4.02 Percent increase and decrease
Introduction
Percentages are used in everyday life to represent proportions and to show rates of increases and decreases. Some examples of percentages include:
- A discount of 30\%
- An interest rate of 2\% per year
- The amount of battery charge on a device at 64\% remaining
In this lesson, we will use proportional relationships to solve percent problems involving percent increase or decrease.
Increase by a percentage
Let's say we wanted to increase 40 by 5\%.
We could first find 5\% of 40, which we can work out by expressing the percentage as the fraction \dfrac{5}{100}:
| \displaystyle \frac{5}{100}\times 40 | \displaystyle = | \displaystyle \frac{200}{100} |
| \displaystyle \frac{200}{100} | \displaystyle = | \displaystyle 2 |
The final amount would then be the original amount 40 plus this extra 5\% of 40. That is, the final amount would be40+2=42
There is a quicker way to do this, however, if we think completely in percentages.
To find the total amount after the increase, we are essentially finding 100\%+5\%=105\% \text{ of } 40.
Therefore, we can work out the increase this way:
| \displaystyle \frac{105}{100}\times 40 | \displaystyle = | \displaystyle \frac{420}{100} |
| \displaystyle \frac{420}{100} | \displaystyle = | \displaystyle 42 |
This gives us the same amount as before, but involved less steps.
Examples
Example 1
We want to increase 200 by 10\% by following the steps below.
First find 10\% of 200.
Add the percentage increase to the original amount to find the amount after the increase.
Calculate 110\% of 200.
Is increasing an amount by 10\% equivalent to finding 110\% of that amount?
Example 2
Mikee currently earns \$70\,000 in a year. He got promoted and received a 7\% raise. Determine Mikee's new annual salary.
To increase x by y\%, we can calculate: x\times \left(100+y\right)\%
Decrease by a percentage
Decreasing by a percentage has a similar shortcut.
If we want to decrease 60 by 25\% we can multiply (100\%-25\%) of 60.
That is:
| \displaystyle \dfrac{75}{100} \times 60 | \displaystyle = | \displaystyle \dfrac{4500}{100} |
| \displaystyle \dfrac{75}{100} \times 60 | \displaystyle = | \displaystyle 45 |
So to decrease an amount by a percentage, we can multiply the amount by (100\%-\text{ percentage}).
Examples
Example 3
We want to decrease 1600 by 45\% by following the steps outlined below.
Find 45\% of 1600.
Subtract the percentage decrease from the original amount to find the amount after the decrease.
Find 55\% of 1600.
Is decreasing an amount by 45\% equivalent to finding 55\% of that amount?
Example 4
A bag of biscuits weighs 120\text{ kg}. If the weight of the bag decreases by 40\%, find the new weight of the bag.
To decrease x by y\%, we can calculate: x\times(100-y)\%