Percentage of Some Quantity :
A percentage tells us how much of a quantity is being considered out of 100 equal parts.
General Rule :
To find \(p\%\) of a quantity \(Q\), \(p\%\) means \(p\) out of \(100\), \(p\%\) of a quantity \(Q\) is found by multiplying \(Q\) by \(\frac{p}{100}\)
(i.e.,)
Example :
Find \(30\%\) of \(250\).
\(30\%\) of \(250\) \(=\)
Free - Hand Computations :
Useful Percenatge Facts:
- \(50 \% = \) \(\frac{1}{2}\). So, \(50 \%\) of \(80 = 40 \).
- \(25 \% = \) \(\frac{1}{4}\). So, \( 25 \%\) of \(80 = 20 \).
- \(10 \% = \) \(\frac{1}{10}\). So, \(10 \%\) of \(80 = 8\).
- \(5 \% = \) \(\frac{1}{20}\). So, \(5 \%\) of \(80 = 4\).
For Larger Percentages:
We can combine with familiar percentages,
- \(15 \% = 10 \% + 5 \%\)
Therefore, \(15 \%\) of \(200 = \) \(10 \%\) of \(200 + 5 \%\) of \(200 = 20 +10 = 30 \)
- \(75 \% = 50 \% + 25 \%\)
Therefore, \(75 \%\) of \(200 = 100 + 50 = 150 \)
A Useful Mental Strategy:
Instead of calculating \(70\%\) directly, think \(70 \% = 50 \% + 20 \%\)
Estimate before you calculate:
Estimation gives us an idea of the expected size of the answer.
Example: \(49 \%\) of \(200\) should be close to \(50\%\) of \(200\)
\(50 \%\) of \(200 = 100\)
So, \(49\%\) of \(200\) should be slightly less than \(100\).
FDP - Trio
Fractions, Decimals and Percentages
Fractions, decimals and percentages are three different ways of describing a part of a quantity. They are closely connected and can often be converted from one form to another.
Example :
All three representations describe the same quantity.
Converting a Fraction into a Percentage:
To convert a fraction into a percentage, multiply it by \(100\).
Example :
Converting a Percentage into a Fraction:
Remove the percentage symbol and write the number over \(100\).
Example :
Converting a Percentage into a Decimal:
To convert a percentage into a decimal, divide by \(100\) and simplify.
Example :
Converting a Decimal into a Percenatge:
To convert a decimal to a percent, multiply the number by \(100\) and add a percent sign \(\%\)
Example :
Important!
Remember :
Fraction <-------> Decimal <---------> Percentage
The form may change, but the value remains the same.
Percentages Greater Than 100:
A percentage does not always have to be less than or equal to \(100\%\).
\(100 \%\) represents the whole quantity. A percentage greater than \(100\%\) means that the quantity is greater than the whole quantity.
Understanding \(100\%\)
We consider the whole quantity as \(100 \%\). Therefore, \(100\% = 1 \)
If a quantity is twice the original quantity, \(200\% = 2\)
If it is three times the original quantity, \(300\% = 3\)
Connecting Percentages Greater then 100 with Fractions and Decimals:
For Example,
\(150 \% =\) Thus, \(150 \% = 1.5\)
\(225 \% =\) Thus, \(225 \% = 2.25\)
What does the percenatge tells us?
| Percentage | Fraction | Decimal | Meaning |
| \(50 \%\) | \(0.5\) | Half | |
| \(100\%\) | \(1\) | \(1\) | Whole |
| \(125\%\) | \(1.25\) | \(25 \%\) more than the whole | |
| \(150\%\) | \(1.5\) | One and a half times the whole | |
| \(200\%\) | \(2\) | \(2\) | Twice the whole |
Important!
Quick Concept Summary:
- Percenatge of a Quantity :
- Fraction ----------> Percenatge
\( \text{Fraction} \times 100 = \text {Percentage}\)
- Percenatge ------------> Fraction
\( p \% = \frac{p}{100}\) Then, Simplify the fraction if possible.
- Decimal ------------> Percentage
\(\text{Decimal} \times 100 = \text{Percenatge}\)
- Percentage -------------> Decimal
\(\text{Percentage} \div 100 = \text{Decimal}\)
- Percentage Greater than 100
\(100\% = \text{whole quantity}\)
\(150\% = 1.5 \,\text{times the whole}\)
\(200\% = 2\)\(\,\text{times the whole}\)