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# Percentages | Lesson

## What are percentages?

A percentage is a ratio whose second term is $100$. Percent means parts per hundred.
$p\mathrm{%}$ is equivalent to:
• $p:100$
• $\frac{p}{100}$
We can use equivalent forms of percentages interchangeably and choose the one(s) that best suit our purpose. Typically, the fraction and decimal equivalents of percentages are best suited for calculations.
A shortcut for converting percentages to decimals is to remove the $\mathrm{%}$ symbol and move the decimal point $2$ places to the left.
Percentages are useful because we tend to have a better intuitive understanding of something out of $100$ than fractions and decimals. That's good because in the real world, we're surrounded by percent calculations: a waiter's tip, income tax, results of surveys, etc.

### What skills are tested?

• Using percent relationships to calculate percentages and values
• Solving word problems involving percentages
• Solving word problems involving percent increases and decreases

## How do we calculate percentages?

For two numbers in a percent relationship, we can find either number or the percentage given the other two values. If $a$ is $p\mathrm{%}$ of $b$, then:
$p=\frac{a}{b}×100$
This equation can be rearranged to show $a$ or $b$ in terms of the other values:
$\begin{array}{rl}a& =\frac{p}{100}×b\\ \\ b& =\frac{a}{\left(\frac{p}{100}\right)}=\frac{100×a}{p}\end{array}$
In word problems involving percentages, remember that the sum of all parts of the whole is $100\mathrm{%}$. For example, if a teacher has graded $60\mathrm{%}$ of an assignment, then they have not graded $100-60\mathrm{%}=40\mathrm{%}$ of the assignment. $60\mathrm{%}$ and $40\mathrm{%}$ are complementary percentages: they add up to $100\mathrm{%}$.

## How do we calculate percent changes?

We're often asked to calculate by what percent a quantity changes relative to an initial value: the percent discount on jeans, the percent increase in the amount of potato chips in a bag, etc. When calculating a percent change from an initial value to a final value:
1. Find the difference between the initial and final values.
2. Divide the difference by the initial value.
3. Convert the quotient to a percentage.
$\mathrm{%}\phantom{\rule{0.167em}{0ex}}\text{change}=\frac{\text{final}-\text{initial}}{\text{initial}}×100$
We can calculate the percent change, the initial value, or the final value given the other two. To do so, we:
1. Write an equation that relates the initial and final values using a percentage.
2. Plug in the known values.
3. Solve for the unknown quantity.

TRY: TAKING A PERCENTAGE
What is $30\mathrm{%}$ of $40$ ?

TRY: CALCULATING A PERCENTAGE
What percent of $20$ is $12$ ?
$\mathrm{%}$

TRY: USING COMPLEMENTARY PERCENTAGES
Tara has read $95\mathrm{%}$ of the books she owns. If Tara owns $160$ books, how many of her books has she not read?

TRY: CALCULATING PERCENT INCREASE
The price of a particular model of headphones was $\mathrm{}7$ in $2016$. In $2018$, the price of the same model of headphones was $\mathrm{}10$. What is the approximate percent increase in the price of the headphones?

## Things to remember

Percent means parts per hundred.
$p\mathrm{%}=\frac{p}{100}$
A shortcut for converting percentages to decimals is to remove the $\mathrm{%}$ symbol and move the decimal point left $2$ places.
If $a$ is $p\mathrm{%}$ of $b$, then:
$\begin{array}{rl}p& =\frac{a}{b}×100\\ \\ a& =\frac{p}{100}×b\\ \\ b& =\frac{a}{\left(\frac{p}{100}\right)}=\frac{100×a}{p}\end{array}$
The sum of all parts of the whole is $100\mathrm{%}$.
When calculating a percent change from an initial value to a final value:
1. Find the difference between the initial and final values.
2. Divide the difference by the initial value.
3. Convert the quotient to a percentage.
$\mathrm{%}\phantom{\rule{0.167em}{0ex}}\text{change}=\frac{\text{final}-\text{initial}}{\text{initial}}×100$
To calculate the percent change, the initial value, or the final value given the other two, we:
1. Write an equation that relates the initial and final values using a percentage.
2. Plug in the known values.
3. Solve for the unknown quantity.

## Want to join the conversation?

• shouldn't the equation be initial - final
- —————— x 100
initial
• can some one exsplain this to me
• umm i got something wrong but its right
• 😐no, you got it wrong😐
• why is it so hard