Percentages

A percentage is a way to express a quantity. From the Latin term meaning ‘by the hundred’, a percentage expresses a quantity out of 100.

Converting between percentages and fractions or decimals

\(5\%\) is equivalent to \(\frac{5}{100}\) or \(0.05\).

  • To convert a fraction or a decimal to a percentage, multiply by 100%.
  • To convert a percentage to a fraction or a decimal, divide by 100%.

Worked Example

Express \(0.15\) as a percentage:

\[0.15\times100\%=15\%\]

Express \(\frac{3}{40}\) as a percentage:

\[\frac{3}{40}\times 100\% = 7.5\%\]

Express \(24\%\) as a decimal:

\[24\% \div 100 \% = 0.24\]

Express \(24\%\) as a fraction:

\[24\% \div 100\% = \frac{24}{100} = \frac{6}{25} \]

Percentages of a quantity

To find one quantity as a percentage of another quantity, the first is divided by the second then multiplied by 100%.

Worked Example

In a class of \(24\) students, three students wear glasses. Determine the percentage of students who wear glasses.

Write the \(3\) as a fraction out of \(24\) and multiply by \(100\%\).

\[\frac{3}{24}\times100\%=12.5\%\]

So, \(12.5\%\) of the class wear glasses.

Expressing one value as a percentage of another

One quantity can be expressed as a percentage of another quantity. To do this, both quantities must be in the same units. To determine quantity \(A\) as a percentage of quantity \(B\) simply calculate \(\frac{A}{B}\times100\%\).

Worked Example

A \(2\text{L}\) fruit drink is made by combining \(240\text{mL}\) of fruit concentrate with water. What percentage of the drink is fruit concentrate?

Convert both quantities to the same unit.

\[2\text{L}=2000\text{mL}\]

\(240\text{mL}\) as a percentage of \(2000\text{mL}\) is \(\frac{240}{2000}\times100\%=12\%\).

Percentages are reversible

Meaning that \(16\%\) of \(40\) is the same as \(40\%\) of \(16\).

Algebraically, this is because \(A\%\) of \(B=\frac{A}{100}\times B=\frac{B}{100}\times A=B\%\) of \(A\).

This trick can be exploited to be calculate a percentage more simply.

Worked Example

\(48\%\) of \(25\) is quite difficult to work out mentally.

But \(48\%\) of \(25\) is the same as \(25\%\) of \(48\).

As \(25\%\) is \(\frac{1}{4}\), a quarter of \(48\) is easily determined to be \(12\).

Hence, \(48\%\) of \(25\) is \(12\).

Percentage increase and decrease

Often useful with money, values can increase or decrease by a percentage .

A car’s value may depreciate over time, or a painting’s value may increase over time.
Perhaps a retail outlet is having a sale and applying a percentage discount to items.

When determining the new value of a quantity after a percentage increase or decrease there are two methods:

  • Work out the amount from the percentage, and add/subtract this to the original value.
  • Determine the new percentage after the increase/decrease and work out this percentage of the value.

Worked Example

Price increase

A painting originally purchased for \(\$15,000\) has increased in value by \(18\%\) since its purchase. What is its new value?

Method 1

Work out the amount to be added.

\[18\% \text{ of } \$15000=0.18\times15000=\$2700\]

Add this from the original amount to find the new value.

\[\$15000+\$2700 =\$17700\]

Method 2

Determine the new value as a percentage.

\(18\%\) more means the value will be \(118\%\) of the original.

Work out this percentage of the original value.

\[118\% \text{ of }\$15000=1.18\times15000=\$17700\]

Price decrease

A retail outlet is having a sale and applying a \(30\%\) discount to all items. One item has an original cost of \(\$45\). What is the sale price of this item?

Method 1

Work out the amount to be discounted.

\[30\% \text{ or } \$45=0.3\times45=\$13.50\]

Subtract this from the original amount.

\[\$45-\$13.50 = \$31.50\]

Method 2

Determine what the new price will be as a percentage.

\(30\%\) less means the new price will be \(70\%\) of the original amount.

Work out this percentage of the original value.

\[70\% \text{ of }\$45=0.7\times45=\$31.50\]