1.9 Percent Change
Learning Objectives
By the end of this section, you will be able to:
- Make calculations with the Percent Change Formula
- Make calculations with the Rate of Change Over Time Formula
For this section you will need the following:
Symbols Used
- [latex]\%C=[/latex] percent change
- [latex]V_\textrm{i}=[/latex] original value
- [latex]V_\textrm{f}=[/latex] updated value
- [latex]RoC=[/latex] rate of change
- [latex]n=[/latex] total number of time periods
Formulas Introduced
-
Formula 1.9a – Percent Change
[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex]
-
Formula 1.9b – Rate of Change Over Time
[latex]\begin{align*}RoC=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}}\right)\times 100\end{align*}[/latex]
Introduction
On your way to work, you notice that the price of gasoline is about [latex]10\%[/latex] higher than it was last month. At the office, reports indicate that input costs are down [latex]5.4\%[/latex] and sales are up [latex]3.6\%[/latex] over last year. Your boss asks you to analyze the year-over-year change in industry sales and submit a report. During your coffee break, you look through the day’s e-flyers in your inbox. Home Depot is advertising that all garden furniture is [latex]40\%[/latex] off this week; Safeway’s ad says that Tuesday is [latex]10\%[/latex] off day; and a Globe and Mail story informs you that the cost of living has risen by [latex]3\%[/latex] since last year. You then log in to your financial services account, where you are happy to find that the mutual funds in your RRSP are up [latex]6.7\%[/latex] from last year. What are you going to do with all this information?
Understanding how data changes from one period to the next is a critical business skill. It allows for quick assessment as to whether matters are improving or getting worse. It also allows for easy comparison of changes in different types of data over time. In this section, the concept of percent change is explored, which allows for the calculation of change between two points in time. Then a rate of change over time is introduced, which allows you to determine the change per period when multiple points in time are involved in the calculation.
Make calculations with the Percent Change Formula
It can be difficult to understand a change when it is expressed in absolute terms. Can you tell at a glance how good a deal it is to buy a [latex]\$359[/latex] futon for [latex]\$215.40[/latex]? It can also be difficult to understand a change when it is expressed as a percentage of its end result. Are you getting a good deal if that [latex]\$215.40[/latex] futon is [latex]60\%[/latex] of the regular price? What most people do find easier to understand is a change expressed as a percentage of its starting amount. Are you getting a good deal if that [latex]\$359[/latex] futon is on sale at [latex]40\%[/latex] off? A percent change expresses in percentage form how much any quantity changes from a starting period to an ending period.
Formula 1.9a: Percent Change
[latex]{\color{red}{\%C}}\text{ is Percent Change:}[/latex] The change in the quantity is always expressed in percent format.
[latex]{\color{blue}{V_\text{f}}}\text{ is Final Quantity Value:}[/latex] This is the value that the quantity has become, or the number that you want to compare against a starting point.
[latex]{\color{green}{V_\text{i}}}\text{ is Initial Quantity Value:}[/latex] This is the value that the quantity used to be, or the number that you want all others to be compared against. Notice how the formula is structured:
1. First calculate “[latex]V_\textrm{f}–V_\textrm{i}[/latex],” the change in the quantity. This is the numerator.
2. Divide the change by “[latex]V_\textrm{i}[/latex]” to express the quantity change first as a fraction, then convert it into its decimal format by performing the division.
[latex]{\color{chocolate}{\times100}}\text{ is Percent Conversion:}[/latex] Recall from Section 1.8 that you convert a decimal to a percentage by multiplying it by [latex]100[/latex]. As the language suggests, percent changes are always percentages; therefore, you must include this component in the formula.
—
To calculate the percent change in a variable, you need to know the starting number and the ending number. Once you know these, Formula 1.9a represents how to express the change as a percentage. Remember two critical concepts about percent change:
Do Not Include the Original Quantity in the Change
Percent change represents the changes in the quantities, not the values of the quantities themselves. The original quantity does not count toward the percent change. Therefore, if any quantity doubles, its percent change is [latex]100\%[/latex], not [latex]200\%[/latex]. For example, if the old quantity was [latex]25[/latex] and the new quantity is [latex]50[/latex], note that the quantity has doubled. However, [latex]25[/latex] of the final [latex]50[/latex] comes from the original amount and therefore does not count toward the change. We subtract it out of the numerator through calculating [latex]50−25=25[/latex]. Therefore, the change as a percentage is:
The same applies to a tripling of quantity. If our new quantity is [latex]75[/latex] (triple the old quantity of [latex]25[/latex]), then:
The original value of [latex]25[/latex] is once again subtracted out of the numerator. The original [latex]100\%[/latex] is always removed from the formula.
