---
title: "Modelling: cheat sheet"
subtitle: "Number Foundations"
---

# Modelling

> The words and methods from this unit, with an example of each.

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

$$1 : 50 \Rightarrow 3\text{ cm} \to 150\text{ cm}$$

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

$$\text{known} \to \text{unknown}$$

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

$$\text{up } p\%: \times\left(1 + \tfrac{p}{100}\right) \qquad \text{down } p\%: \times\left(1 - \tfrac{p}{100}\right)$$

_$100 up 20%, then down 20%: $96_

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

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

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**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.

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

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

$$\dfrac{\text{profit}}{\text{cost}} \times 100\%$$

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**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.

$$200 \times 0.8 \times 0.9 = 144$$

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