Applied and word problems: modelling, motion, work, mixtures, optimisation and estimation
Translate, solve, interpret
Word problems are modelling problems: name the unknown, state the units, write the relation, solve, and check that the answer is meaningful in context (positive, integral where required, inside the stated range).
Motion: distance = speed × time. For meeting problems add speeds; for pursuit subtract them; on a round trip the times add, not the speeds. Upstream/downstream problems use (v ± current).
Work: rates add. If one worker finishes in a hours and another in b hours, together they need ab/(a + b) hours, because 1/a + 1/b = 1/t.
Mixtures: track the amount of pure substance, not the percentage. Mixing m₁ litres at p₁% with m₂ litres at p₂% gives concentration (m₁p₁ + m₂p₂)/(m₁ + m₂).
Optimisation and estimation: express the target quantity as a function of one variable, then use the vertex of a parabola or monotonicity; remember that an optimum often sits at an endpoint or at an integer value.
Traps. Mixing units (minutes with hours, centimetres with metres). Adding times instead of rates. Averaging percentages instead of weighting by mass. Accepting a negative or fractional answer where the context forbids it.
Worked example. One pipe fills a tank in 6 h, another in 12 h. Together: 1/6 + 1/12 = 1/4, so the tank fills in 4 hours.