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Modelling: cheat sheet

Number Foundations · The words and methods from the lessons, with an example of each.

scale

A ratio of the drawing (or model) to the real thing. Each 1 unit on the drawing is 50 units in real life.

1:50⇒3 cm→150 cm1 : 50 \Rightarrow 3\text{ cm} \to 150\text{ cm}

percentage change multiplier

One change after another: multiply the multipliers. Up 20% then down 20% is × 0.96.

up p%:×(1+p100)down p%:×(1−p100)\text{up } p\%: \times\left(1 + \tfrac{p}{100}\right) \qquad \text{down } p\%: \times\left(1 - \tfrac{p}{100}\right)
$100 up 20%, then down 20%: $96
$100 up 20%, then down 20%: $96

percentages in two steps

Each percentage is of a different amount, so they cannot be added: 20% off then 10% off is not 30% off.

200×0.8×0.9=144200 \times 0.8 \times 0.9 = 144

known, unknown, relationship

The first step in modelling: say what you know, what you need, and how they are linked. Then calculate.

known→unknown\text{known} \to \text{unknown}

tax brackets

The marginal rate is the rate on the last dollar; the effective rate is total tax ÷ income.

each part of the income×its bracket’s rate\text{each part of the income} \times \text{its bracket's rate}

model assumptions

Every model assumes something (steady speed, no fees, no starting charge). When it disagrees with real life, name the assumption and refine the model.

fare=4.20+2.10×km\text{fare} = 4.20 + 2.10 \times \text{km}

profit as a percentage

A percentage profit or change is out of the original amount, the cost, not the selling price.

profitcost×100%\dfrac{\text{profit}}{\text{cost}} \times 100\%