Growth Factor

The growth factor is the ratio between the old price and the new price, once sales tax is included.

Rule

Calculating with $a %$ Price Change

For discounts, you have that

\begin{array}{l} New price \\ = old price - discount \\ = 1 \times old price - \frac{n}{100} \times old price \\ = (1 - \frac{n}{100}) \times old price \end{array}

\begin{array}{l} New price & = old price - discount \\ = 1 \times old price - \frac{n}{100} \times old price \\ = (1 - \frac{n}{100}) \times old price \end{array}

For an increase in price, you have that

\begin{array}{l} New price \\ = old price + price increase \\ = 1 \times old price + \frac{n}{100} \times old price \\ = (1 + \frac{n}{100}) \times old price \end{array}

\begin{array}{l} New price & = old price + price increase \\ = 1 \times old price + \frac{n}{100} \times old price \\ = (1 + \frac{n}{100}) \times old price \end{array}

The factors

(1 + \frac{n}{100})

and

(1 - \frac{n}{100})

are called growth factors.

It’s important to not get the old price and the new price mixed up. If the new price is less than the old price, the item is cheaper. That means the growth factor is less than 1. If the growth factor is greater than 1, the price has increased, and the new price is higher than the old price. That’s an important difference!

Example 1

A monthly bus ticket used to cost $$ 32$ , but the price has increased by $15 %$ . Find the new price.

Let’s find the new price by using the growth factor:

\begin{array}{l} New price & = (1 + \frac{15}{100}) \times old price \\ = 1.15 \cdot old price \end{array}

When you insert the given numbers, you get

New price = 1.15 \cdot $ 32 = $ 36.8

Note! The term growth factor is also used for discounts, even if the new price is less than the old price.

Example 2

Growth Factors and the Total Percentage

A $20$ % discount yields the growth factor

(1 - \frac{20}{100}) = 1 - 20 % = 80 %

This shows that a discount and its growth factor make up a whole. When you’re doing calculations with discounts, you are essentially doing the same thing you were doing when you found the part of the whole.

You can also use the growth factor to show that when the sales tax is $5$ %, it’s always $4.75$ % of the price with tax.

Rule