Percents

Percentages are used to express how large or small one quantity is relative to another quantity.

Learning Objectives

Use percents to solve real-world problems

Key Takeaways

Key Points

• In mathematics, a percentage is a number or ratio expressed as a fraction of 100.
• A percentage usually represents a part of, or a change in, a quantity.
• Due to inconsistent usage, it is not always clear from the context what a percentage is relative to. When speaking of a “10% rise” or a “10% fall” in a quantity, the usual interpretation is that this is relative to the initial value of that quantity.

Key Terms

• percent: A number or ratio expressed as a fraction of 100.

In mathematics, a percentage is a number or ratio as a fraction of 100. It is often denoted using the percent sign or the abbreviation “pct.” For example, 45% (read as “forty-five percent”) is equal to 45/100, or 0.45.

Percentages are used to express how large or small one quantity is relative to another quantity. For example, an increase of $0.15 on a price of$2.50 is an increase by a fraction of 0.15/2.50 = 0.06. 0.06 is read as “6 hundredths”; therefore, you know it is 6 parts out of a hundred parts. Expressed as a percentage, this is a 6% increase.

Calculating Percentage

To calculate a percentage, you turn the numbers into a ratio as a fraction of 100. For example, if someone says that 8 out of 15 students are boys, you can calculate the percentage of students who are boys as follows:

\begin{align} \displaystyle \frac{8}{15} &=.53 \\ &=\frac{53}{100} \\ &=53\% \end{align}

Therefore, 53% of the students are boys.

To calculate a percentage of another percentage, you convert both percentages to decimals and multiply them, at which point you can translate the decimal result back to a fraction and then a percentage. For example, 50% of 40% is:

\begin{align} \displaystyle \frac{50}{100}\cdot\frac{40}{100} &= 0.50 \cdot 0.40 \\ &= 0.20 \\ &= \frac{20}{100} \\ &= 20\% \end{align}

Mixture Problems

Mixture problems may involve combining two or more substances or objects and require you to find the percent of one of those substances out of the entire mixture. The percent value is computed by multiplying the numeric value of the ratio by 100.

For example, to find the percentage of 50 green apples out of 1,250 red and green apples in a barrel, first compute the ratio:

$\dfrac{50}{1250} =.04$

and then multiply by 100:

$.04 \times 100 = 4\%$

Therefore, 4% of the apples in the barrel are green.

The Importance of Specificity

Whenever we talk about a percentage, it is important to specify what the percentage is relative to. When speaking of a “10% rise” or a “10% fall” in a quantity, for example, the usual interpretation is that this is relative to the initial value of that quantity. For example, if an item is initially priced at $200 and the price rises 10% (an increase of$20), the new price will be $220. Note that this final price is actually 110% of the initial price (100% + 10% = 110%)—hence the potential confusion. The following problem illustrates this point more fully. Example In a certain college, 60% of all students are female, and 10% of all students are computer science majors. If 5% of female students are computer science majors, what percentage of computer science majors are female? Here, we have been asked to compute the ratio of female computer science majors to all computer science majors. We know that 60% of all students are female, and among these 5% are computer science majors, so we conclude that: $\dfrac{60}{100} \times \dfrac{5}{100} = \dfrac{3}{100} = 3 \%$ Therefore, 3% of all students are female computer science majors. We now need to divide by the proportion of all students that are computer science majors, which we have been told is 10%: $\dfrac{3\%}{10\%}=\dfrac{30}{100}=30\%$ Therefore, 30% of all computer science majors at this school are female. Percentages Greater than 100 Although percentages are usually used to express numbers between zero and one, any ratio can be expressed as a percentage. For instance, 1.11 can also be written as 111%, and -0.0035 can also be written as -0.35%. A percentage greater than 100 can represent growth, as in product sales increasing by 110%. Conversely, a negative percent value can represent a decrease in a value, as in a -5% change in sales, which would indicate a 5% drop. Averages The arithmetic mean, or average, of a set of numbers indicates the “middle” or “typical” value of a data set. Learning Objectives Calculate the average of a set of numbers in a real-world context Key Takeaways Key Points • An average is a measure of the “middle” or “typical” value of a data set. • An arithmetic mean is the sum of a collection of numbers divided by the number of numbers in that collection and is often called the “average.” • There are many real-world applications for calculating averages. Key Terms • average: A measure of the “middle” or “typical” value of a data set. • arithmetic mean: The measure of central tendency of a set of values computed by dividing the sum of the values by the number of values; commonly called the “average.” The arithmetic mean, or “average” is a measure of the “middle” or “typical” value of a data set. It is the sum of a collection of numbers divided by the number of numbers in that collection. While it is often referred to simply as “mean” or “average,” the term “arithmetic mean” is preferred in some contexts because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. The arithmetic mean is used frequently not only in mathematics and statistics but also in fields such as economics, sociology, and history. For example, per capita income is the arithmetic mean income of a nation’s population. Suppose we have a data set containing the values $a_1, \dots, a_n$. The arithmetic mean $A$ is defined via the expression: $\displaystyle A = \frac{1}{n} \sum_{i=1}^n a_i = \frac{1}{n}(a_1 + \cdots + a_n)$ This is simply a mathematical way of writing “the mean equals the sum of all of the values in the set, divided by the number of values in the set.” Example 1 To see how this applies to an actual set of numbers, consider the following set: $\{3,5,10\}$. In order to find the average, we must first find the sum of the numbers: $3 + 5 + 10 = 15$ Next, divide their sum by 3, the number of values in the set: $\dfrac{15}{3} = 5$ Therefore, the average of the set of numbers $\{3,5,10\}$ is 5. Example 2 Find the average of the following set of numbers: $\{12, 25, 34, 17, 8, 42\}$ We need to add the values together and then divide that sum by the total number of values, which is 6. $\dfrac{12 + 25 + 34 + 17 + 8 + 42}{6} = \dfrac{136}{6} = 23$ The average of this set is 23. Example 3 Consider the following real-life situation. A small company has 8 employees. Two of those employees are paid$35 per hour, two of them are paid $27 per hour, and four are paid$25 per hour. What is the average hourly wage of these 8 employees?

We need to add together each of the hourly salaries and then divide by 8, the number of employees:

$\dfrac{35 + 35 + 27 + 27 + 25 + 25 + 25 + 25}{8}= \dfrac{224}{8} = 28$

Therefore, employees of this company are paid an average hourly wage of \$28.