Direct Proportion

Lesson Aims To understand what is meant by direct proportion.

Keywords Proportionality Direct proportion Constant of proportionality

An Example of Direct Proportion If I were to go shopping in Poundland and buy 5 items, the cost would be £5. How much would it cost for 10 items? £10 …. and so on

So what can we say about the relationship between the price & number of items bought? The more items that are bought the more it costs. (No.of items bought = price paid). Therefore, the price and number of items are said to be directly proportional.

Upmarket shopping! This time we go shopping in Costalotmore where each item is £5. No. of Items 1234 Price (£’s) Is this relationship directly proportional? How can we find out?

Look at Ratios We can look at the number of items and the cost of them in terms of their ratios. 1:5, 2:10, 3:15 & 4:20. What is the simplest form of each ratio? 1:5 As the price increases at the same rate as the number of items, they are directly proportional.

Look at a graph

y=x Look at a graph Poundland

Look at a graph y=x Poundland Costalotmore y=5x

Notation α (alpha) α means proportional to or varies directly with. y α x means that y is directly proportional to, or varies with x.

Using the Notation In the first graph example y= x, this can be written as: y α x

In the second example y=5 x can also be written as: y α x Why?

The general equation for direct proportion is: y=kx where k is a constant amount and called the constant of proportion. General Equation

Example If y α x and y=2 when x =3, find the equation connecting y to x.

Solution using y = kx 1. Substitute values for x & y 2 = k x 3 2. Re-arrange k = Put it all back: y = 2 x 3

Summary When 2 quantities are in direct proportion, as one quantity increases the other quantity increases at the same rate. y α x can be expressed as: y = k x

Non linear The examples so far have been linear, ie of the form: y=k x For non linear proportion, the general form becomes: y=k x n or y α k x n