# Compound Percentage Change Calculator

The Compound Percentage Change Calculator will calculate:

1. The compound percentage change of a quantity if the initial value, rate of change and the number of compounding are known.

 🖹 Normal View🗖 Full Page View Calculator Precision (Decimal Places)0123456789101112131415 Initial amount (A0) Rate of percentage change (r) Number of compounding times (n)
Compound percentage increase/decrease formula and calculations The Compound percentage increase, An is % [percent] The Compound percentage decrease, An is % [percent] An = A0 ∙ (1 + r)nAn = ∙ (1 + )An = ∙ (1 + )An = ∙ ()An = ∙ An = An = A0 ∙ (1 - r)nAn = ∙ (1 - )An = ∙ (1 - )An = ∙ ()An = ∙ An = Initial amount (A0) Rate of percentage change (r) Number of compounding times (n)

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each compound percentage change calculation, the Compound Percentage Change Calculator will automatically calculate the results and update the formula elements with each element of the compound percentage change calculation. You can then email or print this compound percentage change calculation as required for later use.

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## Theoretical description

Unlike in simple percentage change where all changes are calculated based on the original amount, a compound percentage change is a kind of recurrent change, where all successive amounts are calculated based on the previous amounts and not on the original ones. Thus, if we have a compound percentage increase by 30% of an amount A0, that is applied twice, the final amount A0 will be

A1 = A0 + 30% ∙ A0 = 130% ∙ A0
A2 = A1 + 30% ∙ A1 = 130% ∙ A1 = 130% ∙ 130% ∙ A0 = 169% ∙ A0

However, calculating the recurrent values one by one as above is not suitable, as it includes a number of steps. Therefore, we use a shorter way for this, by applying the formula

An = A0 ∙ (1 ± r)n

where

r = the percentage change (as decimal),

n = number of compounding times

The sign 'minus' is used for compound percentage increase while the sign 'plus' for percentage decay.

For example, if the number of bacteria in a sample is 100 (A0 = 100) and if their number increases by 50% every hour (r = 50% = 0.5), the number of bacteria in the sample after 7 hours (n = 7) is

A7 = 100 ∙ (1 + 0.5)7
= 100 ∙ 1.57
= 1,700 bacteria

Let's consider another example, but this time with compound percentage decay. Thus, if the number of non-decayed nuclei in a radioactive sample is 400 (A0 = 400) and the radioactive sample decays at a rate of 10% in a day (r = 10% = 0.1), we have for the number of non-decayed nuclei (A5) left in the sample after 5 days (5 compounding cycles therefore, or n = 5):

An = A0 ∙ (1 - r)n
A5 = 400 ∙ (1 - 0.1)5
A5 = 400 ∙ 0.95
= 236 nuclei left

## Percentages Math Tutorials associated with the Compound Percentage Change Calculator

The following Math tutorials are provided within the Percentages section of our Free Math Tutorials. Each Percentages tutorial includes detailed Percentages formula and example of how to calculate and resolve specific Percentages questions and problems. At the end of each Percentages tutorial you will find Percentages revision questions with a hidden answer that reveal when clicked. This allows you to learn about Percentages and test your knowledge of Math by answering the revision questions on Percentages.

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