Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

What price maximises profit at our school bake sale?

AA SLAI SLAA HLAI HL Solid Also in: Calculus, Statistics, Optimisation

Research question

Using a demand curve estimated from a survey of students, what price maximises profit for [a product], and how does the optimum change if costs rise?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

A real sale gives you real decisions. Estimating demand from survey data, building cost and revenue functions, and optimising with calculus is a textbook topic done with your own data — and you can test it at the next sale.

The mathematics you'll need

Linear or exponential demand models from survey data
Revenue R = pq and profit P = R − C
Differentiation to maximise profit
Price elasticity (optional)

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Survey 60+ students on the most they would pay; record actual costs of ingredients and packaging.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Design the survey carefully (avoid leading questions).
Fit a demand model.
Build revenue and profit functions and maximise.
Test the price at a real sale if possible.
Reflect on survey honesty and sell-outs.

Pitfalls that cost marks

Leading survey questions.
Ignoring fixed costs.
Treating stated willingness to pay as real behaviour.

Showing personal engagement

Run the sale and compare predictions with sales.
Compare two products.
Reflect on how you would change the survey.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Linear or exponential demand models from survey data; Revenue R = pq and profit P = R − C
AA HL	Good fit	Linear or exponential demand models from survey data; Revenue R = pq and profit P = R − C
AI SL	Good fit	Linear or exponential demand models from survey data; Revenue R = pq and profit P = R − C
AI HL	Good fit	Linear or exponential demand models from survey data; Revenue R = pq and profit P = R − C

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: leading survey questions; and ignoring fixed costs. Say how each one affects your answer.

Use of mathematics (E)

SL: linear or exponential demand models from survey data; revenue R = pq and profit P = R − C; differentiation to maximise profit; price elasticity (optional), each explained in your own words and carried out correctly.

How finance explorations reach the top bands in general: Finance & economics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model demand with two variables (price and portion size) and discuss the trade-off.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL finance exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete finance exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Linear or exponential demand models from survey data, revenue R = pq and profit P = R − C, differentiation to maximise profit and price elasticity (optional).

Where can I get data for this IA?

Survey 60+ students on the most they would pay; record actual costs of ingredients and packaging. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “What price maximises profit at our school bake sale?” filled in: the question, the maths and the data sources.

Plan this IA free

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