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## Section3.3Review Percent Change

### Subsection3.3.1Meaning of Percent and its Decimal Equivalent

Percent literally means “for every $100$”. So $15 \%$ means “15 for every 100”.

In order to use a percent in calculations, we must change the percent number into a decimal number. For instance, in order to use the expression $15 \%$ in a calculation we must convert it into the decimal number $0.15$ .

The decimal number $0.15$ is called the decimal equivalent of $15 \%\text{.}$ In fact, the decimal equivalent of any percent value is that value divided by 100.

In the future we will use the variable $r$ to represent the decimal equivalent of a percent.

\begin{equation*} r = \frac{\text{Pecent Value}}{100} \end{equation*}
###### Example3.3.2

Converting pecent values into their decimal equivalents

1. $0.05 \%$ has a decimal equivalent of $r = \frac{0.05}{100} = 0.0005$

2. $0.5 \%$ has a decimal equivalent of $r = \frac{0.5}{100} = 0.005$

3. $5 \%$ has a decimal equivalent of $r = \frac{5}{100} = 0.05$

4. $50 \%$ has a decimal equivalent of $r = \frac{50}{100} = 0.5$

WeBWorK Exercise

### Subsection3.3.2Growth Factor: Percent Increase ($b \gt 1$)

When someone gets a $10 \%$ raise it means that their salary was divided into pieces and one of those pieces is added onto the original salary. Since $10 \%$ has a decimal equivalent of $0.10$ the increase (raise) means $0.10$ of their salary is added back onto their original.

Think of it like this

New Salary = Old Salary + $0.10 \cdot$ Old Salary

If we let $N$ represent the new salary and $s$ represent the old salary then we can short-cut all the words and we can write a “math sentence” instead. You know, because this is a math book.

\begin{align*} N \amp= s + 0.1s \end{align*}

Notice when we look at the statement as an equation, the variable $s$ is factor of both terms on the right. Therefore $s$ can be factored out and the equation can be rewritten as

\begin{align*} N \amp= (1 + 0.1)s\\ N \amp= 1.1s \end{align*}

We are now able to think of a $10 \%$ increase as multiplying by $1.1$ where the $1$ represents the $100 \%$ you already had and the $0.1$ represents the $10 \%$ increase.

In the equation

\begin{equation*} N = 1.1s \end{equation*}

the number “$1.1$” is called the growth factor and is usually denoted with letter $b\text{.}$

A growth factor is the number we use to multiply in order to get a percent increase or percent decrease. Unfortunately, it is somewhat confusing to refer to this number as a “growth” factor because we use it for both increasing and decreasing by a percent. Although sometimes you may see it written as a “decay” factor to specifically refer to a percent decrease.

In our salary example multiplying by the growth factor $b = 1.1$ makes the initial value $s$ bigger by $10 \%\text{.}$

In general, for percent increase the growth factor is defined as

\begin{equation*} b = 1 + r \end{equation*}

Notice a percent increase means the growth factor is greater than $1$ ($b \gt 1)\text{.}$

###### Exercise3.3.4Identifying the Growth Factor
WeBWorK Exercise

We should now be able to use the growth factor to increase a value by a percent and we should be able to determine the percent growth between two values by calculating the growh factor and extracting the percent change.

###### Example3.3.5

1. The price of milk last year at a local grocery store was $\$3.43 per gallon. Today the price of milk is $7.5 \%$ higher. What is the new price of milk? Round your answer to the nearest penny.

2. The value of a stock was at $\ 21.32$ per share yesterday. Today that same stock is valued at $\ 24.07$ per share. What is the percent increase in the value of the stock? Round your answer to 2 decimal places.

WeBWorK Exercise

### Subsection3.3.3Growth Factor: Percent Decrease ($0 \lt b \lt 1$)

Just like something can increase by a percent, it is also possible to decrease by a percent.

Complete the following example to determine on your own how to decrease a value by a percent.

###### Example3.3.7

A pair of shorts on sale for “$15 \%$ off”.

###### Exercise3.3.8
WeBWorK Exercise

Notice that our price reduction was achieved by multipying by a growth factor $b \lt 1\text{.}$

Something we need to consider about a percent decrease is, “How much can we decrease?”

Let's think about that for a moment.

If you have 10 apples, you can eat $50 \%$ of them, that's half the apples. You can eat $75 \%$ of them, that's $7 \frac{1}{2}$ apples. You can even eat all the apples, that's $100 \%$ of the apples!

But, can you eat more than all the apples you have? Hint: If you only have $10$ apples, you cannot eat more than $10$ apples

It is not possible to decrease by more than $100 \%$ of what you have.

Therefore when it comes to a percent loss, the growth factor is less than $1$ for sure, but it cannot be less than $0\text{.}$

Furthermore, since multiplying by zero is zero, it makes the “growth” very uninteresting. From now on, a growth factor is always strictly greater than zero.

The only choices are $b$ is greater than $1$ or $b$ is between $0$ and $1$

In math we say

Percent growth means $b \gt 1$

Percent decay means $0 \lt b \lt 1$

Let's finish this activity by putting together some of the skills we have reviewed thus far.

###### Exercise3.3.9Increase or Decrease by a Percent
WeBWorK Exercise

Consider a study on the reduction of asthma.

###### Example3.3.10

In a study of 658 adults, most of whom had mild to moderate asthma, reviewers found that oral vitamin D supplements ranging from 400 to 4,000 units a day reduced the risk of attacks requiring medication by 37 percent.

—New York Times, Vitamin D May Reduce Asthma Attacks, Nicholas Bakalar, Sept. 8 2016

WeBWorK Exercise