Algebra: AQA A Level Biology Exam Questions by Topic

AQAMaths SkillsExam Questions
Written by Tom Whitburn

Algebra in AQA A Level Biology means using equations you are given, or ones you are expected to know, in a biological context. The MS 2 skills in the specification cover the symbols =, <, >, ∝ and ~, changing the subject of an equation, substituting values with the right units, solving equations such as cardiac output = stroke volume × heart rate and the Hardy–Weinberg equation, and using logarithms for numbers that range over several orders of magnitude, such as the growth of a yeast population.

The examiners' reports show that students rarely lose these marks because the maths is hard. They lose them by dividing before they subtract, stopping at r² instead of r, using the diameter instead of the radius, taking N as the number of species, treating 10% affected as q rather than q², ignoring the significant figures asked for, and saying a log scale is used 'to fit on the graph'.

Write out the equation, rearrange it before you put any numbers in, and show every step: method marks are common. If you'd like help with maths skills, my AQA masterclasses run online on Tuesdays at 5.15pm for Year 12 and on Wednesdays or Thursdays at 6.30pm for Year 13.

What you need from GCSE

Each question on this page assumes you're secure on these GCSE ideas. If any of them feels shaky, look it up before you start.

  • GCSE: substituting numbers into a formula, and rearranging a formula by doing the same operation to both sides.
  • GCSE: the order of operations: brackets, then powers, then multiplying and dividing, then adding and subtracting.
  • GCSE: the area of a circle is πr², and the radius is half the diameter.
  • GCSE: the symbols <, > and ∝ (directly proportional to), and rounding to a given number of significant figures.
  • GCSE: standard form, and converting between units such as mm and µm, or cm³ and dm³.

Key terms

These words come up in the questions and mark schemes. Examiners expect you to use them exactly.

  • Subject of an equation: the quantity on its own on one side of the equation, such as heart rate in heart rate = cardiac output ÷ stroke volume.
  • Changing the subject: rearranging an equation so the quantity you want is on its own, by doing the inverse operation to both sides in the reverse order.
  • Substitution: putting numerical values, in the units the equation needs, in place of the letters.
  • Solving an equation: working out the value of the unknown quantity, for example q from q² in the Hardy–Weinberg equation.
  • ∝, << and >>: ∝ means directly proportional to; << means much less than and >> means much greater than; ~ means approximately.
  • Radius: half the diameter; formulae such as πr², 4πr² and 4/3πr³ use the radius.
  • Index of diversity: a measure of diversity that uses the number of species and the number of individuals of each; N is the total number of organisms of all species.
  • Hardy–Weinberg equation: p + q = 1 and p² + 2pq + q² = 1, where q² is the frequency of the homozygous recessive genotype and 2pq the heterozygotes.
  • Logarithm (log₁₀): the power to which 10 is raised to give a number; each whole number up a log₁₀ scale is ten times larger.
  • Log scale: a scale used when values range over several orders of magnitude, such as numbers of bacteria or yeast cells.

What these questions cover (AQA maths skills MS 2: algebra)

The questions test these mathematical skills from the AQA AS and A-level Biology specification (7401/7402):

  • MS 2.1 Understand and use the symbols: =, <, <<, >>, >, ∝, ~ (the only MS 2.1 part available; it asks you to write an equation for percentage loss)
  • MS 2.2 Change the subject of an equation
  • MS 2.3 Substitute numerical values into algebraic equations using appropriate units for physical quantities
  • MS 2.4 Solve algebraic equations
  • MS 2.5 Use logarithms in relation to quantities that range over several orders of magnitude (both questions ask why a log scale is used; no available part asks you to calculate a log)

How to use this page

Write your answer on paper before you open the mark scheme. Allow about 1.3 minutes per mark: the 17 questions are worth 31 marks, so about 40 minutes in total. Then mark your answer strictly against the mark scheme and read the examiners' tip underneath it. If you'd rather print the questions, download the PDF pack.

Symbols and writing equations (MS 2.1)

Question 1

AS Paper 2 · June 2018 · Q07.5 · 1 mark · Hard

Write the equation for calculating the percentage loss in mass of cellulose.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Earlier part of the same exam question, giving the information this part uses. Table 4: mass of cellulose remaining after 48 hours from 15 g at the start: endocellulase 11.9 g, exocellulase 14.8 g, both enzymes 9.2 g; question asks for the equation for percentage loss in mass, 1 mark.
Show the mark scheme Mark scheme: (initial mass − final mass) ÷ initial mass × 100; equivalent forms accepted, such as (1 − final ÷ initial) × 100.

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From the examiners' report: Many knew this, but marks went on leaving out × 100, subtracting the mass lost instead of the final mass, or working out the percentage remaining. Use the words initial and final, not 'new' and 'old'.