Negative Changes
[latex]\begin{align*}\frac{15-20}{20}×100=-25\%\end{align*}[/latex].
Be careful in expressing a negative percent change. There are two correct ways to do this properly:
-
- “The change is [latex]−[/latex]25%.”
- “It has decreased by 25%.”
Note in the second statement that the word “decreased” replaces the negative sign. To avoid confusion, do not combine the negative sign with the word “decreased” — recall that two negatives make a positive, so “It has decreased by [latex]−25\%[/latex]” would actually mean the quantity has increased by [latex]25%[/latex].
How to
Solve Any Question About Percent Change
Step 1: Notice that there are three variables in the formula. Identify the two known variables and the one unknown variable.
Step 2: Solve for the unknown variable using Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex].
Assume last month your sales were [latex]\$10,000[/latex], and they have risen to [latex]\$15,000[/latex] this month. You want to express the percent change in sales.
The known variables are [latex]V_\textrm{i}=\$10,000[/latex] and [latex]V_\textrm{f}=\$15,000[/latex]. The unknown variable is percent change, or [latex]\%C[/latex].
Using Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex]:
[latex]\begin{align*}\%C=\frac{\$15,000-\$10,000}{\$10,000}\times 100=0.5\times 100 =50\%\end{align*}[/latex]
Observe that the change in sales is [latex]\$5,000[/latex] month-over-month. Relative to sales of [latex]\$10,000[/latex] last month, this month’s sales have risen by [latex]50\%[/latex].
Key Concepts
To access the percent change function on your BAII Plus calculator, press [latex]\text{2nd}\Delta\%[/latex] located above the number [latex]5[/latex] on your keypad. Always clear the memory of any previous question by pressing 2nd CLR Work once the function is open. Use the ↑ (up) and ↓ (down) arrows to scroll through the four lines of this function. To solve for an unknown variable, key in three of the four variables and then press Enter. With the unknown variable on your display, press CPT. Each variable in the calculator is as follows:
- OLD = The old or original quantity; the number that represents the starting point
- NEW = The new or current quantity; the number to compare against the starting point
- %CH = The percent change, or [latex]\%C[/latex] in its percent format without the [latex]\%[/latex] sign
- #PD = Number of consecutive periods for change. By default, it is set to [latex]1[/latex]. For now, do not touch this variable. Later in this section, when we cover rate of change over time, this variable will be explained.
Things To Watch Out For
Watch out for two common difficulties involving percent changes.
- Rates versus Percent Changes. Sometimes you may be confused about whether questions involve rates (Section 3.1) or percent changes. Recall that a rate expresses the relationship between a portion and a base. Look for some key identifiers, such as “is/are” (the portion) and “of” (the base). For percent change, key identifiers are “by” or “than.” For example, “[latex]x[/latex] has increased by [latex]y\%[/latex]” and “[latex]x[/latex] is [latex]y\%[/latex] more than last year” both represent a percent change.
- Mathematical Operations. You may imagine that you are supposed to add or subtract percent changes, but you cannot do this. Remember that percentages come from fractions. According to the rules of algebra, you can add or subtract fractions only if they share the same base (denominator). For example, if an investment increases in value in the first year by [latex]10\%[/latex] and then declines in the second year by [latex]6\%[/latex], this is not an overall increase of [latex]10\%−6\%=4\%[/latex]. Why? If you originally had [latex]\$100[/latex], an increase of [latex]10\%[/latex] (which is [latex]\$100\times 10\%=\$10[/latex]) results in [latex]\$110[/latex] at the end of the first year. You must calculate the [latex]6\%[/latex] decline in the second year using the [latex]\$110[/latex] balance, not the original [latex]\$100[/latex]. This is a decline of [latex]\$110\times (−6\%)=-\$6.60[/latex], resulting in a final balance of [latex]\$103.40[/latex]. Overall, the percent change is [latex]3.4\%[/latex].
A percent change extends the rate, portion, and base calculations introduced in Section 1.8. The primary difference lies in the portion. Instead of the portion being a part of a whole, the portion represents the change of the whole. Putting the two formulas side by side, you can calculate the rate using either approach.
