2.02 Percentage increase and decrease
Introduction
A percentage increase or decrease is a measure of how much some value has changed compared to its original amount. If a value is greater than its original amount, it has increased. If that value is less than its original amount, it has decreased.
Ideas
Adding or subtracting percentage changes
There are two main parts to any percentage increase or decrease; the original amount and the percentage change.
The original amount is equal to 100\% of itself and has not been increased or decreased yet. The percentage change is the amount that we are either increasing or decreasing the original amount by. We can calculate this as some percentage of the original amount.
To find the result of a percentage increase, we add the percentage change to the original amount. For example: if we increase 40 by 25\% then we are adding 25\% of 40 to the original 40. The final result will be equal to the expression:40+25\%\times 40
Since 25\% of 40 is equal to 10, we find that the result of the percentage increase is 50.
Similarly, we can calculate a percentage decrease by subtracting the percentage change from the original amount. As such, if we decrease 40 by 25\% the result will be 30.
Examples
Example 1
Bob wants to decrease 110 by 60\%, so he calculates 110-(60\%\times 110).
What was his result?
To find the result of a percentage increase, we add the percentage change to the original amount.
To find the result of a percentage decrease, we subtract the percentage change from the original amount.
Finding percentage changes directly
It is mentioned above that the original amount is equal to 100\% of itself. This fact is particularly useful if we want to find the result of a percentage increase or decrease directly.
To increase 40 by 25\% using the addition method, we are effectively adding 25\% of 40 to 100\% of 40. This is the same as finding 125\% of 40, which we can see gives the same result:125\%\times 40=50
We can do the same for percentage decreases. To decrease 40 by 25\%, we take 25\% away from the original 100\%, leaving only 75\% of 40. This gives us:75\%\times 40=30
Examples
Example 2
Sandy starts with the number 110, and then calculates 110\times160\%.
What was her result?
Which is the best description of her result?
To increase a quantity by x\%, multiply the quantity by (100+x)\%.
To decrease a quantity by y\%, multiply the quantity by (100-y)\%.
What percentage change was required?
Now that we are able to calculate the result of percentage increases and decreases, we can also do the reverse.
Suppose we want to increase an amount by 70\%, by what percentage would we multiply the original price?
Since we want to increase the amount, we will be adding to the original 100\%. Since we want to increase by 70\%, that is how much we will be adding. As such, to increase the original amount by 70\%, we multiply the original amount by 170\%.
We can do the same for flat changes by adding a couple of extra steps.
Examples
Example 3
In training for her next marathon, Sally increased her practise route from 7000 metres to 7910 metres.
By what percentage has Sally increased the distance of her practice route?
To find the increase as a percentage of the original quantity, write it as a fraction and multiply by 100\%: \text{Percentage increase}=\dfrac{\text{Increase}}{\text{Original quantity}}\times 100\%
To find the decrease as a percentage of the original quantity, write it as a fraction and multiply by 100\%: \text{Percentage decrease}=\dfrac{\text{Decrease}}{\text{Original quantity}}\times 100\%