Changing the subject (MS 2.2)

Question 2

A level Paper 3 · June 2019 · Q01.3 · 1 mark · Easy

Rearrange y = 0.72x + 4 to find the loop of Henle length for a given urine concentration.

Question text: urine concentration in arbitrary units (y) and mean length of the loop of Henle in mm (x) are related by y = 0.72x + 4; calculate x when y is 16.56, 1 mark.
Show the mark scheme Mark scheme: 17.4 (mm); accept any number of fours after the decimal point.

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From the examiners' report: About 80% scored. Undo the + 4 first, then divide by 0.72. Dividing first gives 19, which is wrong.

Question 3

A level Paper 2 · June 2022 · Q05.4 · 1 mark · Medium

Rearrange the mark-release-recapture equation to find how many lemurs in the second sample were marked.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Question text: 30 lemurs were marked and released; a second sample of 25 was captured; the population was calculated as 250; calculate how many lemurs in the second sample were marked, 1 mark.
Show the mark scheme Mark scheme: 3.

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From the examiners' report: Only 45% got this: recalling and rearranging the equation caused more problems than expected. Number marked in the second sample = first sample × second sample ÷ population.

Question 4

AS Paper 2 · November 2020 · Q03.4 · 2 marks · Medium

Use cardiac output = heart rate × stroke volume to find an athlete's heart rate after exercise.

Table 1: before exercise, heart rate 62 beats minute⁻¹, stroke volume 80 cm³, cardiac output 4960 cm³ minute⁻¹. After exercise stroke volume rose by 30% and cardiac output was 13 832 cm³ minute⁻¹; give heart rate to 2 significant figures.
Show the mark scheme Mark scheme: 130 beats minute⁻¹ for 2 marks; 1 mark for stroke volume = 104.

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From the examiners' report: Over 90% could do the maths, but only 25% got both marks because over half didn't give 2 significant figures: 133 becomes 130.

Question 5

AS Paper 2 · June 2018 · Q03.2 · 2 marks

Rearrange surface area = 4πr² to find the mean diameter of a frog egg.

Question text: mean surface area of spherical frog eggs is 9.73 mm²; surface area = 4πr² with π = 3.14; calculate the mean diameter, 2 marks.
Show the mark scheme Mark scheme: 1.75 to 1.76 mm for 2 marks; 1 mark for a radius of 0.88 mm or a correct rearrangement, r = √(surface area ÷ 4π).

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From the examiners' report: Most rearranged the equation correctly. Marks were lost by stopping at r², or by forgetting to double the radius to get the diameter.

Question 6

A level Paper 1 · June 2024 · Q03.4 · 2 marks

Use magnification to calculate the mean diameter of alveoli in micrometres.

Table 2: alveolus diameters measured on the image in mm: 4, 2, 5, 1, 2, 3, 5, 2; the magnification is × 40; calculate the mean actual diameter in µm, 2 marks.
Show the mark scheme Mark scheme: 75 µm for 2 marks; 1 mark for actual = image ÷ magnification, ÷ 40, a mean of 3000 µm, × 1000, or 0.075 (mm).

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From the examiners' report: This discriminated well and many got both marks. Rearrange to actual size = image size ÷ magnification, and convert mm to µm by multiplying by 1000.

Substituting into equations (MS 2.3)

Question 7

A level Paper 1 · June 2022 · Q01.4 · 2 marks · Easy

Substitute shoot biomass data into d = 1 − Σ(n/N)² to find an index of diversity.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Table 1: total shoot biomass in g m⁻² of seven plant species: 2, 4, 5, 7, 15, 36 and 51; equation d = 1 − Σ(n/N)², where n is the biomass of each species and N the total.
Show the mark scheme Mark scheme: 0.7 to 0.71 for 2 marks; 1 mark for 0.29 to 0.3 (not subtracted from 1) or a total biomass of 120.

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From the examiners' report: Over 90% got at least 1 mark. Use the equation the question gives you, not the one you learned for the specification, and remember to subtract from 1.

Question 8

A level Paper 2 · June 2025 · Q06.4 · 2 marks · Easy

Substitute into an equation for lean body weight to calculate a drug dose to 3 significant figures.

Question text: atropine dose is 0.01 mg kg⁻¹ of lean body weight; LBW = (9.27 × 10³ × TBW) ÷ ((6.68 × 10³) + (216 × BMI)); TBW = 115.2 kg, BMI = 41; give the dose to 3 significant figures, 2 marks.
Show the mark scheme Mark scheme: 0.687 mg for 2 marks; 1 mark for the right digits with the wrong order of magnitude, the wrong number of significant figures, or adding instead of multiplying by BMI.

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From the examiners' report: About 55% got both marks. The usual single error was an answer ten or a hundred times out, or not giving 3 significant figures. Work out the bottom line first, then multiply by 0.01.

