Maths Skills in Biology

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.

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Units and SI Prefixes

Biology works across an enormous range of sizes, so you must move confidently between units.

Prefix	Symbol	Meaning	Example
kilo	k	$\times 10^{3}$×103	$1 \text{ kJ} = 1000 \text{ J}$1 kJ=1000 J
(base)	—	$\times 1$×1	metre, gram, second
milli	m	$\times 10^{-3}$×10−3	$1 \text{ mm} = 0.001 \text{ m}$1 mm=0.001 m
micro	µ	$\times 10^{-6}$×10−6	$1 \text{ }\mu\text{m} = 0.001 \text{ mm}$1 μm=0.001 mm
nano	n	$\times 10^{-9}$×10−9	$1 \text{ nm} = 0.001 \text{ }\mu\text{m}$1 nm=0.001 μm

The length chain you use most: $1 \text{ m} = 1000 \text{ mm}$1 m=1000 mm, $1 \text{ mm} = 1000 \text{ }\mu\text{m}$1 mm=1000 μm, $1 \text{ }\mu\text{m} = 1000 \text{ nm}$1 μm=1000 nm. Going to smaller units, multiply; going to larger units, divide — each step is $\times$× or $\div 1000$÷1000.

Worked example: Convert $0.05 \text{ mm}$0.05 mm to micrometres.

$0.05 \text{ mm} \times 1000 = 50 \text{ }\mu\text{m}$0.05 mm×1000=50 μm

Exam Tip: A converting-direction error is the classic maths slip. A $0.5 \text{ mm}$0.5 mm bacterium is $500 \text{ }\mu\text{m}$500 μm wide (multiply), not $0.0005$0.0005. When in doubt, sanity-check: micrometres are smaller, so there should be more of them.

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Standard Form

Standard form writes very large or very small numbers as $A \times 10^{n}$A×10n where $1 \le A < 10$1≤A<10.

• $150\,000 = 1.5 \times 10^{5}$150000=1.5×105
• $0.00032 = 3.2 \times 10^{-4}$0.00032=3.2×10−4
• A red blood cell $\approx 7 \text{ }\mu\text{m} = 7 \times 10^{-6} \text{ m}$≈7 μm=7×10−6 m

Multiplying: multiply the $A$A values, add the powers.

$(3 \times 10^{4}) \times (2 \times 10^{3}) = 6 \times 10^{7}$(3×104)×(2×103)=6×107

Dividing: divide the $A$A values, subtract the powers.

$(8 \times 10^{6}) \div (4 \times 10^{2}) = 2 \times 10^{4}$(8×106)÷(4×102)=2×104

Exam Tip: Bacterial-growth questions love standard form. After 10 divisions a single cell becomes $2^{10} = 1024 \approx 1.02 \times 10^{3}$210=1024≈1.02×103 cells — practise expressing such answers in standard form to the requested significant figures.

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Worked example (large number in standard form): A culture contains $4\,500\,000$4500000 bacteria. Write this in standard form.

Move the decimal point so the first part is between 1 and 10: $4\,500\,000 = 4.5 \times 10^{6}$4500000=4.5×106 (the point moved 6 places to the left).

Worked example (small number in standard form): A ribosome is about $0.000\,000\,025 \text{ m}$0.000000025 m across. Write this in standard form.

$0.000\,000\,025 = 2.5 \times 10^{-8} \text{ m}$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.

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Decimals and Significant Figures

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:

• $0.00452$0.00452 to 2 s.f. $= 0.0045$=0.0045
• $3476$3476 to 2 s.f. $= 3500$=3500
• $1.256$1.256 to 3 s.f. $= 1.26$=1.26

Decimal places — count digits after the point:

• $3.456$3.456 to 1 d.p. $= 3.5$=3.5
• $0.0783$0.0783 to 2 d.p. $= 0.08$=0.08

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.

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Percentages

A percentage of a total:

$\text{percentage} = \frac{\text{part}}{\text{total}} \times 100$percentage=totalpart​×100

Worked example: Of 250 students, 45 have brown eyes. What percentage is this?

$\frac{45}{250} \times 100 = 18\%$25045​×100=18%

A percentage of an amount:

Worked example: 60% of 1500 bacteria survive treatment. How many survive?

$\frac{60}{100} \times 1500 = 900 \text{ bacteria}$10060​×1500=900 bacteria

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Percentage Change

One of the most frequently examined calculations — used heavily in the osmosis practical.

