2.4: Averages
Formula & Symbol Hub
For this section you will need the following:
Symbols Used
 [latex]\sum=[/latex] Summation
 [latex]\%C=[/latex] Percent change
 [latex]\text{GAvg}=[/latex] Geometric average
 [latex]n=[/latex] Number of pieces of data
 [latex]\text{SAvg}=[/latex] Simple average
 [latex]w=[/latex] Weight factor for a piece of data
 [latex]\text{WAvg}=[/latex] Weighted average
 [latex]x=[/latex] A piece of data
Formulas Used

Formula 2.4a – Simple Average
[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex]

Formula 2.4b – Weighted Average
[latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex]

Formula 2.4c – Geometric Average
[latex]\begin{align*}\text{GAvg}=\left(\left[\left(1 +\%C_1\right)\times\left(1+\%C_2\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\end{align*}[/latex]

Formula 3.1b – Rate, Portion, Base (see Section 3.1)
[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex]
Introduction
No matter where you go or what you do, averages are everywhere. Let’s look at some examples:
 Threequarters of your student loan is spent. Unfortunately, only half of the first semester has passed, so you resolve to squeeze the most value out of the money that remains. But have you noticed that many grocery products are difficult to compare in terms of value because they are packaged in different sized containers with different price points?
 For example, one tube of toothpaste sells in a [latex]125\;\text{mL}[/latex] size for [latex]\$1.99[/latex] while a comparable brand sells for [latex]\$1.89[/latex] for [latex]110\;\text{mL}[/latex]. Which is the better deal? A fair comparison requires you to calculate the average price per millilitre.
 Your local transit system charges [latex]\$2.25[/latex] for an adult fare, [latex]\$1.75[/latex] for students and seniors, and [latex]\$1.25[/latex] for children. Is this enough information for you to calculate the average fare, or do you need to know how many riders of each kind there are?
 Five years ago you invested [latex]\$8,000[/latex] in Roller Coasters Inc. The stock value has changed by [latex]9\%[/latex], [latex]−7\%[/latex], [latex]13\%[/latex], [latex]4\%[/latex], and [latex]−2\%[/latex] over these years, and you wonder what the average annual change is and whether your investment kept up with inflation.
 If you participate in any sport, you have an average of some sort: bowlers have bowling averages; hockey or soccer goalies have a goals against average (GAA); and baseball pitchers have an earned run average (ERA).
Averages generally fall into three categories. This section explores simple, weighted, and geometric averages.
Simple Averages
An average is a single number that represents the middle of a data set. It is commonly interpreted to mean the “typical value.” Calculating averages facilitates easier comprehension of and comparison between different data sets, particularly if there is a large amount of data. For example, what if you want to compare yearoveryear sales? One approach would involve taking company sales for each of the [latex]52[/latex] weeks in the current year and comparing these with the sales of all [latex]52[/latex] weeks from last year. This involves [latex]104[/latex] weekly sales figures with [latex]52[/latex] points of comparison. From this analysis, could you concisely and confidently determine whether sales are up or down? Probably not. An alternative approach involves comparing last year’s average weekly sales against this year’s average weekly sales. This involves the direct comparison of only two numbers, and the determination of whether sales are up or down is very clear.
In a simple average, all individual data share the same level of importance in determining the typical value. Each individual data point also has the same frequency, meaning that no one piece of data occurs more frequently than another. Also, the data do not represent a percent change. To calculate a simple average, you require two components:
 The data itself—you need the value for each piece of data.
 The quantity of data—you need to know how many pieces of data are involved (the count), or the total quantity used in the calculation.
[latex]\boxed{2.4\text{a}}[/latex] Simple Average
[latex]\color{blue}{\sum}\color{black}{\text{ is Summation:}}[/latex] This symbol is known as the Greek capital letter sigma. In mathematics it denotes that all values written after it (to the right) are summed.
[latex]\color{red}{\text{SAvg}}\color{black}{\text{ is Simple Average:}}[/latex] A simple average for a data set in which all data has the same level of importance and the same frequency.
[latex]\color{green}{n}\color{black}{\text{ is Total Quantity:}}[/latex] This is the physical total count of the number of pieces of data or the total quantity being used in the average calculation. In business, the symbol [latex]n[/latex] is a common standard for representing counts.
[latex]\color{purple}{x}\color{black}{\text{ is Any Piece of Data:}}[/latex] In mathematics this symbol is used to represent an individual piece of data.
