Editing Percentage (section)

==Compounding percentages==

Percent changes applied sequentially ''do not add up'' in the usual way. 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), then the final price will be $198—''not'' the original price of $200. The reason for this apparent discrepancy is that the two percent changes (+10% and −10%) are measured relative to different initial values ($200 and $220, respectively), and thus do not "cancel out".

In general, if an increase of {{math|''x''}} percent is followed by a decrease of {{math|''x''}} percent, and the initial amount was {{math|''p''}}, the final amount is {{nowrap|1= {{math|''p''}}(1 + 0.01{{math|''x''}})(1 − 0.01{{math|''x''}}) = {{math|''p''}}(1 − (0.01{{math|''x''}}){{sup|2}})}}; hence the net change is an overall decrease by {{math|''x''}} percent ''of'' {{math|''x''}} 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 {{nowrap|1= {{math|''x''}} = 10 percent}}, the final amount, $198, was 10% of 10%, or 1%, less than the initial amount of $200. The net change is the same for a decrease of {{math|''x''}} percent, followed by an increase of {{math|''x''}} percent; the final amount is {{nowrap|1= {{math|''p''}}(1 - 0.01{{math|''x''}})(1 + 0.01{{math|''x''}}) = {{math|''p''}}(1 − (0.01{{math|''x''}}){{sup|2}})}}.

This can be expanded for a case where one does not have the same percent change. If the initial amount {{math|''p''}} leads to a percent change {{math|''x''}}, and the second percent change is {{math|''y''}}, then the final amount is {{nowrap|1= {{math|''p''}}(1 + 0.01{{math|''x''}})(1 + 0.01{{math|''y''}})}}.  To change the above example, after an increase of {{nowrap|1= {{math|''x''}} = 10 percent}} and decrease of {{nowrap|1= {{math|''y''}} = −5 percent}}, the final amount, $209, is 4.5% more than the initial amount of $200.

As shown above, percent changes can be applied in any order and have the same effect.

In the case of [[interest rate]]s, a very common but ambiguous way to say that an interest rate rose from 10% per annum to 15% per annum, for example, is to say that the interest rate increased by 5%, which could ''theoretically'' mean that it increased from 10% per annum to 10.5% per annum. It is clearer to say that the interest rate increased by 5 [[percentage point]]s (pp). The same confusion between the different concepts of percent(age) and percentage points can potentially cause a major misunderstanding when journalists report about election results, for example, expressing both new results and differences with earlier results as percentages. For example, if a party obtains 41% of the vote and this is said to be a 2.5% increase, does that mean the earlier result was 40% (since 41 = <span style="padding-right:0.1em;">{{nowrap|40 × (1 + {{sfrac|2.5|100}})}}</span>) or 38.5% (since 41 = {{nowrap|38.5 + 2.5}})?

In financial markets, it is common to refer to an increase of one percentage point (e.g. from 3% per annum to 4% per annum) as an increase of "100 basis points".