# Percentages » Growth factors

See also Calculating faster with percentages.

## Contents:

1. Growth factors with increase2. Growth factors with decrease

## 1. Growth factors with increase

An increase of 7% is the same result as multiplying with 1.07.

We call 1.07 a growth factor.

With an increase of 7% you will get 107%.

The growth factor that corresponds with 107% = 107 : 100 = 1.07.

Written as a formula it look like this:

With an increase of `p`% the growth factor is

`g` = 100 + `p`100 or shorter: `g` = 1 + `p`100.

Vice versa, the growth factor `g` gives a percentage increase of `p` = 100(`g` – 1).

*Example 1*

Stan earns € 820.

He gets a 2.5% raise for three consecutive years.

Every year is salary will be multiplied with a growth factor 1 + 2.5100 = 1.025, so after three years he will earn 820 × 1.025^{3} = € 883.05.

*Example 2*

Margot earns € 935.

After getting a `p`% raise for two consecutive years, she earns € 1030.84.

935 × g^{2} = | 1030.84^{ } |

g^{2} = | 1030.84935 |

g = | = 1.05 |

Thus `p` = 100(1.05 – 1) = 5%.

*Example 3*

A growing plant increases its height every week with 20%.

How many per cent is the increase per day?

To answer this question, we have to do the following three steps:

1) Calculate the growth factor per week.

2) Then calculate the growth factor per day.

3) Finally, calculate the percentage per day.

Answer:

1) The growth factor per week is 1 + 20100 = 1.2.

2) The growth factor per day is ≈ 1.026.

3) The increase per day is (1.026 – 1) × 100 = 2.6%.

## 2. Growth factors with decrease

A decrease of 7% gives the same result as multiplying with 0.93.

We call 0.93 in that case also a growth factor.

Because the growth factor is between 0 and 1, you have a decrease.

The rules are:

With a decrease of `p`% the growth factor is

`g` = 100 – `p`100 or shorter: `g` = 1 – `p`100.

Vice versa, the growth factor `g` gives a percentage decrease of `p` = 100(1 – `g`).

*Example 1*

A pair of pants of € 80 is reduced by 24%, what will be the new price?

The growth factor is `g` = 1 – 24100 = 0.76.

The pair of pants will cost 80 × 0.76 = 60.80 euros.

*Example 2*

A jacket of € 120 is reduced in price to € 105, what percentage was the price reduction?

The growth factor has been `g` = 105 : 120 = 0.875.

The percentage decrease is therefore 100(1 – 0,875) = 12.5%.