As expressed in Formula 2.4a, you calculate a simple average by adding together all of the pieces of data then taking that total and dividing it by the quantity.
HOW TO
Calculate a Simple Average
The steps required to calculate a simple average are as follows:
Step 1: Sum every piece of data.
Step 2: Determine the total quantity involved.
Step 3: Calculate the simple average using Formula 2.4a[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex].
Assume you want to calculate an average on three pieces of data: [latex]95[/latex], [latex]108[/latex], and [latex]97[/latex]. Note that the data are equally important and each appears only once, thus having the same frequency. You require a simple average.
Step 1: Sum all data:
[latex]\sum x=95+108+97=300[/latex]
Step 2: There are three pieces of data, or [latex]n= 3[/latex].
Step 3: Apply Formula 2.4a[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex]:
[latex]\begin{align*}\text{SAvg}=\frac{300}{3}=100\end{align*}[/latex]
The simple average of the data set is [latex]100[/latex].
Key Takeaway
Although mentioned earlier, it is critical to stress that a simple average is calculated only when all of the following conditions are met:
 All of the data shares the same level of importance toward the calculation.
 All of the data appear the same number of times.
 The data does not represent percent changes or a series of numbers intended to be multiplied with each other.
If any of these three conditions are not met, then either a weighted or geometric average is used depending on which of the above criteria failed. We discuss this later when each average is introduced.
Try It
It is critical to recognize if you have potentially made any errors in calculating a simple average. Review the following situations and, without making any calculations, determine the best answer.
1) The simple average of [latex]15[/latex], [latex]30[/latex], [latex]40[/latex], and [latex]45[/latex] is:
 lower than [latex]20[/latex]
 between [latex]20[/latex] and [latex]40[/latex], inclusive
 higher than [latex]40[/latex]
Solution
The best answer is b. because a simple average should fall in the middle of the data set, which appears spread out between [latex]15[/latex] and [latex]45[/latex], so the middle would be around [latex]30[/latex]).
Try It
2) If the simple average of three pieces of data is [latex]20[/latex], which of the following data do not belong in the data set? Data set: [latex]10[/latex], [latex]20[/latex], [latex]30[/latex], [latex]40[/latex]
 [latex]10[/latex]
 [latex]20[/latex]
 [latex]30[/latex]
 [latex]40[/latex]
Solution
The data set that does not belong is d. If the number [latex]40[/latex] is included in any average calculation involving the other numbers, it is impossible to get a low average of [latex]20[/latex].
Example 2.4.1
First quarter sales for Buzz Electronics are as indicated in the table below.

Martha needs to prepare a report for the board of directors comparing yearoveryear quarterly performance. To do this, she needs you to do the following:
 Calculate the average sales in the quarter for each year.
 Express the [latex]2014[/latex] sales as a percentage of the [latex]2013[/latex] sales, rounding your answer to two decimals.
Solution
Step 1: What are we looking for?
You need to calculate a simple average, or [latex]SAvg[/latex], for the first quarter in each of [latex]2013[/latex] and [latex]2014[/latex]. Then convert the numbers into a percentage.
Step 2: What do we already know?
You know the three quarters annually:
[latex]\begin{align*}2013:\;\;x_1=\$413,200\;\;x_2=\$328,986\;\;x_3=\$350,003\\[2ex]2014:\;\;x_1=\$455,876\;\; x_2=\$334,582\;\;x_3=\$312,777\end{align*}[/latex]
Additionally, you know that the simple average can be obtained using Formula 2.4a[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex] and that using Formula 3.1b[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex] you can calculate [latex]2014[/latex] sales as a percentage of [latex]2013[/latex] sales by treating [latex]2013[/latex] average sales as the base and [latex]2014[/latex] average sales as the portion.
Step 3: Make substitutions using the information known above.
