Definition of Percentages

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5.1Definition of Percentages


In these revision notes for Definition of Percentages, we cover the following key points:

  • What are percentages?
  • What other methods of expressing numbers are equivalent to percentages?
  • How to write a number as a percentage of another number?
  • How to find the percentage of an amount?
  • How to calculate the percentage of another percentage of an amount?
  • How to add/subtract/multiply/divide percentages?
  • What does the word "of" means when dealing with percentages?

Definition of Percentages Revision Notes

Percentages represent a third method of expressing non-whole amounts in addition to decimals and fractions. In symbols, percentages are expressed as

ab/100 = 0.ab = ab %

where a represent the tens of ab while b represents the units.

We can express a number A as a percentage of another number B by dividing A by B and express the corresponding result (either fraction or decimal) as a percentage. In general

A = x% of B ⟹ x = A × 100/B

Another situation involving percentages is when we want to find a given percentage of an amount. In this case, we simply multiply the percentage to the amount after expressing the percentage as a decimal fraction. In symbols, we have

x = A% of B ⟹ x = A% × B = A/100 ×B = A × B/100

When calculating the percentage of another percentage of an amount, we have to find a compound percentage. A compound percentage is calculated by multiplying the two percentages and then, the total percentage is used as the term A in the formula of the previous paragraph. Thus, we have

x% of y% of A = x/100 × y/100 × A = x × y × A/10,000

We can complete operations with percentages in the same way as we do with fractions and decimals. Yet, with percentages, we can apply more types of additions and subtractions than with fractions or decimals. Thus, if amounts expressed in percentages are independent from each other, we can add or subtract the percentages as in addition or subtraction of real numbers. The same is true for subtraction as well. However, we must be careful to make the correct distinction between independent events and dependent ones where one percentage determines the other, i.e. there are situations where one percentage is calculated based on another percentage.

We can multiply or divide percentages in the same way as we multiply or divide fractions or decimals.

In percentages, and decimals, the term "of" means "multiplication", just like when dealing when dealing with fractions.

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