[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\end{align*}[/latex]
Try It
1) It has been five years since Juan went shopping for a new car. On his first visit to a car lot, he had sticker shock when he realized that new car prices had risen by about [latex]20\%[/latex]. What does this situation involve?
- Percent Change
- Rate, portion, Base
Solution
a. (the question involves how car prices have changed; note the keyword “by”)
Try It
2) Manuel had his home custom built in [latex]2006[/latex] for [latex]\$300,000[/latex]. In [latex]2014[/latex] he had it professionally appraised at [latex]\$440,000[/latex]. He wants to figure out how much his home has appreciated. How would he do so?
- The [latex]2006[/latex] price is [latex]V_\textrm{f}[/latex] and the [latex]2014[/latex] price is [latex]V_\textrm{i}[/latex].
- The [latex]2006[/latex] price is [latex]V_\textrm{i}[/latex] and the [latex]2014[/latex] price is the [latex]V_\textrm{f}[/latex].
Solution
b. (the [latex]2006[/latex] price is what the house used to be worth, which is [latex]V_\textrm{i}[/latex]; the [latex]2014[/latex] price represents the new value of the home, or [latex]V_\textrm{f}[/latex]).
Example 1
In [latex]1982[/latex], the average price of a new car sold in Canada was [latex]$10,668[/latex]. By [latex]2009[/latex], the average price of a new car had increased to [latex]\$25,683[/latex]. By what percentage has the price of a new car changed over these years?
Solution
Step 1: What are you looking for?
You are trying to find the percent change in the price of the new car, or [latex]\%C[/latex].
Step 2: What do you already know?
You know the old and new prices for the cars: [latex]V_\textrm=\$10,668[/latex] and [latex]V_\textrm{f}=\$25,683[/latex]. You also know that you can apply Formula 3.2a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex] to find [latex]\%C[/latex].
Step 3: Make substitutions using the information known above.
[latex]\begin{align*}\%C&=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\\[1ex]\%C&=\frac{\$25,683-\$10,668}{\$10,668}\times 100\\[1ex]\%C&=\frac{\$15,015}{\$10,668}\times 100\\[1ex]\%C&=140.748\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
From [latex]1982[/latex] to [latex]2009[/latex], average new car prices in Canada have increased by [latex]140.748\%[/latex].
Example 2
When you purchase a retail item, the tax increases the price of the product. If you buy a [latex]\$799.00[/latex] Bowflex Classic Home Gym machine in Ontario, it is subject to [latex]13\%[/latex] HST. What amount do you pay for the Bowflex including taxes?
Solution
Step 1: What are you looking for?
You are looking for the price of the Bowflex after increasing it by the sales tax. This is the “Final” price for the Bowflex.
Step 2: What do you already know?
You know the original price of the machine and how much to increase it by: [latex]V_\textrm{i}=\$799.00[/latex] and [latex]\%C=13\%[/latex]. You also know that, given these values, you can apply Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex] to find the value of [latex]V_\textrm{f}[/latex].
Step 3: Make substitutions using the information known above.
[latex]\begin{align*}\%C&=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\\[1ex]13\%&=\frac{V_\textrm{f}-\$799}{\$799}\times 100\\[1ex]0.13&=\frac{V_\textrm{f}-\$799}{\$799}\\[1ex]\$103.87&=V_\textrm{f}-\$799\\\$902.87&=V_\textrm{f}\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
The price of a Bowflex, after increasing the price by the taxes of [latex]13\%[/latex], is [latex]\$902.87[/latex].
Example 3
Consumers often object to price changes in many daily products, even though inflation and other cost changes may justify these increases. To reduce the resistance to a price increase, many manufacturers adjust both prices and product sizes at the same time. For example, the regular selling price for a bottle of shampoo was [latex]\$5.99[/latex] for [latex]240\;\text{mL}[/latex]. To account for cost changes, the manufacturer decided to change the price to [latex]\$5.79[/latex], but also reduce the bottle size to [latex]220\;\text{mL}[/latex]. What was the percent change in the price per millilitre?
Solution
Step 1: What are you looking for?
You need to find the percent change in the price per millilitre, or [latex]\%C[/latex].
Step 2: What do you already know?
