 Hong Kong
Stage 4 - Stage 5

# Proportion and Variation

Lesson

In maths, particularly in algebra and equations, we look at how to write relationships between two variables. Each equation has an independent variable ($x$x) and a dependent variable ($y$y). There are lots of of different kinds of equations but in this entry we are going to focus on proportions and variations.

## Proportions

A proportion is an equation stating that two ratios are equal. Two ratios are said to be equal if, when written as fractions, the fractions are equivalent. It is usually written as an equality between two ratios,e.g. $\frac{a}{b}=\frac{c}{d}$ab=cd. These are linear equations, so when we graph them, they look like straight lines.

## Variations

Variations are a special kind of proportional relationship that explain how one quantity changes with respect to another quantity. There are two primary types of variation: direct variation and inverse variation.

### Direct Variation

In Growing Together, we looked at direct proportion. If two amounts are directly proportional, it means that as one amount increases, the other amount increases at a constant rate. Mathematically, we write this as $y=kx$y=kx

### Inverse Variation

In One Goes Up, One Goes Down, we looked at inverse proportion. Inverse proportion means that as one amount increases at the same rate that another amount decreases. Mathematically, we write this as $y=\frac{k}{x}$y=kx.

## Summary

Proportions are equations involving ratios and variations are problems written as proportions.

#### Worked Examples

##### Question 1

The value of $y$y varies as $x$x varies. If the value of $y$y can be determined from the value of $x$x, then $x$x is:

1. Dependent

A

Independent

B

##### Question 2

Luke works $3$3 days a week and earns $\$900900 per day.

1. How much does he earn in $5$5 weeks?

2. How much does Luke earn if he works for $7$7 days continuously?

3. Write down the formula for the amount earned, $y$y, if Luke works for $x$x days.

##### Question 3

An empty box weighs $1$1kg.

1. How much will it weigh if it contains a load of $4$4kg?

2. How much will it weigh if it contains a load of $5$5kg?

3. How much will it weigh if it contains a load of $7$7kg?

4. How much will it weigh if it contains a load of $9$9kg?

5. Write down the formula for the total weight of the box ($w$w) if it has a $l$lkg load.

6. Which graph represents the formula of the total weight of the box found above ?

A

B