If a regular haircut costs $5.00 in 1965, and now approximately 50 years later the same haircut costs $20.00, is this increase a 200% increase or a 300% increase?
Which calculation should be used to determine the correct percentage?
1. ($20.00 - $5.00)/$5.00 = $15.00/$5.00 = 3 = 300% or expressed as $5.00 + (300% * $5.00) = $20
2. $20.00/$5.00 = 4 = 400%; 400% * $5.00 = $20.00
What you are asking is about linguistic usage.
If the average price of a haircut was $5 and $20 in 1965 and 2015 respectively, it is obvious that the later price is four times the earlier price. No one disputes that.
It is also clear in that case that the actual
increase in the average price of a haircut is $15.
But for many purposes, we want to compare
relative increases. Suppose over the same period the average price of a manicure increased from $60 to $75. That is also an actual increase of $15. But 75 is not four times times 60. And suppose the average price of a bunch of bananas went from $1 to $4. That is an actual increase of only $3, not $15, yet the later price is 4 times the earlier.
In fact, it is so common to refer to relative change that we frequently leave off the word "relative," which can lead to confusion.
How do measure the actual change?
We subtract the earlier price from the later price. In the haircut case, where the price went from $5 to $20, the change is 20 - 5 = 15. The increase does not include the original price. No one would say that the price increased by $20 if it went from $5 to $20.
The same principal applies to measuring relative change. We divide the difference in prices by the original price to figure out what the
change is relative to the original.
\(\displaystyle \dfrac{20 - 5}{5} = \dfrac{15}{5} = 300\% = \text { relative change in average price of haircut.}\)
\(\displaystyle \dfrac{75 - 60}{60} = \dfrac{15}{60} = 25\% = \text { relative change in average price of manicure.}\)original.
\(\displaystyle \dfrac{4 - 1}{1} = \dfrac{3}{1} = 300\% = \text { relative change in average price of bananas.}\)