorium-231
January 2021, Paper 1, Q5(a) (1 mark): Give a safety precaution that would reduce the teacher’s exposure to radiation when working with the radioactive source. (1)
January 2021, Paper 1, Q5(b) (1 mark): Radium-226 can be represented using the symbol 226 Ra 88 How many neutrons are in the nucleus of an atom of radium-226? (1) A 88 B 138 C 226 D 314
January 2021, Paper 1, Q5(c)(i) (1 mark): Give the name of a piece of apparatus that detects ionising radiation. (1)
January 2021, Paper 1, Q5(c)(ii) (1 mark): The teacher finds that the radiation emitted from the radioactive source is not detected when the detector is more than 5 cm away from the source. State the typ
January 2021, Paper 1, Q5(d)(i) (2 marks): State what is meant by the term half-life. (2)
January 2021, Paper 1, Q5(d)(ii) (1 mark): The radioactive source contains 2.66 × 1021 atoms of radium-226. Approximately how many atoms of radium-226 will remain in the source after 800 years? (1) A 0.6
January 2021 (R), Paper 1, Q10(c) (3 marks): Iodine-129 is an isotope found in radioactive waste from nuclear power stations. Iodine-129 has a half-life of approximately 15 million years. A sample of iodin
January 2021 (R), Paper 1, Q10(d) (5 marks): Some radioactive waste from nuclear power stations has a very long half-life. Discuss precautions that must be taken when disposing of this radioactive waste. (
January 2021 (R), Paper 2, Q5(b)(i) (2 marks): Explain what is meant by background radiation. (2)
January 2021 (R), Paper 2, Q5(b)(ii) (3 marks): Suggest why there is a limit to the number of hours that an airline pilot can fly at maximum height. (3)
November 2020, Paper 1, Q10(a)(i) (2 marks): Describe the difference between the nuclei of technetium-98 and technetium-99 (2)
November 2020, Paper 1, Q10(a)(ii) (2 marks): Technetium-99 is formed when the element molybdenum-99 decays. Complete the nuclear equation for the decay of molybdenum into technetium-99 (2) 99 Mo → 99 Tc +
November 2020, Paper 1, Q10(b) (5 marks): Describe an experiment that a scientist could use to demonstrate that the emission from technetium-99 is gamma radiation. Include details of a safety precaution
November 2020, Paper 1, Q10(c) (4 marks): A scientist measures the count rate at different distances from the technetium source. The graph shows how the count rate changes with distance from the technet
November 2020 (R), Paper 1, Q2(c) (1 mark): Uranium-235 is radioactive and emits alpha radiation. Which of these statements describes alpha radiation? (1) A an electromagnetic wave B a helium nucleus C a
November 2020 (R), Paper 1, Q2(d) (1 mark): Some of the products of nuclear fission are radioactive because their nuclei have too many neutrons. These nuclei become more stable when a neutron changes into
November 2020 (R), Paper 2, Q7(c)(i) (2 marks): State what is meant by the term half-life. (2)
November 2020 (R), Paper 2, Q7(c)(ii) (2 marks): In 1990, the SC stopped being used at CERN, a research centre. In 1990, the largest number of barium-133 atoms were found 40 cm deep in the concrete walls. Sinc
January 2020, Paper 1, Q9(a) (1 mark): Name a suitable radiation detector that the teacher could use. (1)
January 2020, Paper 1, Q9(b) (1 mark): State the independent variable in the teacher’s investigation. (1)
January 2020, Paper 1, Q9(c) (2 marks): Explain why every absorbing material used in the investigation has a thickness of 1 cm. (2)
January 2020, Paper 1, Q9(d) (1 mark): Suggest one improvement the teacher could make to this method. (1)
January 2020, Paper 1, Q9(e)(i) (3 marks): On the grid, plot a bar chart of the mean count rate for each absorbing material. (3)