$\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100$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 \text{ g}$2.5 g becomes $2.8 \text{ g}$2.8 g. Find the percentage change.

Change $= 2.8 - 2.5 = 0.3 \text{ g}$=2.8−2.5=0.3 g, so:

$\frac{0.3}{2.5} \times 100 = +12\%$2.50.3​×100=+12%

Worked example (decrease): A population of $800$800 falls to $600$600.

Change $= 800 - 600 = 200$=800−600=200, so:

$\frac{200}{800} \times 100 = -25\%$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.

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Ratios

A ratio compares quantities. Simplify by dividing both sides by their highest common factor.

Worked example: In a sample there are $15$15 tall plants and $5$5 short plants. Express this as a ratio.

$15 : 5 = 3 : 1 \quad (\text{dividing both by } 5)$15:5=3:1(dividing both by 5)

Ratios convert to fractions and percentages: a $3:1$3:1 ratio means $\frac{3}{4}$43​ (75%) show one trait and $\frac{1}{4}$41​ (25%) the other.

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Magnification

From the microscope practical; appears on either paper.

$\text{magnification} = \frac{\text{image size}}{\text{actual size}}$magnification=actual sizeimage size​

Rearranged:

$\text{actual size} = \frac{\text{image size}}{\text{magnification}} \qquad \text{image size} = \text{magnification} \times \text{actual size}$actual size=magnificationimage size​image size=magnification×actual size

Worked example — finding actual size: An image of a cell is $45 \text{ mm}$45 mm long at a magnification of $\times 500$×500. Find the actual size.

$\text{actual size} = \frac{45 \text{ mm}}{500} = 0.09 \text{ mm} = 90 \text{ }\mu\text{m}$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.

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Surface-Area-to-Volume Ratio

This explains why small organisms exchange substances by diffusion but large ones need specialised exchange and transport systems.

Cube side	Surface area $(6 \times \text{side}^2)$(6×side2)	Volume $(\text{side}^3)$(side3)	SA:V
$1 \text{ cm}$1 cm	$6 \text{ cm}^2$6 cm2	$1 \text{ cm}^3$1 cm3	$6:1$6:1
$2 \text{ cm}$2 cm	$24 \text{ cm}^2$24 cm2	$8 \text{ cm}^3$8 cm3	$3:1$3:1
$3 \text{ cm}$3 cm	$54 \text{ cm}^2$54 cm2	$27 \text{ cm}^3$27 cm3	$2:1$2:1

Worked example: Find the SA:V ratio of a cube of side $2 \text{ cm}$2 cm.

Surface area $= 6 \times 2^{2} = 24 \text{ cm}^2$=6×22=24 cm2; volume $= 2^{3} = 8 \text{ cm}^3$=23=8 cm3.

$\text{SA:V} = 24 : 8 = 3 : 1$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.

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Rate Calculations

Rate questions appear in enzyme, photosynthesis and respiration contexts.

$\text{rate} = \frac{\text{change in quantity}}{\text{time}} \qquad \text{or} \qquad \text{rate} = \frac{1}{\text{time}}$rate=timechange in quantity​orrate=time1​

Worked example: $30 \text{ cm}^3$30 cm3 of gas is collected in $5 \text{ minutes}$5 minutes. Find the rate.

$\text{rate} = \frac{30 \text{ cm}^3}{5 \text{ min}} = 6 \text{ cm}^3/\text{min}$rate=5 min30 cm3​=6 cm3/min

Worked example ($\frac{1}{\text{time}}$time1​): Amylase digests starch in $40 \text{ s}$40 s. Find the rate.

$\text{rate} = \frac{1}{40} = 0.025 \text{ s}^{-1}$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.

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Probability in Genetics

Genetic crosses give predictable ratios, which convert to probabilities.

Worked example: Cross two heterozygous brown-eyed parents, $Bb \times Bb$Bb×Bb ($B$B = brown, dominant; $b$b = blue).

	$B$B	$b$b
$B$B	$BB$BB	$Bb$Bb
$b$b	$Bb$Bb	$bb$bb

Brown $(BB + Bb + Bb) = 3$(BB+Bb+Bb)=3; blue $(bb) = 1$(bb)=1, giving a $3:1$3:1 ratio.

$P(\text{blue-eyed child}) = \frac{1}{4} = 0.25 = 25\%$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 $\frac{1}{4}$41​ each time.

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Rearranging the Magnification Formula