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Understanding percentages

Understanding percentages

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Tutor: Toby

Summary

Understanding percentages

In a nutshell

The symbol that represents percent is %\%​.  Per cent means "out of a hundred", and 100%100\% represents the whole amount.  Percentages can be written as fractions with a denominator of 100100​, or can be converted into decimals.


Curiosity: The word "percent" comes from the language Latin, which was spoken in ancient Rome. "Per" means "each" or "every", and "cent" means "one hundred". The word centimetre comes from the same root word - there are 100100 centimetres in a metre!



Relating percentages and fractions

Percentages can be written as fractions by placing the percentage number over a denominator of 100100. The means that the percentage (%\%) sign is effectively another way of writing 100\dfrac{}{100}.  Any percentage can be converted to a fraction with a denominator of 100.100.​​


Procedure

1.
Recognise the percentage sign (%)(\%) and recall that this means "out of one hundred".
2.
Write out a fraction with a denominator of 100.100.​​
3.
Write the number in front of the percentage as the numerator of the fraction.

Example 1

Write 15%15\% as a fraction.


Recognise the %\%​ and remember this relates to a fraction out of 100.100.

%=100\%= \dfrac{}{100}


Write the number in front of the percentage as the numerator of the fraction.

15%=1510015\%= \underline{\dfrac{15}{100}}​​


Note: As you may have spotted, this fraction can be simplified, as 15100=320\dfrac{15}{100} = \dfrac{3}{20}.  Make sure to fully simplify the fraction if the question asks you to.


Example 2

Are 40%40\% and 2050\dfrac{20}{50} equivalent?


Convert 40%40 \% into a fraction out of 100.100.

40%=4010040 \% = \dfrac{40}{100}​​


Expand the other fraction so that it has a denominator of 100100, to check whether the fractions are equivalent.

2050×22=40100\dfrac{20}{50}\times \dfrac{2}{2}=\dfrac{40}{100}​​


Therefore:

Yes,\underline{\text{Yes}}, ​ 40%40 \%​ and 2050\dfrac{20}{50}are equivalent.



Relating percentages and decimals

To change a percentage into a decimal, you divide by 100.100.  To change a decimal into a percentage (%\%), you apply the inverse operation and multiply by 100.100.​​


Example 3

Change 42%42 \% into a decimal.


Divide the number in front of the percentage sign by 100100.

42%42÷100=0.4242 \% \longrightarrow 42 \div 100 = 0.42​​


Therefore:

42%=0.4242\% = \underline{0.42}​​


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Exercises

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FAQs - Frequently Asked Questions

What is a percentage?

How are percentages and fractions related?

How do I change a percentage into a decimal?

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