Percentage applications

A percentage (percent) means “per hundred”. It is a way to describe a ratio out of 100, written with the symbol %.

Percentages are used to compare parts to a total in finance, science, population data, discounts, tax, growth, and error rates.

What percent means

\[ p\% = \frac{p}{100}. \]

For example:

\[ 45\%=\frac{45}{100}=0.45. \]

So 45% means “45 out of every 100.”

Percent, fraction, and decimal forms

\[ 35\%=\frac{35}{100}=0.35. \]

• Percent \(\to\) decimal: divide by 100.
• Decimal \(\to\) percent: multiply by 100 and add \(%\).
• Fraction \(\to\) percent: convert to denominator 100 or decimal, then multiply by 100.

Core examples

Example 1: class composition

If 50% of students in a class are male, that means 50 out of every 100 students are male. If there are 500 students:

\[ 50\%\text{ of }500 = 0.50\times500 = 250. \]

So 250 students are male.

Example 2: price increase

A price rises by \$0.15 from \$2.50. The fractional increase is:

\[ \frac{0.15}{2.50}=0.06. \]

So the price increase is 6%.

Short history

Before decimal notation became standard, many calculations were done with fractions out of 100. In Roman contexts, some taxes were set as hundredth fractions.

Through the late Middle Ages and early modern period, denominator-100 arithmetic became common in trade, profit/loss, and interest calculations. By the 17th century, quoting rates in hundredths was standard in many settings.

Percent sign (%)

The word “percent” comes from Latin per centum, meaning “by the hundred.” Over time, written abbreviations of “per cento” contracted into the modern % symbol.

Finding part and whole

To find \(p\%\) of a quantity \(Q\):

\[ \frac{p}{100}\times Q. \]

\[ 18\%\text{ of }250 = 0.18\times250 = 45. \]

To find what percent one number is of another:

\[ \text{percent}=\frac{\text{part}}{\text{whole}}\times100\%. \]

Example: 50 apples as a percentage of 1250 apples:

\[ \frac{50}{1250}=0.04,\quad 0.04\times100\%=4\%. \]

Percent increase and decrease

Percent change compares new and original values.

\[ \%\text{ change}=\frac{\text{new}-\text{original}}{\text{original}}\times100\%. \]

New values can also be found with multipliers:

• Increase by \(r\%\): multiply by \(1+\frac{r}{100}\).
• Decrease by \(r\%\): multiply by \(1-\frac{r}{100}\).

Percentage of a percentage

Convert both percentages to fractions (or decimals), then multiply:

\[ 50\%\text{ of }40\%=\frac{50}{100}\times\frac{40}{100}=0.20=20\%. \]

Avoid double-dividing by 100. For example:

\[ 25\%=0.25,\text{ not }\frac{25\%}{100}. \]

Always state “of what?”

A percentage must be tied to a reference total (what counts as 100%).

Example:

• 60% of all students are female.
• 10% of all students are computer science majors.
• 5% of female students are computer science majors.

Female CS majors as a percentage of all students:

\[ 60\%\times5\%=3\%. \]

As a percentage of CS majors:

\[ \frac{3\%}{10\%}=30\%. \]

So 30% of CS majors are female.

Real-world applications

Discounts, tax, markups, interest, and error rates are all percentage models.

Reversing expressions does not change multiplication results:

\[ 50\%\text{ of }20=10,\qquad 20\%\text{ of }50=10. \]

Checkpoint: 25% off a US$80 item

\(\text{Discount}=0.25\times80=20\), so sale price is US$60.

Checkpoint: increase 200 by 12%

\(200\times1.12=224\).

Checkpoint: find 30% of 450

\(0.30\times450=135\).

Checkpoint: what percent is 18 out of 120?

\(\frac{18}{120}\times100\%=15\%\).