Word Problems and Modelling
What the domain tests
Multi-step problems set in real life: shopping and discounts, percentages and interest, speed and travel, sharing in a ratio, scale drawings, mixtures and simple work rates. The exam is testing whether you can translate text into mathematics, carry the calculation through with correct units, and return to the context with a sensible answer.
A reliable routine
- Read the whole task once and underline what is actually asked.
- Name the unknown with a letter and a unit, for example t hours or x euros.
- Write the relationship as an equation, table or proportion.
- Solve, keeping units attached to every intermediate value.
- Answer in a sentence and sanity-check the size of the result.
Core rules
- Percentages: part = whole · p/100. An increase of p% multiplies by (1 + p/100); a decrease multiplies by (1 − p/100). To reverse a change, divide: original = new/(1 ± p/100).
- Speed: s = v·t. Average speed is total distance divided by total time - never the average of the two speeds unless the times are equal.
- Ratio sharing: add the parts, divide the total by the number of parts, then multiply.
- Scale: a scale of 1 : n means real length = drawing length × n; areas scale by n².
- Rates working together: combined rate = sum of the individual rates.
- Simple interest: interest = principal × rate × time (in years).
Worked example
A car covers 120 km at 80 km/h, then a further 120 km at 120 km/h. Find the average speed for the whole journey. Time for part one: 120/80 = 1,5 h. Time for part two: 120/120 = 1 h. Total distance 240 km, total time 2,5 h. Average speed = 240/2,5 = 96 km/h. The naive average of 80 and 120 gives 100 km/h, which is wrong because the car spent longer at the slower speed.
Traps that cost marks
- Taking the percentage of the wrong base, for example applying a discount to the already discounted price.
- Averaging two speeds instead of dividing total distance by total time.
- Mixing minutes and hours, grams and kilograms, or centimetres and metres inside one calculation.
- Solving correctly but answering a different question from the one asked (price before discount instead of after, or per unit instead of in total).
- Rounding money or measurements too early, and forgetting the unit in the final answer.