The big idea: Paper 1B and your IA constantly ask you to process data — to turn the numbers you read off a graph or table into a meaningful result.
Almost every one of these is built from just four moves:
Difference — how much bigger is one value than another? (just subtract)
Percentage change — how big is that change compared to where it started?
Ratio / index — what fraction of the whole is one part?
Rate — how fast is something changing? This is the gradient (steepness) of a graph.
Learn the four formulas and you can handle nearly any 'Calculate…' question.
- Difference
- How far apart two values are: the larger value minus the smaller value.
- Percentage change
- The change in a value expressed as a percentage of the ORIGINAL value: (new − old) ÷ old × 100.
- Ratio
- A comparison of one quantity with another, written A : B, or as a fraction/percentage of the whole.
- Range
- The spread of a set of data: the largest value minus the smallest value.
- Rate
- How much a quantity changes per unit of time; on a graph it is the gradient of the line.
- Gradient
- The steepness of a line: the change up the y-axis (rise) divided by the change along the x-axis (run).
Percentage change vs a percentage OF: These two are easy to mix up:
Percentage change asks how much something went up or down: divide the change by the old value, .
Percentage of asks what share one part is of a total: divide the part by the whole, e.g. 54 chips supply 1.2 g of a 6 g daily limit → .
Read the question: 'change/increase/decrease' → percentage change; 'what percentage of' → percentage of the whole.
Here is each method with its formula and a fully worked number — follow the units all the way through.
1 · Difference and percentage change: A pondweed releases 38 cm³ of oxygen in an hour at low light and 50 cm³ at high light.
Difference = larger − smaller = .
Percentage change uses the ORIGINAL (low-light) value as the base:
(a 31.6 % increase).
If instead it fell from 50 to 38, the change is — a 24 % decrease. Notice the base value is different, so the percentage is different too.
2 · Ratio and index: On a root-tip slide you count 80 cells, of which 12 are dividing (in mitosis).
The mitotic index is the fraction of cells that are dividing:
(or 15 %).
A ratio can also be written A : B — e.g. 12 dividing to 68 not-dividing is , which simplifies to about . Always say which two things you are comparing.
3 · Rate = the gradient of a graph: A rate is how fast something changes — per second, per minute, per day. On a graph of amount against time, the rate is the gradient (the steepness) of the line:
.
Read TWO points off the straight part of the line, find the rise (change up the y-axis) and the run (change in time), then divide — and write the units.
A steeper line means a faster rate; a line that levels off (gradient → 0) means the rate has slowed to zero.
| Step | What to do | On our example |
|---|---|---|
| 1 | Pick two clear points on the straight part of the line | (10 s, 20 cm³) and (40 s, 80 cm³) |
| 2 | Find the rise = change on the y-axis | 80 − 20 = 60 cm³ |
| 3 | Find the run = change on the x-axis (the time) | 40 − 10 = 30 s |
| 4 | rate = rise ÷ run, and write the UNITS | 60 ÷ 30 = 2 cm³ s⁻¹ |
Worked rate — oxygen from photosynthesis: Oxygen collected from pondweed reads 20 cm³ at 10 s and 80 cm³ at 40 s on a straight line.
Rise = . Run = .
.
Same idea for breathing: 30 breaths in 2 minutes gives .
| What you're asked | Formula | Worked mini-example |
|---|---|---|
| Difference / change | difference = larger − smaller | From 38 to 50 → 50 − 38 = 12 |
| Percentage change | % change = (new − old) ÷ old × 100 | 38 → 50 → (50−38)/38 ×100 = +31.6 % |
| Percentage decrease | % decrease = (old − new) ÷ old × 100 | 50 → 38 → (50−38)/50 ×100 = 24 % fall |
| Ratio / index | ratio = part ÷ whole (or A : B) | 12 dividing of 80 cells → 12 ÷ 80 = 0.15 |
| Range | range = largest − smallest | Values 9–27 °C → 27 − 9 = 18 °C |
| Rate (from a graph) | rate = change in quantity ÷ time = gradient | 60 cm³ in 30 s → 60 ÷ 30 = 2 cm³ s⁻¹ |
Watch the base and the units: Two marks are lost again and again:
Dividing the change by the new value instead of the old one (percentage change is always measured against where you started).
Quoting a rate without units — a rate is meaningless until you write per second / per minute.
| Slip-up | Why it's wrong | Do this instead |
|---|---|---|
| Dividing the change by the NEW value | Percentage change is always measured against where you started | Divide (new − old) by the OLD value |
| Calling a fall a positive % change | Going down is a decrease | Show the minus sign or write '% decrease' |
| Reading the rate straight off the y-axis | A single y-value is an amount, not a rate | Use TWO points: rate = rise ÷ run (the gradient) |
| Forgetting the units of the rate | A rate without units is incomplete and loses the mark | Always write per unit time, e.g. cm³ s⁻¹ or breaths min⁻¹ |
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How this is tested: On Paper 1B these are short Calculate items (1–2 marks) hung off a data set: read two values, then find a difference, percentage change, ratio or rate.
Show your working and your units — examiners award the method even if the final number slips, and a rate with no units does not score.
For a rate, never read a single y-value — take two points and divide the rise by the run (the gradient).
IB-style question — process the photosynthesis data
A student measures the volume of oxygen released by pondweed. At a light intensity of 20 units the weed releases 25 cm³ in 5 minutes; at 60 units it releases 90 cm³ in 5 minutes, rising along a straight line. (a) Calculate the percentage increase in oxygen released when the light goes from 20 to 60 units. (b) Calculate the rate of oxygen release at 60 units, in cm³ min⁻¹. [4]
Worked solution
- (a) Pick the formula. Percentage change is measured against the ORIGINAL value: .
- (a) Substitute. Old = 25 cm³, new = 90 cm³: .
- (a) Answer. — a 260 % increase. (Mark 1: divide the change by the old value. Mark 2: 260 %.)
- (b) Pick the formula. A rate is the gradient: .
- (b) Substitute + units. . (Mark 3: divide volume by time. Mark 4: correct value with units, cm³ min⁻¹.)
Final answer
(a) (90 − 25)/25 × 100 = 260 % increase. (b) rate = 90 cm³ ÷ 5 min = 18 cm³ min⁻¹.
✓ Why this scores full marks: Part (a) divides the change by the old value (25), not the new one, and labels it an increase. Part (b) takes the gradient (volume ÷ time) and — crucially — writes the units cm³ min⁻¹.
The classic dropped mark is dividing by 90 in part (a) (giving 72 %) or leaving '18' with no units in part (b).