Calculate simple averages for [latex]2013[/latex] and [latex]2014[/latex] using Formula 2.4a:
[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex]
Simple average for [latex]2013[/latex]:
[latex]\begin{align*}\text{SAvg}_{2013}&=\frac{\$413,200 + \$328,986 + \$350,003}{3}\\[1ex]\text{SAvg}_{2013}&=\frac{\$1,092,189}{3}\\[1ex]\text{SAvg}_{2013}&=\$364,063\end{align*}[/latex]
Simple average for [latex]2014[/latex]:
[latex]\begin{align*}\text{SAvg}_{2014}&=\frac{\$455,876 + \$334,582 + \$312,777}{3}\\[1ex]\text{SAvg}_{2014}&=\frac{\$1,103,235}{3}\\[1ex]\text{SAvg}_{2014}&=\$367,745\end{align*}[/latex]
Finally, apply , substituting [latex]\text{SAvg}_{2013}[/latex] for [latex]\text{Base}[/latex] and [latex]\text{SAvg}_{2014}[/latex] for [latex]\text{Portion}[/latex] and multiplying by [latex]100[/latex] to obtain percentage. Round the result to two decimal places:
[latex]\begin{align*}\%&=\frac{\text{Portion}}{\text{Base}}\times 100\\[1ex]\%&=\frac{\$367,745}{\$364,063}\times 100\\[1ex]\%&=101.01\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
The average monthly sales in [latex]2013[/latex] were [latex]\$364,063[/latex] compared to sales in [latex]2014[/latex] of [latex]\$367,745[/latex]. This means that [latex]2014[/latex] sales are [latex]101.01\%[/latex] of [latex]2013[/latex] sales.
Weighted Averages
Have you considered how your grade point average (GPA) is calculated? Your business program requires the successful completion of many courses. Your grades in each course combine to determine your GPA; however, not every course necessarily has the same level of importance as measured by your course credits.
Perhaps your math course takes one hour daily while your communications course is only delivered in onehour sessions three times per week. Consequently, the college assigns the math course five credit hours and the communications course three credit hours. If you want an average, these different credit hours mean that the two courses do not share the same level of importance, and therefore a simple average cannot be calculated.
In a weighted average, not all pieces of data share the same level of importance or they do not occur with the same frequency. The data cannot represent a percent change or a series of numbers intended to be multiplied with each other. To calculate a weighted average, you require two components:
 The data itself—you need the value for each piece of data.
 The weight of the data—you need to know how important each piece of data is to the average. This is either an assigned value or a reflection of the number of times each piece of data occurs (the frequency).
[latex]\boxed{2.4\text{b}}[/latex] Weighted Average
[latex]{\color{red}{\text{WAvg}}}{\color{black}{\text{ is Weighted Average:}}}[/latex] An average for a data set where the data points may not all have the same level of importance or they may occur at different frequencies.
[latex]\color{blue}{\sum}\color{black}{\text{ is Summation:}}[/latex] This symbol is known as the Greek capital letter sigma. In mathematics it denotes that all values written after it (to the right) are summed.
[latex]\color{green}{w}\color{black}{\text{ is Weighting Factor:}}[/latex] A number that represents the level of importance for each piece of data in a particular data set. It is either predetermined or reflective of the frequency for the data.
[latex]\color{purple}{x}\color{black}{\text{ is Any Piece of Data:}}[/latex] In mathematics this symbol is used to represent an individual piece of data.
As expressed in Formula 2.4b, calculate a weighted average by adding the products of the weights and data for the entire data set and then dividing this total by the total of the weights.
HOW TO
Calculate a Weighted Average
The steps required to calculate a weighted average are:
Step 1: Sum every piece of data multiplied by its associated weight.
Step 2: Sum the total weight.
Step 3: Calculate the weighted average using Formula 2.4b[latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex].
Let’s stay with the illustration of the math and communications courses and your GPA. Assume that these are the only two courses you are taking. You finish the math course with an A, translating into a grade point of [latex]4.0[/latex]. In the communications course, your C+ translates into a [latex]2.5[/latex] grade point. These courses have five and three credit hours, respectively. Since they are not equally important, you use a weighted average.
Step 1: In the numerator, sum the products of each course’s credit hours (the weight) and your grade point (the data). This means:
[latex]\small(\text{math credit hours}\times\text{math grade point})+(\text{communications credit hours}\times\text{communications grade point})[/latex].
Numerically, this is:
[latex]\begin{align*}\sum wx=\left(5\times 4\right)+\left(3\times 2.5\right)=27.5\end{align*}[/latex]
Step 2: In the denominator, sum the weights. These are the credit hours. You have:
[latex]\begin{align*}\sum w=5+3=8\end{align*}[/latex]
Step 3: Apply Formula 2.4b[latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex] to calculate your GPA.
