Mathematical Skills for Combined Science

At least 10% of the marks across the six OCR Gateway Combined Science A papers require mathematical skills — calculations, unit work, standard form, significant figures, percentages, ratios and graph work. That is a substantial block of marks, and they are among the most reliable in the whole qualification because the method is fixed: there is a right answer and a clear route to it. Students who treat science as a "no-maths" subject routinely give these marks away. This lesson drills the core maths techniques, with worked examples from all three sciences, so that the calculation questions become automatic.

By the end of this lesson you should be able to rearrange equations, convert units, use standard form and significant figures, and handle percentages and ratios — showing full working every time.

These maths skills are almost entirely AO2 (applying a fixed method to reach a number), and they secure the guaranteed 10%-plus of maths marks spread across the six papers.

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The Universal Three-Step Method

Almost every calculation, in any science, follows the same three steps:

1. Write the equation (from memory, or select it from the physics equations sheet).
2. Substitute the values — after converting units so they match.
3. Calculate and give the answer with the correct unit and sensible significant figures.

Following this structure earns method marks even if your final number is wrong. Only ever writing the final answer risks scoring zero if that answer is incorrect.

Exam Tip: Lay your working out on separate lines — equation, then substitution, then answer. It is easier for the examiner to award method marks, and easier for you to spot your own slips.

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Rearranging Equations

Many students lose marks not on the arithmetic but on the rearrangement. Take the density relationship:

$\rho = \frac{m}{V}$ρ=Vm​

To make each quantity the subject:

$m = \rho \times V \qquad V = \frac{m}{\rho}$m=ρ×VV=ρm​

The rule is simple: whatever you do to one side, do to the other. If a quantity is dividing, multiply both sides by it; if it is multiplying, divide both sides by it.

Worked example (physics). A block has density 2700 kg/m³ and volume 0.0025 m³. Calculate its mass.

$m = \rho \times V = 2700 \times 0.0025 = 6.75 \text{ kg}$m=ρ×V=2700×0.0025=6.75 kg

Worked example (chemistry). Rearrange $n = \dfrac{m}{M_r}$n=Mr​m​ to find the mass of 0.25 mol of CaCO₃ (Mr = 100).

$m = n \times M_r = 0.25 \times 100 = 25 \text{ g}$m=n×Mr​=0.25×100=25 g

Exam Tip: If rearranging under pressure worries you, substitute first, then rearrange the numbers. For $F = ma$F=ma with $F = 12$F=12 and $m = 3$m=3: write $12 = 3 \times a$12=3×a, then $a = 12 \div 3 = 4$a=12÷3=4. Working with numbers is often easier than working with symbols.

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Unit Conversions

Questions often give values in non-standard units, and you must convert before substituting. This is one of the most common reasons calculation marks are lost.

Conversion	How
cm → m	÷ 100
mm → m	÷ 1000
km → m	× 1000
g → kg	÷ 1000
mg → g	÷ 1000
mA → A	÷ 1000
kJ → J	× 1000
cm³ → dm³	÷ 1000
minutes → seconds	× 60
µm → mm	÷ 1000

Worked example (biology). A cell measures 60 µm across. Express this in mm and in m.

$60\,\mu\text{m} = 60 \div 1000 = 0.060 \text{ mm} = 0.060 \div 1000 = 6.0 \times 10^{-5} \text{ m}$60μm=60÷1000=0.060 mm=0.060÷1000=6.0×10−5 m

Exam Tip: Write the required unit next to each value before you substitute. If the equation needs metres and the question gives centimetres, convert first — a magnification or density answer out by a factor of 100 or 1000 is almost always an un-converted unit.

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Standard Form

Standard form writes very large or very small numbers as a value between 1 and 10 multiplied by a power of 10. It appears in all three sciences.

Number	Standard form	Context
6700	$6.7 \times 10^{3}$6.7×103	general
0.0045	$4.5 \times 10^{-3}$4.5×10−3	general
300 000 000	$3.0 \times 10^{8}$3.0×108	speed of light (m/s), physics
0.000 001	$1.0 \times 10^{-6}$1.0×10−6	1 micrometre (m), biology cell sizes
602 000 000 000 000 000 000 000	$6.02 \times 10^{23}$6.02×1023	Avogadro constant, chemistry

How to convert: move the decimal point until you have a number between 1 and 10; count the moves. Moving left gives a positive power; moving right gives a negative power.

Worked example. Write 0.000 82 in standard form. Move the point 4 places right: $8.2 \times 10^{-4}$8.2×10−4.

Exam Tip: Learn to enter standard form on your calculator using the EXP or ×10ˣ key, not by typing "× 10 ^". Mis-keying powers of ten is a frequent silent error, especially with negative indices.

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Significant Figures

If a question does not state how many significant figures to use, give your answer to 3 significant figures (or match the least precise data in the question).

Number	Significant figures	To 3 s.f.
0.004562	4	0.00456
1234	4	1230
45.678	5	45.7
0.10500	5	0.105

Exam Tip: Only round at the final step. If you round intermediate values, small errors accumulate and your final answer can drift outside the accepted range. Keep full precision on your calculator until the end.

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Percentages and Percentage Change

Percentages appear in biology (osmosis, populations), chemistry (percentage yield, atom economy) and physics (efficiency).

$\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100\%$percentage change=originalchange​×100%

Worked example (biology — osmosis). A potato chip starts at 2.50 g and ends at 2.80 g. Find the percentage change in mass.

$\text{change} = 2.80 - 2.50 = 0.30 \text{ g}$change=2.80−2.50=0.30 g

$\text{percentage change} = \frac{0.30}{2.50} \times 100\% = 12\%$percentage change=2.500.30​×100%=12%

Worked example (physics — efficiency). A motor is supplied with 500 J and usefully transfers 350 J. Find its efficiency.

$\text{efficiency} = \frac{\text{useful output}}{\text{total input}} \times 100\% = \frac{350}{500} \times 100\% = 70\%$efficiency=total inputuseful output​×100%=500350​×100%=70%

Exam Tip: Efficiency and percentage yield can never exceed 100%. If you calculate more than 100%, you have almost certainly divided the wrong way round or mixed up the input and output — go back and check.

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Ratios

Ratios appear most in biology (genetic crosses) and chemistry (mole ratios from balanced equations).

Worked example (biology — genetic cross). A cross predicts a 3 : 1 ratio of tall to short plants. There are 240 offspring. How many of each are expected?

$\text{tall} = \frac{3}{4} \times 240 = 180 \qquad \text{short} = \frac{1}{4} \times 240 = 60$tall=43​×240=180short=41​×240=60

Worked example (chemistry — mole ratio). In $H_2SO_4 + 2NaOH \rightarrow Na_2SO_4 + 2H_2O$H2​SO4​+2NaOH→Na2​SO4​+2H2​O, how many moles of NaOH react with 0.15 mol of sulfuric acid? The ratio is 1 : 2, so:

$n(\text{NaOH}) = 2 \times 0.15 = 0.30 \text{ mol}$n(NaOH)=2×0.15=0.30 mol

Exam Tip: To simplify a ratio, divide both parts by their highest common factor. To scale a ratio to a total, add the parts (3 + 1 = 4), then take the correct fraction of the total. Getting the "add the parts" step wrong is the classic ratio slip.

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A Full Cross-Science Worked Calculation