You know the old price and bottle size, as well as the planned price and bottle size:
[latex]\begin{align*}\text{Old price}&=\$5.99&\text{Old size}=240mL\\\text{New price}&=\$5.79&\text{New size}=220mL\end{align*}[/latex]
You can convert the price and size to a price per millilitre by taking the price and dividing by the size, and then find the percent change per millilitre by applying Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex].
Step 3: Make substitutions using the information known above.
First, find price per millilitre for Old and New prices / sizes:
[latex]\begin{align*}\frac{\text{Old price}}{\text{Old size}}=\frac{\$5.99}{240mL}=\$0.024958/mL\end{align*}[/latex]
[latex]\begin{align*}\frac{\text{New price}}{\text{New size}}=\frac{\$5.79}{220mL}=\$0.026318/mL\end{align*}[/latex]
Next, substitute into Formula 1.9a:
[latex]\begin{align*}\%C&=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\\[1ex]\%C&=\frac{\$0.026318- \$0.024958}{\$0.024958}\times 100\\[1ex]\%C&=\frac{\$0.001359}{\$0.024958}\times 100\\[1ex]\%C&=5.4485\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
The price per mL has increased by [latex]5.4485\%[/latex]. Note that to the consumer, it would appear as if the price has been lowered from [latex]\$5.99[/latex] to [latex]\$5.79[/latex], as most consumers would not notice the change in the bottle size.
Make calculations with the Rate of Change Over Time Formula
The percent change measures the change in a variable from start to end overall. It is based on the assumption that only a single change occurs. What happens when the ending number results from multiple changes and you want to know the typical value of each change? For example, the population of the Toronto census metropolitan area (CMA) has grown from [latex]4,263,759[/latex] in [latex]1996[/latex] to [latex]5,113,149[/latex] in [latex]2006[/latex]. What annual percentage growth in population does this reflect? Notice that we are not interested in calculating the change in population over the [latex]10[/latex] years; instead, we want to determine the percentage change in each of the [latex]10[/latex] years. The rate of change over time measures the percent change in a variable per time period.
[latex]\boxed{3.2\text{b}}[/latex] Rate of Change Over Time
[latex]{\color{purple}{n}}\text{ is Total Number of Periods:}[/latex] The total number of periods reflects the number of periods of change that have occurred between the Old and New quantities.
[latex]{\color{red}{RoC}}\text{ is Rate of Change per time period:}[/latex] This is a percentage that expresses how the quantity is changing per time period. It recognizes that any change in one period affects the change in the next period.
[latex]{\color{green}{V_\textrm{f}}}\text{ is Final Quantity Value:}[/latex] What the quantity has become.
[latex]{\color{blue}{V_\textrm{i}}}\text{ is Initial Quantity Value:}[/latex] What the quantity used to be.
[latex]{\color{Mahogany}{\times 100}}\text{ is Percent Conversion:}[/latex] Rates of change over time are always expressed as percentages.
Calculating the rate of change over time is not as simple as dividing the percent change by the number of time periods involved, because you must consider the change for each time period relative to a different starting quantity. For example, in the Toronto census example, the percent change from [latex]1996[/latex] to [latex]1997[/latex] is based on the original population size of [latex]4,263,759[/latex]. However, the percent change from [latex]1997[/latex] to [latex]1998[/latex] is based on the new population figure for [latex]1997[/latex]. Thus, even if the same number of people were added to the city in both years, the percent change in the second year is smaller because the population base became larger after the first year. As a result, when you need the percent change per time period, you must use Formula 3.2b.
How to
Work With Rate of Change Over Time
When you work with any rate of change over time, follow these steps:
Step 1: Identify the three known variables and the one unknown variable.
Step 2: Solve for the unknown variable using Formula 1.9b[latex]\begin{align*}RoC=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}}\right)\times 100\end{align*}[/latex].
Key Concepts
On your calculator, calculate the rate of change over time using the percent change ([latex]\%C[/latex]) function. Previously, we had ignored the #PD variable in the function and it was always assigned a value of [latex]1[/latex]. In rate of change, this variable is the same as [latex]n[/latex] in our equation. Therefore, if our question involved [latex]10[/latex] changes, such as the annual population change of the Toronto CMA from [latex]1996[/latex] to [latex]2006[/latex], then this variable is set to [latex]10[/latex].
Paths To Success
You may find it difficult to choose which formula to use: percent change or rate of change over time. To distinguish between the two, consider the following:
- If you are looking for the overall rate of change from beginning to end, you need to calculate the percent change.