[latex]\begin{align*}\text{WAvg}&=\frac{\sum wx}{\sum w}\\[1ex]\text{WAvg}&=\frac{27.5}{8}\\[1ex]\text{WAvg}&=3.44\text{ (GPAs have two decimals)}\end{align*}[/latex]
Note that your GPA is higher than if you had just calculated a simple average:
[latex]\begin{align*}\text{SAvg}&=\frac{\sum x}{n}\\[1ex]\text{SAvg}&=\frac{4 + 2.4}{2}\\[1ex]\text{SAvg}&=2.25\end{align*}[/latex]
This happens because your math course, in which you scored a higher grade, was more important in the calculation.
Things To Watch Out For
The most common error in weighted averages is to confuse the data with the weight. If you have the two backwards, your numerator is still correct; however, your denominator is incorrect. To distinguish the data from the weight, notice that the data forms a part of the question. In the above example, you were looking to calculate your grade point average; therefore, grade point is the data. The other information, the credit hours, must be the weight.
Paths To Success
The formula used for calculating a simple average is a simplification of the weighted average formula. In a simple average, every piece of data is equally important. Therefore, you assign a value of 1 to the weight for each piece of data. Since any number multiplied by 1 is the same number, the simple average formula omits the weighting in the numerator as it would have produced unnecessary calculations. In the denominator, the sum of the weights of 1 is no different from counting the total number of pieces of data. In essence, you can use a weighted average formula to solve simple averages.
Try It
Determine which information is the data and which is the weight.
3) Rafiki operates a lemonade stand during his garage sale today. He has sold [latex]13[/latex] small drinks for [latex]\$0.50[/latex], [latex]29[/latex] medium drinks for [latex]\$0.90[/latex], and [latex]21[/latex] large drinks for [latex]\$1.25[/latex]. What is the average price of the lemonade sold?
Solution
The price of the drinks is the data, and the number of drinks is the weight.
Try It
Determine which information is the data and which is the weight.
4) Natalie received the results of a market research study. In the study, respondents identified how many times per week they purchased a bottle of CocaCola. Calculate the average number of purchases made per week.

Solution
The purchases per week is the data, and the number of people is the weight.
Example 2.4.2
A mark transcript received by a student at a local college:

This chart shows how each grade translates into a grade point:

Calculate the student’s grade point average (GPA). Round your final answer to two decimals.
Solution
Step 1: What are we looking for?
The courses do not carry equal weights as they have different credit hours. Therefore, to calculate the GPA you must find a weighted average, or WAvg.
Step 2: What do we already know?
Since the question asked for the grade point average, the grade points for each course are the data, or [latex]x[/latex]. The corresponding credit hours are the weights, or [latex]w[/latex]. This information can be substituted into Formula 2.4b[latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex] to find the weighted average.
Step 3: Make substitutions using the information known above.
Use the secondary table above to convert each course grade into its grade point:

Sum every piece of data multiplied by its associated weight:
[latex]\begin{align*}\sum wx&= \sum(\text{Credit Hours}\times\text{Grade Point})\\\sum wx&=(4\times 3.0)+(5\times 4.0)+(3\times 3.5)+(4\times 2.0)+(3\times 4.5)+(4\times 1.0)\\\sum wx&=68\end{align*}[/latex]
Sum the total weight:
[latex]\begin{align*}\sum w&=4+5+3+4+3+4\\\sum w&=23\end{align*}[/latex]
Substitute into Formula 2.4b:
[latex]\begin{align*}\text{WAvg}&=\frac{\sum wx}{\sum w}\\[1ex]\text{WAvg}&=\frac{68}{23}\\[1ex]\text{WAvg}&=2.96\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
The student’s GPA is [latex]2.96[/latex]. Note that math contributed substantially (almost onethird) to the student’s grade point because this course was weighted heavily and the student performed well.
Example 2.4.3
Angelika started the month of March owing [latex]\$20,000[/latex] on her home equity line of credit (HELOC). She made a payment of [latex]\$5,000[/latex] on the fifth, borrowed [latex]\$15,000[/latex] on the nineteenth, and made another payment of [latex]\$5,000[/latex] on the twentysixth. Using each day’s closing balance for your calculations, what was the average balance in the HELOC for the month of March?
Solution
Step 1: What are we looking for?
The balance owing in Angelika’s HELOC is not equal across all days in March. Some balances were carried for more days than others. This means you will need to use the weighted average technique and find [latex]\text{WAvg}[/latex].
Step 2: What do we already know?
