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At least 10% of the total marks in OCR Gateway Biology require mathematical skills — roughly 9 marks per paper, or around 18 marks across the qualification. These are some of the most reliable marks available because each calculation has a fixed method and a definite answer. Yet students lose them constantly by not showing working, forgetting units, or making conversion errors. This lesson works through every type of maths the specification expects, each with a worked example.
By the end of this lesson you should be able to handle units and prefixes, standard form, decimals and significant figures, percentages and percentage change, ratios, magnification, surface-area-to-volume ratio, rate calculations and probability in genetics.
Exam Tip: A scientific calculator is permitted on both papers. Practise with your calculator before the exam — especially the standard-form button (often labelled EXP, EE or ×10ˣ). Fumbling the calculator in the exam costs time and accuracy.
Biology works across an enormous range of sizes, so you must move confidently between units.
| Prefix | Symbol | Meaning | Example |
|---|---|---|---|
| kilo | k | ×103 | 1 kJ=1000 J |
| (base) | — | ×1 | metre, gram, second |
| milli | m | ×10−3 | 1 mm=0.001 m |
| micro | µ | ×10−6 | 1 μm=0.001 mm |
| nano | n | ×10−9 | 1 nm=0.001 μm |
The length chain you use most: 1 m=1000 mm, 1 mm=1000 μm, 1 μm=1000 nm. Going to smaller units, multiply; going to larger units, divide — each step is × or ÷1000.
Worked example: Convert 0.05 mm to micrometres.
0.05 mm×1000=50 μm
Exam Tip: A converting-direction error is the classic maths slip. A 0.5 mm bacterium is 500 μm wide (multiply), not 0.0005. When in doubt, sanity-check: micrometres are smaller, so there should be more of them.
Standard form writes very large or very small numbers as A×10n where 1≤A<10.
Multiplying: multiply the A values, add the powers.
(3×104)×(2×103)=6×107
Dividing: divide the A values, subtract the powers.
(8×106)÷(4×102)=2×104
Exam Tip: Bacterial-growth questions love standard form. After 10 divisions a single cell becomes 210=1024≈1.02×103 cells — practise expressing such answers in standard form to the requested significant figures.
Worked example (large number in standard form): A culture contains 4500000 bacteria. Write this in standard form.
Move the decimal point so the first part is between 1 and 10: 4500000=4.5×106 (the point moved 6 places to the left).
Worked example (small number in standard form): A ribosome is about 0.000000025 m across. Write this in standard form.
0.000000025=2.5×10−8 m
(The point moved 8 places to the right, so the power is negative.)
Exam Tip: A positive power means a large number (point moves left); a negative power means a small number (point moves right). Counting the places carefully is the whole skill — get the direction of the sign right and standard-form marks are reliable.
Questions often tell you how many significant figures (s.f.) or decimal places (d.p.) to give.
Significant figures — count from the first non-zero digit:
Decimal places — count digits after the point:
If the question does not specify, give your answer to the same number of significant figures as the data.
Exam Tip: Getting the calculation right but rounding wrongly still loses the final mark. Do the whole calculation first, then round once at the end — rounding part-way through introduces errors.
A percentage of a total:
percentage=totalpart×100
Worked example: Of 250 students, 45 have brown eyes. What percentage is this?
25045×100=18%
A percentage of an amount:
Worked example: 60% of 1500 bacteria survive treatment. How many survive?
10060×1500=900 bacteria
One of the most frequently examined calculations — used heavily in the osmosis practical.
percentage change=originalchange×100
The denominator is always the original value, not the final value.
Worked example (increase): A potato cylinder of mass 2.5 g becomes 2.8 g. Find the percentage change.
Change =2.8−2.5=0.3 g, so:
2.50.3×100=+12%
Worked example (decrease): A population of 800 falls to 600.
Change =800−600=200, so:
800200×100=−25%
Exam Tip: State whether the change is positive (increase) or negative (decrease), and always divide by the original value — dividing by the final value is the single most common percentage-change error.
A ratio compares quantities. Simplify by dividing both sides by their highest common factor.
Worked example: In a sample there are 15 tall plants and 5 short plants. Express this as a ratio.
15:5=3:1(dividing both by 5)
Ratios convert to fractions and percentages: a 3:1 ratio means 43 (75%) show one trait and 41 (25%) the other.
From the microscope practical; appears on either paper.
magnification=actual sizeimage size
Rearranged:
actual size=magnificationimage sizeimage size=magnification×actual size
Worked example — finding actual size: An image of a cell is 45 mm long at a magnification of ×500. Find the actual size.
actual size=50045 mm=0.09 mm=90 μm
Exam Tip: Make both sizes the same unit before dividing, and remember magnification has no units (it is a ratio). A handy memory aid is the formula triangle with image size on top, and magnification and actual size below.
This explains why small organisms exchange substances by diffusion but large ones need specialised exchange and transport systems.
| Cube side | Surface area (6×side2) | Volume (side3) | SA:V |
|---|---|---|---|
| 1 cm | 6 cm2 | 1 cm3 | 6:1 |
| 2 cm | 24 cm2 | 8 cm3 | 3:1 |
| 3 cm | 54 cm2 | 27 cm3 | 2:1 |
Worked example: Find the SA:V ratio of a cube of side 2 cm.
Surface area =6×22=24 cm2; volume =23=8 cm3.
SA:V=24:8=3:1
Key principle: as an organism gets larger its SA:V decreases, making simple diffusion insufficient and explaining adaptations such as alveoli, villi and gills.
Rate questions appear in enzyme, photosynthesis and respiration contexts.
rate=timechange in quantityorrate=time1
Worked example: 30 cm3 of gas is collected in 5 minutes. Find the rate.
rate=5 min30 cm3=6 cm3/min
Worked example (time1): Amylase digests starch in 40 s. Find the rate.
rate=401=0.025 s−1
Exam Tip: Always give rate with units — the unit follows the measurement (cm³/min, s⁻¹, bubbles per minute). A correct number with no unit usually loses the final mark.
Genetic crosses give predictable ratios, which convert to probabilities.
Worked example: Cross two heterozygous brown-eyed parents, Bb×Bb (B = brown, dominant; b = blue).
| B | b | |
|---|---|---|
| B | BB | Bb |
| b | Bb | bb |
Brown (BB+Bb+Bb)=3; blue (bb)=1, giving a 3:1 ratio.
P(blue-eyed child)=41=0.25=25%
Exam Tip: You may be asked for the answer as a ratio, a fraction, a decimal or a percentage — be ready to convert between all four. The probability for each child is independent: two brown-eyed parents can still have a blue-eyed child with probability 41 each time.
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