Percentage - Percentage Increase and Decrease

Percentage Increase and Decrease

Sometimes due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of $20), the new price will be $220. Note that this final price is 110% of the initial price (100% + 10% = 110%).

Some other examples of percent changes:

• An increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial); in other words, the quantity has doubled.
• An increase of 800% means the final amount is 9 times the original (100% + 800% = 900% = 9 times as large).
• A decrease of 60% means the final amount is 40% of the original (100% − 60% = 40%).
• A decrease of 100% means the final amount is zero (100% − 100% = 0%).

In general, a change of percent in a quantity results in a final amount that is percent of the original amount (equivalently, times the original amount).

It is important to understand that percent changes, as they have been discussed here, do not add in the usual way, if applied sequentially. For example, if the 10% increase in price considered earlier (on the $200 item, raising its price to $220) is followed by a 10% decrease in the price (a decrease of $22), the final price will be $198, not the original price of $200. The reason for the apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to different quantities ($200 and $220, respectively), and thus do not "cancel out".

Another common mistake is thinking that working 50% faster means taking 50% less time to complete the task. On this account, 100% faster means twice the speed, so half the time. For example, if one traveled at 50 mph, 100% faster would be 100 mph (taking 50% less time). And 50% faster speed means 33.33% less time to travel the same distance.

In general, if an increase of percent is followed by a decrease of percent, and the initial amount was, the final amount is ; thus the net change is an overall decrease by percent of percent (the square of the original percent change when expressed as a decimal number). Thus, in the above example, after an increase and decrease of percent, the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200.

This can be expanded for a case where you do not have the same percent change. If the initial percent change is and the second percent change is, and the initial amount was, then the final amount is . To change the above example, after an increase of and decrease of percent, the final amount, $209, is 4.5% more than the initial amount of $200.

In the case of interest rates, it is a common practice to state the percent change differently. If an interest rate rises from 10% to 15%, for example, it is typical to say, "The interest rate increased by 5%" — rather than by 50%, which would be correct when measured as a percentage of the initial rate (i.e., from 0.10 to 0.15 is an increase of 50%). Such ambiguity can be avoided by using the term "percentage points". In the previous example, the interest rate "increased by 5 percentage points" from 10% to 15%. If the rate then drops by 5 percentage points, it will return to the initial rate of 10%, as expected.