Question 9

AS Paper 2 · June 2023 · Q07.4 · 2 marks · Medium

Use P = F × V to calculate the speed of a chromatid in nm s⁻¹.

Question text: during anaphase the spindle exerts 3 × 10⁻¹¹ N on each chromatid, generating 6 × 10⁻¹⁹ W of power; use P = F × V (V in m s⁻¹) to find the speed in nm s⁻¹, 2 marks.
Show the mark scheme Mark scheme: 20 nm s⁻¹ for 2 marks; 1 mark for 2 × 10⁻⁸ m s⁻¹.

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From the examiners' report: Almost three-quarters scored at least 1 mark. Rearrange to V = P ÷ F, then convert m s⁻¹ to nm s⁻¹ by multiplying by 10⁹.

Question 10

A level Paper 2 · June 2018 · Q03.3 · 2 marks · Medium

Use πr²l to calculate the volume of gas produced by yeast in cm³ per hour.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Figure 3: respirometer with yeast culture in glucose, alkaline pyrogallol to absorb oxygen, and coloured liquid in capillary tubing against a scale. Liquid moved 1.5 cm in 24 hours; lumen diameter 1 mm; volume = πr²l with π = 3.14.
Show the mark scheme Mark scheme: 4.9 × 10⁻⁴ to 4.91 × 10⁻⁴ cm³ hour⁻¹ for 2 marks; 1 mark for dividing by 24, or for 1175 to 1178, or 49 with the decimal point anywhere.

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From the examiners' report: Fewer than a third got both marks, but over 52% got 1 mark for valid working, often for dividing by 24. Show your working, and use the radius, not the diameter.

Question 11

A level Paper 1 · June 2023 · Q05.2 · 2 marks · Medium

Use the volume of a cylinder to calculate the minimum depth of a well cut in agar.

Question text: a well of diameter 6 mm is cut in agar and 50 mm³ of cinnamon oil is added; using volume of a cylinder = πr² × l with π = 3.14, calculate the minimum depth, 2 marks.
Show the mark scheme Mark scheme: 1.768 to 1.8, or 2, mm for 2 marks; 1 mark for 28.26 (π × 3²), for 0.4423 (using the diameter) or for 1.76 (wrong rounding).

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From the examiners' report: 60% got both marks. Use the radius (3 mm), not the diameter, work out πr² first, then rearrange to l = volume ÷ πr².

Question 12

A level Paper 3 · June 2017 · Q04.1 · 2 marks · Hard

Calculate the index of diversity for a grassland site using d = N(N − 1) ÷ Σn(n − 1).

Table 1: numbers of individuals of seven plant species at one site: 3, 19, 3, 3, 10, 16 and 6; d = N(N − 1) ÷ Σn(n − 1).
Show the mark scheme Mark scheme: 4.92 for 2 marks (accept 4.916, 4.917 or 4.9); 1 mark for N(N − 1) = 3540 or Σn(n − 1) = 720.

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From the examiners' report: 58% got it right. N is the total number of individuals of all species (60), not the number of species, and n(n − 1) is worked out for each species before you add them.

Solving equations (MS 2.4)

Question 13

A level Paper 2 · June 2018 · Q10.2 · 2 marks · Medium

Use the Hardy–Weinberg equation to find the percentage of heterozygotes on an island where 10% are affected.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Passage: complete achromatopsia is caused by an autosomal recessive allele; on the Pacific island of Pingelap ten percent of the population are affected; calculate the percentage who are heterozygous, 2 marks.
Show the mark scheme Mark scheme: 42 to 44% for 2 marks; 1 mark for showing 2pq is the heterozygotes or carriers. Accept 1 − (p² + q²).

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From the examiners' report: 70% got at least 1 mark but only 39% got both. The affected people are q² = 0.1, so q = 0.32. Annotate your working; the bare equation scores nothing.

Question 14

A level Paper 3 · June 2025 · Q06.3 · 2 marks

Use the helium dilution equation to calculate lung volume in dm³.

Question text: lung volume V2 = V1(C1 − C2) ÷ C2; spirometer volume 3250 cm³, initial helium 10%, final helium 3.78%; calculate lung volume in dm³, 2 marks.
Show the mark scheme Mark scheme: 5 (5.347883598) dm³ for 2 marks; 1 mark for wrong rounding, an answer in cm³ (5347.88), the wrong order of magnitude, or 20 215 for the numerator.

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From the examiners' report: About 60% could substitute and convert units. The usual lost mark was rounding: divide cm³ by 1000 for dm³ and round correctly.

Question 15

AS Paper 1 · June 2022 · Q04.2 · 3 marks · Medium

Read a dissociation curve, rearrange an equation and calculate the volume of dissolved oxygen in lugworm blood.

