4.06 Percentage increase and decrease
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.
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%$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$40 by $25%$25% then we are adding $25%$25% of $40$40 to the original $40$40. The final result will be equal to the expression:
$40+25%\times40$40+25%×40
Since $25%$25% of $40$40 is equal to $10$10, we find that the result of the percentage increase is $50$50.
Similarly, we can calculate a percentage decrease by subtracting the percentage change from the original amount. As such, if we decrease $40$40 by $25%$25% the result will be $30$30.
Practice question
Question 1
Bob wants to decrease $110$110 by $60%$60%, so he calculates $110-\left(60%\times110\right)$110−(60%×110).
What was his result?
Finding percentage changes directly
It is mentioned above that the original amount is equal to $100%$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$40 by $25%$25% using the addition method, we are effectively adding $25%$25% of $40$40 to $100%$100% of $40$40. This is the same as finding $125%$125% of $40$40, which we can see gives the same result:
$125%\times40=50$125%×40=50
We can do the same for percentage decreases. To decrease $40$40 by $25%$25%, we take $25%$25% away from the original $100%$100%, leaving only $75%$75% of $40$40. This gives us:
$75%\times40=30$75%×40=30
By adding or subtracting at the percentage level, we can more clearly see how the original amount changes.
Practice question
Question 2
Sandy starts with the number $110$110, and then calculates $110\times160%$110×160%.
What was her result?
Which is the best description of her result?
She increased $110$110 by $60%$60%.
AShe decreased $110$110 by $60%$60%.
BShe decreased $110$110 by $160%$160%.
CShe increased $110$110 by $160%$160%.
D
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%$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%$100%. Since we want to increase by $70%$70%, that is how much we will be adding. As such, to increase the original amount by $70%$70%, we multiply the original amount by $170%$170%.
We can do the same for flat changes by adding a couple of extra steps.
Worked Example
What percentage do we need to multiply $220$220 by to get $187$187?
Think: The flat change will be the difference between the original and desired amounts. The percentage change will be this difference as a percentage of the original amount. Since the amount has decreased, we will want to subtract this percentage change from the original $100%$100%.
Do: The flat change is the difference between $220$220 and $187$187, which we can calculate to be $33$33.
By simplifying the expression $\frac{33}{220}\times100%$33220×100% we find that the percentage change required is $15%$15%.
This means that, to get $187$187, we need to decrease $220$220 by $15%$15%. In other words, we can multiply it by $85%$85%.
Reflect: Consider that $\frac{187}{220}\times100%=85%$187220×100%=85%. Since this result is $15%$15% less than $100%$100%, we need to decrease $220$220 by $15%$15% to get $187$187. By reversing the multiplication required we notice that we can also deduce required percentage changes by dividing the desired amount by the original amount.
Practice question
Question 3
In training for her next marathon, Sally increased her practise route from $7000$7000 metres to $7910$7910 metres.
By what percentage has Sally increased the distance of her practice route?
Write the answer as a percentage value.