Nobody fails Chemistry because the maths was too hard. People lose marks because they left an answer as a fraction, averaged in a rough titre, or called a straight line “directly proportional” when it missed the origin. This page is about those marks.
📚 What you need to know
You need decimals, fractions, percentages, ratios, reciprocals, logs and (HL) exponentials.
Final answers must be decimals, not fractions.
The mean is the total divided by the number of values — and anomalies are left out.
For titrations, never include the rough titre, and only average concordant results.
Range = highest − lowest.
Scientific notation is a × 10n, where a is between 1 and 10.
Percentage change, percentage difference and percentage error are three different formulas. Learn which is which.
Directly proportional means a straight line through the origin. Nothing else counts.
The kinds of maths you will actually meet
Type
Where it shows up
Watch out for
Decimals
Nearly every calculation
Rounding too early
Fractions
Uncertainty work
Leaving the answer as a fraction
Percentages
Yield, atom economy, error, uncertainty
Using the wrong formula
Ratios
Moles, when stoichiometry is not 1:1
Reading the equation coefficients wrongly
Reciprocals
Gas laws (1/V), rate graphs (1/T)
Forgetting to invert the units too
Logarithms
pH, pKa, the Arrhenius equation
Mixing up log and ln
Exponentials (HL)
The Arrhenius equation
Calculator in the wrong mode
Section 1 of the data booklet gives you a list of equations, but it is not everything. Percentage yield, percentage error and percentage uncertainty are all expected knowledge and none of them is printed for you.
The mean average
Add all the values, divide by how many there are. The complication in chemistry is deciding which values to include.
An anomalous result is one that clearly does not fit with the others. It gets excluded from the mean, because leaving it in would drag the answer towards a measurement you already believe is wrong.
Averaging titres
Titrations have their own rule, and it is worth knowing exactly.
The rough titre is never included. It is a scouting run, done fast, to find roughly where the end point is.
Only concordant titres are averaged. Concordant means within 0.10 cm3 of each other.
Anything outside that window is treated as anomalous and dropped.
The two green circles stacked at the same place are two identical titres of 23.85 cm3. Repeats landing on the same value is exactly what you want to see.
Range
The range measures how spread out a set of numbers is. Highest minus lowest — remember it as “hi − lo”.
You can quote it either as a span (9.2 to 8.4) or as a single number (0.8). Be careful though: if the highest or lowest value is itself anomalous, the range is misleading.
Scientific notation
Also called standard form. It is a way of writing numbers that are far too big or far too small to write out comfortably.
Standard forma × 10n where 1 ≤ a < 10
Large numbers give a positive power. It counts how many times a is multiplied by 10.
Small numbers give a negative power. It counts how many times a is divided by 10.
The power of ten is never rounded. It is not a measurement, it is a place-value marker.
Orders of magnitude
An order of magnitude is just the power of ten. Something one order of magnitude larger is about ten times larger; two orders of magnitude is about a hundred times.
One catch: orders of magnitude follow rounding rules. The order of magnitude of 3 × 108 is 108, but the order of magnitude of 6 × 108 is 109, because 6 rounds up to 10.
Approximation, estimation and useful assumptions
These sound like sloppiness. They are the opposite — they are decisions about what you can safely ignore.
An approximation is a value close to the real one, used when a precise figure is not needed. Saying “the pH of a strong acid is about 1” is an approximation.
An estimation is a reasoned guess when the true value cannot be measured directly.
The weak acid assumption
Here is the assumption you meet most often. For a weak acid HA:
The exact expressionKa = [H+][A–] ÷ [HA]
Because a weak acid barely dissociates, two things are close enough to true:
[H+] = [A–], since almost every H+ came from the acid rather than from water.
[HA] at equilibrium is roughly the same as the concentration you started with, since so little has reacted.
The simplified expressionKa = [H+]2 ÷ [HA]
The assumption is not equally safe for every acid. The bigger the Ka, the more the acid dissociates and the worse the approximation gets. If a question asks you to comment on validity, that is the point being tested.
The three percentage formulas
These get muddled constantly, so learn what each one compares.
