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

Rule

For discounts, you have that

$$\begin{array}{llll}\hfill & \phantom{=}\text{Newprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{oldprice}-\text{discount}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1\times \text{oldprice}-\frac{n}{100}\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\phantom{\rule{-0.17em}{0ex}}\left(1-\frac{n}{100}\right)\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

$$\begin{array}{llll}\hfill \text{Newprice}& =\text{oldprice}-\text{discount}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1\times \text{oldprice}-\frac{n}{100}\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\phantom{\rule{-0.17em}{0ex}}\left(1-\frac{n}{100}\right)\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

For an increase in price, you have that $$\begin{array}{llll}\hfill & \phantom{=}\text{Newprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\text{oldprice}+\text{priceincrease}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1\times \text{oldprice}+\frac{n}{100}\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{n}{100}\right)\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

$$\begin{array}{llll}\hfill \text{Newprice}& =\text{oldprice}+\text{priceincrease}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1\times \text{oldprice}+\frac{n}{100}\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{n}{100}\right)\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

The factors $\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{n}{100}\right)$ and $\phantom{\rule{-0.17em}{0ex}}\left(1-\frac{n}{100}\right)$ 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 $\text{\$}\text{}32\text{}$, but the price has increased by $\text{}15\text{}\phantom{\rule{0.17em}{0ex}}\text{\%}$. Find the new price. **

Let’s find the new price by using the growth factor: $$\begin{array}{llll}\hfill \text{Newprice}& =\phantom{\rule{-0.17em}{0ex}}\left(1+\frac{15}{100}\right)\times \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =1.15\cdot \text{oldprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

When you insert the given numbers, you get

$$\text{Newprice}=1.15\cdot \text{\$}32=\text{\$}36.8$$ |

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

Example 2

A $20$ % discount yields the growth factor

$$\phantom{\rule{-0.17em}{0ex}}\left(1-\frac{20}{100}\right)=1-20\phantom{\rule{0.17em}{0ex}}\text{\%}=80\phantom{\rule{0.17em}{0ex}}\text{\%}$$ |

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

The growth factor is $1.05$ when sales tax is added to the price before tax. You call the price before tax $x$, which gives you the equation

$$\text{salesprice}=1.05\cdot x$$ |

You solve this equation by dividing both sides with $1.05$ and get that $$\begin{array}{llll}\hfill x& =\frac{1}{1.05}\cdot \text{totalsalesprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \approx \text{}95.25\text{\%ofthetotalsalesprice}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

Because the price before tax is $95.25$ % of the total sales price, that means that the tax is $4.75$ % of the total sales price.

This is true for all items with a sales tax rate of $5$ %.

It’s a good idea to remember this rule whether you like to use algebra or not.

Previous entry

Prices with and Without Taxes

Next entry

Activities 5