You know the following:

Step 3: Make substitutions using the information known above.
Sum every piece of data multiplied by its associated weight:
[latex]\begin{align*}\sum wx&=(4\times \$20,000)+(14\times\$15,000)+(7\times\$30,000)+(6\times\$25,000)\\\sum wx&=\$650,000\end{align*}[/latex]
Sum the total weight:
[latex]\begin{align*}\sum w&=4+14+7+6\\\sum w&=31\end{align*}[/latex]
Calculate the weighted average using Formula 2.4b:
[latex]\begin{align*}\text{WAvg}&=\frac{\sum wx}{\sum w}\\[1ex]\text{WAvg}&=\frac{\$650,000}{31}\\[1ex]\text{WAvg}&=\$20,967.74\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
Over the entire month of March, the average balance owing in the HELOC was [latex]\$20,967.74[/latex]. Note that the balance with the largest weight (March 5 to March 18) and the largest balance owing (March 19 to March 25) account for almost twothirds of the calculated average.
Geometric Averages
How do you average a percent change? If sales increase [latex]100\%[/latex] this year and decrease [latex]50\%[/latex] next year, is the average change in sales an increase of
[latex]\begin{align*}\frac{100\%50\%}{2}=25\%\end{align*}[/latex]
per year? The answer is clearly “no.” If sales last year were [latex]\$100[/latex] and they increased by [latex]100\%[/latex], that results in a [latex]\$100[/latex] increase. The total sales are now [latex]\$200[/latex]. If sales then decreased by [latex]50\%[/latex], you have a [latex]\$100[/latex] decrease. The total sales are now [latex]\$100[/latex] once again. In other words, you started with [latex]\$100[/latex] and finished with [latex]\$100[/latex]. That is an average change of nothing, or [latex]0\%[/latex] per year! Notice that the second percent change is, in fact, multiplied by the result of the first percent change. A geometric average finds the typical value for a set of numbers that are meant to be multiplied together or are exponential in nature.
In business mathematics, you most commonly use a geometric average to average a series of percent changes. Formula 2.4c is specifically written to address this situation.
[latex]\boxed{2.4\text{c}}[/latex] Geometric Average
[latex]\color{green}{n}\color{black}{\text{ is Total Quantity:}}[/latex] The physical total count of how many percent changes are involved in the calculation.
[latex]\color{red}{\text{GAvg }}\color{black}{\text{ is Geometric Average:}}[/latex] The average of a series of percent changes expressed in percent format. Every percent change involved in the calculation requires an additional ([latex]1+\%C[/latex]) to be multiplied under the radical. The formula accommodates as many percent changes as needed.
[latex]\color{purple}{\times 100}\color{black}{\text{ is Percent Conversion:}}[/latex] Because you are averaging percent changes, convert the final result from decimal form into a percentage.
[latex]\color{blue}{\%C}\color{black}{\text{ is Percent Change:}}[/latex] The value of each percent change in the series from which the average is calculated. You need to express the percent changes in decimal format.
HOW TO
Calculate a Geometric Average
To calculate a geometric average follow these steps:
Step 1: Identify the series of percent changes to be multiplied.
Step 2: Count the total number of percent changes involved in the calculation.
Step 3: Calculate the geometric average using Formula 2.4c[latex]\begin{align*}\text{GAvg}=\left(\left[\left(1 +\%C_1\right)\times\left(1+\%C_2\right)\times\text{...}\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\end{align*}[/latex].
Let’s use the sales data presented above, according to which sales increase 100% in the first year and decrease [latex]50\%[/latex] in the second year. What is the average percent change per year?
Step 1: The changes are [latex]\%C_1=+100\%[/latex] and [latex]\%C_2=50\%[/latex].
Step 2: Two changes are involved, or [latex]n=2[/latex].
Step 3: Apply Formula 2.4c:
[latex]\begin{align*}\text{GAvg}&=\left(\left[\left(1+\%C_1\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\\\text{GAvg}&=\left(\left[\left(1+100\%\right)\times\left(150\%\right)\right]^{\frac{1}{2}}1\right)\times 100\\\text{GAvg}&=\left(\left[2\times 0.50\right]^{\frac{1}{2}}1\right)\times 100\\\text{GAvg}&=\left(\left[1\right]^{\frac{1}{2}}  1\right)\times 100\\\text{GAvg}&=0\%\end{align*}[/latex]
The average percent change per year is [latex]0\%[/latex] because an increase of [latex]100\%[/latex] and a decrease of [latex]50\%[/latex] cancel each other out.