January 2020, Paper 1, Q9(e)(ii) (1 mark): Why is a bar chart the correct way to display the results? (1) A absorbing material is a continuous variable B absorbing material is not a continuous variable C
January 2020, Paper 1, Q9(e)(iii) (2 marks): A student concludes that plastic is the best absorber of gamma radiation because plastic gives the largest mean count rate. Evaluate the student’s conclusion. (
January 2020, Paper 1, Q11(a) (3 marks): Technetium-99m has a half-life of 6 hours. A sample of technetium-99m has an initial activity of 160 Bq. Complete the graph to show how the activity of this sam
January 2020, Paper 1, Q11(b)(i) (1 mark): Technetium-99m does not exist naturally. Suggest why technetium-99m is usually made at the hospital where it is used. (1)
January 2020, Paper 1, Q11(b)(ii) (2 marks): Explain why technetium-99m is an effective isotope to use as a medical tracer. (2)
January 2020, Paper 1, Q11(c) (3 marks): The gamma radiation emitted by technetium-99m is potentially harmful to humans. Discuss the risks of using technetium-99m to doctors and to patients. (3)
January 2020, Paper 2, Q7(a)(i) (2 marks): State what information the numbers 92 and 238 give about the nucleus of this isotope of uranium. (2) 92. 238
January 2020, Paper 2, Q7(a)(ii) (2 marks): Uranium‑238 decays by alpha emission. Describe how the nucleus of a uranium‑238 atom changes as a result of alpha emission. (2)
January 2020, Paper 2, Q7(b)(i) (2 marks): Calculate the number of uranium‑238 atoms in the plate. (2) number of atoms =
January 2020, Paper 2, Q7(b)(ii) (3 marks): Uranium‑238 is an alpha emitter and has a half‑life of 4.5 billion years. Explain why it is safe to eat food from the uranium glass plate. (3)
January 2020 (R), Paper 1, Q9(a)(i) (2 marks): Complete the equation that shows the alpha decay of polonium-210 into lead-206. (2) 210 206 84Po → α 82Pb +
January 2020 (R), Paper 1, Q9(a)(ii) (1 mark): The alpha particle ionises air molecules in the room. State what is meant by the term ionisation. (1)
January 2020 (R), Paper 1, Q9(a)(iii) (2 marks): Explain why workers in the same room as the deioniser are not at risk from the alpha radiation it emits. (2)
January 2020 (R), Paper 1, Q9(b)(i) (2 marks): State what is meant by the term half-life. (2)
January 2020 (R), Paper 1, Q9(b)(ii) (3 marks): The initial activity of the polonium-210 source in the deioniser is 70 kBq. Calculate the activity of the source after 420 days. (3) activity = kBq
June 2019, Paper 1, Q3(a) (2 marks): The diagram shows how gamma radiation is used to irradiate fruit stored in a wooden box. The radiation kills bacteria on the fruit. wooden box containing fruit
June 2019, Paper 1, Q3(b) (2 marks): The wooden box has this label. these fruits have been irradiated Explain why fruit irradiated with gamma radiation is safe to eat. (2)
June 2019, Paper 1, Q9(b)(i) (2 marks): State the meaning of the term isotope. (2)
June 2019, Paper 2, Q4(a) (1 mark): Describe the nature of an alpha particle. (1)
June 2019 (R), Paper 1, Q2(a)(iii) (2 marks): The daughter nuclei can cause contamination and irradiation. Describe the difference between contamination and irradiation. (2)
June 2019 (R), Paper 1, Q3(a) (2 marks): State what is meant by the term isotope. (2)
June 2019 (R), Paper 1, Q3(b) (1 mark): How many protons are in the nucleus of lead-210? (1) A 82 B 128 C 210 D 292
June 2019 (R), Paper 1, Q3(c)(i) (2 marks): A sample of lead-210 has an initial activity of 240 Bq. After 66 years, the activity of the sample is 30 Bq. Calculate the half-life of lead-210. (2) half-life
June 2019 (R), Paper 1, Q3(c)(ii) (2 marks): Lead-210 decays into lead-206 through a number of stages. This involves one alpha decay and a number of beta decays. This incomplete equation summarises these s

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