Figure 4: oxyhaemoglobin dissociation curve for a lugworm, percentage saturation against pO₂ from 0 to 4.5 kPa, rising steeply near 1 kPa and levelling at about 95%. Blood is 92% saturated; volume 0.2 cm³; pO₂ = CdO₂ ÷ 0.000 031.
Show the mark scheme Mark scheme: 9.3 × 10⁻⁶ cm³ for 3 marks: reading 1.5 kPa from the graph; CdO₂ = 0.000 031 × pO₂; then × 0.2.

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From the examiners' report: Many read 1.5 kPa from the graph and rearranged the equation, but only 13% got all 3. Finish by multiplying the concentration per cm³ by the 0.2 cm³ of blood.

Logarithms (MS 2.5)

Question 16

A level Paper 2 · June 2018 · Q03.4 · 1 mark · Hard

Explain why a log scale is used for the number of yeast cells in a population growth curve.

Figure 4: typical population growth curve for yeast, log₁₀ number of cells against time, with a lag phase, a steep rise, a plateau and a death phase.
Show the mark scheme Mark scheme: large range, difference or increase in numbers; accept exponential increase. Ignore 'to fit on the scale' or numbers being high.

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From the examiners' report: Only about 31% scored. Say the numbers cover a large range or increase exponentially; 'to fit on the scale', 'numbers are large' and 'more accurate' don't score.

Question 17

A level Paper 3 · June 2017 · Q02.4 · 2 marks · Hard

Use a log scale graph to explain why only some bacterial samples needed diluting before counting.

Information from earlier in this question:

Earlier part of the same exam question, giving the information this part uses. Figure 3: log₁₀ of the mean number of live pathogenic bacteria per cm³ in mice not treated (mean about 8, range 6.6 to 8.8) and treated with bacteriophage (mean about 5.1, range 0.6 to 5.4).
Show the mark scheme Mark scheme: the log scale shows a big range in the number of bacteria; some samples have too many to count, so dilute, but some are countable. Simple statements of log values aren't enough.

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From the examiners' report: Few students understood log scales. Answer from the graph: the numbers differ by orders of magnitude, so some samples had too many bacteria to count and some few enough. About 20% made this point.

Where students lose marks (from AQA examiners' reports and mark schemes)

Each point below is followed by wording that gained marks.

  • Subtract before you divide: About 80% got the loop of Henle length from y = 0.72x + 4. Those who didn't rounded wrongly or divided first and then subtracted, giving 19.

    ✔ Gains credit: "x = (16.56 − 4) ÷ 0.72 = 17.4 mm."

  • Keep going until the subject is r: On the frog egg question some students left the subject of the equation as r², or found the radius and forgot to double it for the diameter.

    ✔ Gains credit: "r = √(9.73 ÷ 12.56) = 0.88 mm, so the diameter is 1.76 mm."

  • Halve the diameter first: Only 30% got both marks for the mitochondria in a polar body; some used diameters instead of radii. In the agar well question, using the 6 mm diameter instead of the radius also cost a mark.

    ✔ Gains credit: "r = 3 mm, so πr² = 28.26 mm²; depth = 50 ÷ 28.26 = 1.77 mm."

  • N counts individuals, not species: 58% got the index of diversity right and 20% scored 1. The commonest error was taking N as the number of species.

    ✔ Gains credit: "N(N − 1) = 3540, Σn(n − 1) = 720, so d = 4.92."

  • Ten per cent affected is q², not q: Only 39% got both marks for the achromatopsia carriers. Many used 0.1 as q, giving 2 × 0.9 × 0.1; others wrote the Hardy–Weinberg equation with no annotation, or used pq for heterozygotes.

    ✔ Gains credit: "q² = 0.1, so q = 0.32 and p = 0.68; 2pq = 0.43, so 43% are heterozygous."

  • Round to the figures the question sets: Over 90% could do the heart rate calculation, but only 25% got full marks: in over half the answers students didn't give 2 significant figures.

    ✔ Gains credit: "Stroke volume = 80 × 1.3 = 104; 13 832 ÷ 104 = 133, so 130 beats minute⁻¹."

  • Start from the initial mass: Students left out the × 100, subtracted the mass lost instead of the final mass, or calculated the percentage of cellulose remaining; 10.7% didn't attempt it.

    ✔ Gains credit: "(Initial mass − final mass) ÷ initial mass × 100."

  • A log scale covers a large range: Only about 31% explained why a log scale is used for yeast numbers. 'To fit on the scale', 'the numbers are large', 'more accurate' and 'more comparable' didn't score.

    ✔ Gains credit: "There is a large range (increase) in the number of cells", or "an exponential increase".

Download the questions

Download the PDF pack (17 questions, mark schemes at the end)

Keep practising

Questions, mark schemes and examiners' report extracts © AQA.

Tom Whitburn
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