Thing To Watch Out For
A critical requirement of the geometric average formula is that every ([latex]1+\%C[/latex]) expression must result in a number that is positive. This means that the [latex]\%C[/latex] cannot be a value less than [latex]100\%[/latex] else Formula 2.4c[latex]\begin{align*}\text{GAvg}=\left(\left[\left(1 +\%C_1\right)\times\left(1+\%C_2\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\end{align*}[/latex] cannot be used.
Paths To Success
An interesting characteristic of the geometric average is that it will always produce a number that is either smaller than (closer to zero) or equal to the simple average. In the example, the simple average of [latex]+100\%[/latex] and [latex]50\%[/latex] is [latex]25\%[/latex], and the geometric average is [latex]0\%[/latex]. This characteristic can be used as an error check when you perform these types of calculations.
Try It
Determine whether you should calculate a simple, weighted, or geometric average.
5) Randall bowled [latex]213[/latex], [latex]245[/latex], and [latex]187[/latex] in his Thursday night bowling league and wants to know his average.
Solution
Simple; each item has equal importance and frequency.
Try It
Determine whether you should calculate a simple, weighted, or geometric average.
6) Cindy invested in a stock that increased in value annually by [latex]5\%[/latex], [latex]6\%[/latex], [latex]3\%[/latex], and [latex]5\%[/latex]. She wants to know her average increase.
Solution
Geometric; these are a series of percent changes on the price of stock.
Try It
Determine whether you should calculate a simple, weighted, or geometric average.
7) A retail store sold [latex]150[/latex] bicycles at the regular price of [latex]\$300[/latex] and [latex]50[/latex] bicycles at a sale price of [latex]\$200[/latex]. The manager wants to know the average selling price.
Solution
Weighted; each item has a different frequency.
Try It
8) Gonzalez has calculated a simple average of [latex]50\%[/latex] and a geometric average of [latex]60\%[/latex]. He believes his numbers are correct. What do you think?
Solution
At least one of the numbers is wrong since a geometric average is always smaller than or equal to the simple average.
Examples 2.4.4
From [latex]2006[/latex] to [latex]2010[/latex], WestJet’s yearoveryear annual revenues changed by [latex]+21.47\%[/latex], [latex]+19.89\%[/latex], [latex]10.55\%[/latex], and [latex]+14.38\%[/latex]. This reflects growth from sales of [latex]\$1.751\;\text{billion}[/latex] in [latex]2006[/latex] to [latex]\$2.609\;\text{billion}[/latex] in [latex]2010[/latex].^{1} What is the average percent growth in revenue for WestJet during this time frame?
Solution
Step 1: What are we looking for?
Note that these numbers reflect percent changes in revenue. Yearoveryear changes are multiplied together, so you would calculate a geometric average, or [latex]\text{GAvg}[/latex].
Step 2: What do we already know?
You know the four percent changes:
[latex]\%C_1=21.47\%[/latex]
[latex]\%C_2=19.89\%[/latex]
[latex]\%C_3=10.55\%[/latex]
[latex]\%C_4=14.38\%[/latex]
You also know that four changes are involved, or [latex]n=4[/latex].
Step 3: Make substitutions using the information known above.
Express the percent changes in decimal format and substitute into Formula 2.4c:
[latex]\begin{align*}\text{GAvg}&=\left(\left[\left(1+\%C_1\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\\\text{GAvg}&=\left(\left[\left(1+0.2147\right)\times\left(1+0.1989\right)\times\left(10.1055\right)\times\left(1+0.1438\right)\right]^{\frac{1}{4}}1\right)\times 100\\\text{GAvg}&=10.483\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
On average, WestJet revenues have grown [latex]10.483\%[/latex] each year from [latex]2006[/latex] to [latex]2010[/latex].
Section 2.4 Exercises
Mechanics
Calculate a simple average for questions 1 and 2.
 [latex]8[/latex], [latex]17[/latex], [latex]6[/latex], [latex]33[/latex], [latex]15[/latex], [latex]12[/latex], [latex]13[/latex], [latex]16[/latex]
 [latex]\$1,500[/latex], [latex]\$2,000[/latex], [latex]\$1,750[/latex], [latex]\$1,435[/latex], [latex]\$2,210[/latex]
Calculate a weighted average for questions 3 and 4.
