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Welcome to our Math lesson on Compound Percentage Change, this is the fourth lesson of our suite of math lessons covering the topic of Applications of Percentage in Banking. Simple and Compound Interest, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Before explaining the other type of interest applied in banking (compound interest), it would be appropriate to explain the general concept of compound percentage change. It is different from the simple percentage change, as a compound percentage change involves a recurrent percentage change applied each time on the actual value rather than on the initial value.
For example, if we have a compound percentage increase (growth) of the number of bacteria in a sample by 20% each hour and the original number of bacteria is 4,000 (A0 = 4,000), the number of bacteria in the sample after 3 hours (A3) will be calculated through repeated calculations based on the actual value, i.e.
As you see, if we continue using this method, it would be a time-consuming procedure, so it is necessary to find an easier and shorter way to calculate compound percentage change. Thus, if we express the percentage change by r (as decimal), we can write
Obviously, our example involved a compound percentage increase, but this method can be used for the percentage decrease (decay) as well. We just put a minus before r instead of plus. Therefore, using this procedure n times, we obtain the general formula for the compound percentage change:
If we express the change r as a percentage instead as decimal, we obtain
The number of radioactive particles in a sample decreases by 5% every day. If the original number of radioactive particles was 30,000, how many particles are left in the sample after one week?
Since the recurrence occurs 7 times in total (once in a week) have the following clues:
A0 = 30,000
r = 5
n = 7
An = ?
Applying the equation of recurring percentage change
where the sign before n must be minus as we have a percentage decay involved in this situation, we obtain the number of radioactive particles left in the sample after the given recurrence:
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