Formula
What it compares
Calculation
Percentage change
A value before and after
(final − initial) ÷ initial × 100
Percentage difference
Two measurements of equal standing
(value 1 − value 2) ÷ mean of the two × 100
Percentage error
Your result against a literature value
(accepted − experimental) ÷ accepted × 100
Percentage uncertainty
Instrument uncertainty against the reading
uncertainty ÷ measured value × 100
The giveaway is the denominator. Percentage change divides by the starting value. Percentage difference divides by the average, because neither value is more trustworthy. Percentage error divides by the accepted value, because that one is assumed right.
Describing data and trends
Qualitative and quantitative
Qualitative data is described in words. “The copper sulfate solution is pale blue.”
Quantitative data uses numbers. “The temperature rose by 2.5°C.”
Discrete quantitative data can only take separate, countable values — stoichiometric coefficients must be whole numbers.
Continuous quantitative data can take any value in a range — temperature, volume, time.
Proportionality, and the trap in it
An inversely proportional curve never touches either axis. It gets closer and closer but never arrives, because you cannot divide by zero.
The trap. A straight line sloping upwards is not automatically directly proportional. If it crosses the y-axis anywhere other than zero, it is proportional but not directly proportional. Examiners test this every year.
Correlation
Positive correlation means as x increases, y increases: the graph slopes upwards. Negative correlation means as x increases, y decreases. Correlation describes the direction of the trend, not its shape, so it applies to curves as well as straight lines.
Rate of change from a table
You do not always need to plot a graph. The average rate of change between two rows of a table is the change in the dependent variable divided by the change in the independent variable.
Average rate of change
change in y ÷ change in x
Worked examples
WORKED EXAMPLE
A titration gives a rough titre of 24.60 cm3, then titres of 23.85, 24.10, 23.90 and 23.85 cm3. Calculate the mean titre.
Throw out the rough titre
It is a scouting run, so 24.60 is never used.
Test the rest for concordance
23.85, 23.90 and 23.85 all sit within 0.05 cm3 of each other. 24.10 is 0.25 away, so it is anomalous.
Average the three that agree(23.85 + 23.90 + 23.85) ÷ 3 = 71.60 ÷ 3 = 23.8666…Mean titre = 23.87 cm3quote it to 2 decimal places, matching the burette readings
WORKED EXAMPLE
An experiment gives the enthalpy of combustion of propan-1-ol as −1.50 × 103 kJ mol–1. The literature value is −2021 kJ mol–1. Calculate the percentage error.
Pick the right formula
We are comparing against a literature value, so this is percentage error, and we divide by the accepted value.
Substitute(2021 − 1500) ÷ 2021 × 100Work it through521 ÷ 2021 × 100 = 25.78…Percentage error = 26%a large error like this usually means heat loss to the surroundings
WORKED EXAMPLE
The volume of gas produced is 3.0 cm3 at 10 s and 10.0 cm3 at 30 s. Calculate the average rate of change over that interval.
Identify the variables
Volume is dependent (y), time is independent (x).
Find both changesΔV = 10.0 − 3.0 = 7.0 cm3Δt = 30 − 10 = 20 sDivide7.0 ÷ 20 = 0.35Rate = 0.35 cm3 s–1this is an average over the interval, not the rate at any single instant
💡 Exam tip
Convert fractions to decimals before writing your final answer. Know where the S↔D button is.
Keep the full calculator value all the way through and round only at the end.
Say which values you excluded from a mean and why. That reasoning is often a mark.
“Directly proportional” needs a straight line through the origin. Otherwise just say “proportional”.
Check the denominator before using any percentage formula.
Section 1 of the data booklet is not a complete list. Learn percentage yield, error and uncertainty separately.
⚠️ Common mix-up
Including the rough titre in the mean. It is never used, no matter how close it looks.
Averaging non-concordant results. Only titres within 0.10 cm3 of each other go in.
Percentage error and percentage uncertainty. Error compares to a literature value; uncertainty comes from the equipment.
Rounding the power of ten. Only a gets rounded in standard form.
Calling any upward line directly proportional. Check the intercept first.
Rounding to 1 significant figure mid-calculation. It introduces errors and can lose marks automatically.
Up next: Units, Symbols and Numerical Values — the SI system, the prefixes, and how many digits you are actually allowed to write down.
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