 [latex]4[/latex], [latex]4[/latex], [latex]4[/latex], [latex]4[/latex], [latex]12[/latex], [latex]12[/latex], [latex]12[/latex], [latex]12[/latex], [latex]12[/latex], [latex]12[/latex], [latex]12[/latex], [latex]15[/latex], [latex]15[/latex]

Calculate a geometric average for exercises 5 and 6. Round all percentages to four decimals.
5. 
[latex]5.4\%[/latex] 
[latex]8.7\%[/latex] 
[latex]6.3\%[/latex] 

6. 
[latex]10\%[/latex] 
[latex]4\%[/latex] 
[latex]17\%[/latex] 
[latex]10\%[/latex] 

Solutions
 [latex]15[/latex]
 [latex]\$1,779[/latex]
 [latex]10[/latex]
 [latex]\$3,795[/latex]
 [latex]6.7910\%[/latex]
 [latex]2.6888\%[/latex]
Applications
 If a [latex]298[/latex] mL can of soup costs [latex]\$2.39[/latex], what is the average price per millilitre?
 Kerry participated in a fundraiser for the Children’s Wish Foundation yesterday. She sold [latex]115[/latex] pins for [latex]\$3[/latex] each, [latex]214[/latex] ribbons for [latex]\$4[/latex] each, [latex]85[/latex] coffee mugs for [latex]\$7[/latex] each, and [latex]347[/latex] baseball hats for [latex]\$9[/latex] each. Calculate the average amount Kerry raised per item.
 Stephanie’s mutual funds have had yearly changes of [latex]9.63\%[/latex], [latex]2.45\%[/latex], and [latex]8.5\%[/latex]. Calculate the annual average change in her investment.
 In determining the hourly wages of its employees, a company uses a weighted system that factors in local, regional, and national competitor wages. Local wages are considered most important and have been assigned a weight of [latex]5[/latex]. Regional and national wages are not as important and have been assigned weights of [latex]3[/latex] and [latex]2[/latex], respectively. If the hourly wages for local, regional, and national competitors are [latex]\$16.35[/latex], [latex]\$15.85[/latex], and [latex]\$14.75[/latex], what hourly wage does the company pay?
 Canadian Tire is having an endofseason sale on barbecues, and only four floor models remain, priced at [latex]\$299.97[/latex], [latex]\$345.49[/latex], [latex]\$188.88[/latex], and [latex]\$424.97[/latex]. What is the average price for the barbecues?
 Calculate the grade point average (GPA) for the following student. Round your answer to two decimals.
Table 2.4.9 Course
Grade
Credit Hours
Grade
Grade Point
Grade
Grade Point
Economics 100
D
5
A+
4.5
C+
2.5
Math 100
B
3
A
4.0
C
2.0
Marketing 100
C
4
B+
3.5
D
1.0
Communications 100
A
2
B
3.0
F
0.0
Computing 100
A+
3
Accounting 100
B+
4
 An accountant needs to report the annual average age (the length of time) of accounts receivable (AR) for her corporation. This requires averaging the monthly AR averages, which are listed below. Calculate the annual AR average.
Table 2.4.10 Month
Monthly AR Average
Month
Monthly AR Average
Month
Monthly AR Average
January
$45,000
May
$145,000
September
$185,000
February
$70,000
June
$180,000
October
$93,000
March
$85,000
July
$260,000
November
$60,000
April
$97,000
August
$230,000
December
$50,000
 From January 2007 to January 2011, the annual rate of inflation has been [latex]2.194\%[/latex], [latex]1.073\%[/latex], [latex]1.858\%[/latex], and [latex]2.346\%[/latex]. Calculate the average rate of inflation during this period.
Solutions
 [latex]\$0.00802/\text{ml}[/latex]
 [latex]\$6.46[/latex]
 [latex]5.0821\%[/latex]
 [latex]\$15.88[/latex]
 [latex]\$314.83[/latex]
 [latex]2.74[/latex]
 [latex]\$125,000[/latex]
 [latex]1.8666\%[/latex]
Challenge, Critical Thinking, & Other Applications
 Gabrielle is famous for her trail mix recipe. By weight, the recipe calls for [latex]50\%[/latex] pretzels, [latex]30\%[/latex] Cheerios, and [latex]20\%[/latex] peanuts. She wants to make a [latex]2[/latex] kg container of her mix. If pretzels cost [latex]\$9.99/\text{kg}[/latex], Cheerios cost [latex]\$6.99/\text{kg}[/latex], and peanuts cost [latex]\$4.95/\text{kg}[/latex], what is the average cost per [latex]100[/latex] g rounded to four decimals?
