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Section3.3Review Percent Change

List3.3.1Topics of Review
  • Meaning of Percent.

  • Decimal Equivalent of Percent.

  • Growth Factor: Percent Increase, Percent Decrease.

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 \(r = \frac{50}{100} = 0.5\)

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 \(1\) (\(b \gt 1)\text{.}\)

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.

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”.

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.

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