 Caruso is the marketing manager for a local John Deere franchise. He needs to compare his average farm equipment sales against his local Case IH competitor’s sales. In the past three months, his franchise has sold six [latex]\$375,000[/latex] combines, eighteen [latex]\$210,000[/latex] tractors, and fifteen [latex]\$120,000[/latex] air seeders. His sales force estimates that the Case IH dealer has sold four [latex]\$320,000[/latex] combines, twentyfour [latex]\$225,000[/latex] tractors, and eleven [latex]\$98,000[/latex] air seeders. Express the Case IH dealer’s average sales as a percentage of the John Deere dealer’s average sales.
 You are shopping for shampoo and consider two brands. Pert is sold in a bundle package of two [latex]940[/latex] mL bottles plus a bonus bottle of [latex]400[/latex] mL for [latex]\$13.49[/latex]. Head & Shoulders is sold in a bulk package of three [latex]470[/latex] mL bottles plus a bonus bottle of [latex]280[/latex] mL for [latex]\$11.29[/latex].
 Which package offers the best value?
 If the Head & Shoulders increases its package size to match Pert at the same price per mL, how much money do you save by choosing the lowest priced package?
 The following are annual net profits (in millions of dollars) over the past four years for three divisions of Randy’s Wholesale:
Cosmetics: [latex]\$4.5[/latex], [latex]\$5.5[/latex], [latex]\$5.65[/latex], [latex]\$5.9[/latex]
Pharmaceutical: [latex]\$15.4[/latex], [latex]\$17.6[/latex], [latex]\$18.5[/latex], [latex]\$19.9[/latex]
Grocery: [latex]\$7.8[/latex], [latex]\$6.7[/latex], [latex]\$9.87[/latex], [latex]\$10.75[/latex]
Rank the three divisions from best performing to worst performing based on average annual percent change.
 You are shopping for a Nintendo Wii gaming console and visit www.shop.com, which finds online sellers and lists their prices for comparison. Based on the following list, what is the average price for a gaming console (rounded to two decimals)?
Table 2.4.11 NothingButSoftware.com
$274.99
eComElectronics
$241.79
NextDayPC
$241.00
Ecost.com
$249.99
Amazon
$169.99
eBay
$165.00
Buy.com
$199.99
HSN
$299.95
Gizmos for Life
$252.90
Toys ‘R’ Us
$169.99
Best Buy
$169.99
The Bay
$172.69
Walmart
$169.00
 Juanita receives her investment statement from her financial adviser at GreatWest Life. Based on the information below, what is Juanita’s average rate of return on her investments?
Table 2.4.12 Investment Fund
Proportion of Entire Portfolio Invested in Fund
Fund Rate of Return
Real Estate
0.176
8.5%
Equity Index
0.073
36.2%
Mid Cap Canada
0.100
−1.5%
Canadian Equity
0.169
8.3%
US Equity
0.099
−4.7%
US Mid Cap
0.091
−5.7%
North American Opportunity
0.063
2.5%
American Growth
0.075
−5.8%
Growth Equity
0.085
26.4%
International Equity
0.069
−6.7%
Solutions
15. [latex]\$0.8082[/latex]
16. [latex]99.0805\%[/latex]
17a. Pert better; Pert=[latex]\$0.005916/\text{ml}[/latex]; H&S=[latex]\$0.006680/\text{ml}[/latex]
17b. Pert saves [latex]\$1.74[/latex]
18. Grocery [latex]11.2853\%[/latex]; Cosmetics [latex]9.4493\%[/latex]; Pharmaceuticals [latex]8.9208\%[/latex]
19. [latex]\$213.64[/latex]
20. [latex]5.9115\%[/latex]
1 WestJet, WestJet Fact Sheet.
THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex][latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex][latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex][latex]\begin{align*}\text{GAvg}=\left(\left[\left(1 +\%C_1\right)\times\left(1+\%C_2\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}1\right)\times 100\end{align*}[/latex]
Attribution
“3.2: Averages” from Business Math: A StepbyStep Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons AttributionNonCommercialShareAlike 4.0 International License unless otherwise noted.