- If you are looking for the rate of change per interval, you need to calculate the rate of change over time.
Ultimately, the percent change formula is a simplified version of the rate of change over time formula where [latex]n = 1[/latex]. Thus you can solve any percent change question using Formula 1.9b[latex]\begin{align*}RoC=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}}\right)\times 100\end{align*}[/latex] instead of Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex].
Try It
For each of the following, distinguish whether you should solve the question by the percent change formula or the rate of change over time formula.
3) When Peewee started five-pin bowling with the Youth Bowling Canada (YBC) in [latex]1997[/latex], his average was [latex]53[/latex]. In [latex]2011[/latex], he finished his last year of the YBC with an average of [latex]248[/latex]. How did his average change from [latex]1997[/latex] to [latex]2011[/latex]?
Solution
Percent change (looking for an overall change)
Try It
4) A stock was priced at [latex]\$4.34[/latex] per share in [latex]2006[/latex] and reached [latex]\$7.15[/latex] per share in [latex]2012[/latex]. What annual return did a shareholder realize?
Solution
Rate of change over time (looking for change per year)
Try It
5) In [latex]2004[/latex], total sales reached [latex]\$1.2\;\text{million}[/latex]. By [latex]2010[/latex], sales had climbed to [latex]\$4.25\;\text{million}[/latex]. What is the growth in sales per year?
Solution
Rate of change over time (looking for change per year)
Example 4
Using the Toronto CMA information, where the population grew from [latex]4,263,759[/latex] in [latex]1996[/latex] to [latex]5,113,149[/latex] in [latex]2006[/latex], calculate the annual percent growth in the population.
Solution
Step 1: What are you looking for?
We need to calculate the percent change per year, which is the rate of change over time, or [latex]RoC[/latex].
Step 2: What do you already know?
We know the starting and ending numbers for the population along with the time frame involved.
[latex]\begin{align*}V_\textrm{i}&=4,263,759\\V_\textrm{f}&=5,113,149\\n&=2006-1996=10\text{ years}\end{align*}[/latex]
Step 3: Make substitutions using the information known above.
[latex]\begin{align*}RoC&=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}} - 1\right)\times 100\\[1ex]RoC&=\left(\left(\frac{5,113,149}{4,263,759}\right)^{\frac{1}{10}}-1\right)\times 100\\[1ex]RoC&=\left(1.199211^{\frac{1}{10}}-1\right)\times 100\\RoC&=\left(1.018332-1\right)\times 100\\RoC&=0.018332\times 100\\RoC&=1.8332\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
Over the [latex]10[/latex] year span from [latex]1996[/latex] to [latex]2006[/latex], the CMA of Toronto grew by an average of [latex]1.8332\%[/latex] per year.
Example 5
Kendra collects hockey cards. In her collection, she has a rookie year Wayne Gretzky card in mint condition. The book value of the card varies depending on demand for the card and its condition. If the estimated book value of the card fell by $84 in the first year and then rose by [latex]\$113[/latex] in the second year, determine the following:
- What is the percent change in each year if the card is valued at [latex]\$1,003.33[/latex] at the end of the first year?
- Over the course of the two years, what was the overall percent change in the value of the card?
- What was the rate of change per year?
Solution
Step 1:What are you looking for?
We need to provide four answers to the questions and find the percent change in [latex]\text{Year}\;1[/latex], or [latex]\%C_1[/latex], then the percent change in [latex]\text{Year}\;2[/latex], or [latex]\%C_2[/latex]. Using these first two solutions, we calculate both the overall percent change across both years, or [latex]\%C_\textrm{overall}[/latex], and the rate of change per year, or [latex]RoC[/latex].
Step 2: What do you already know?
We know the price of the card at the end of the first year as well as how it has changed each year.
[latex]\begin{align*}\text{Price at end of first year}&=\$1,003.33\\\text{Price change in first year}&=-\$84\\\text{Price change in second year}&=\$113\end{align*}[/latex]
Furthermore, we know that we can find the percent change using Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex] to answer questions a and b, and use Formula 1.9b[latex]\begin{align*}RoC=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}}\right)\times 100\end{align*}[/latex] to find the rate of change over time for question c.
Step 3: Make substitutions using the information known above.
First, calculate the price at the beginning of the first year:
[latex]\begin{align*}V_\textrm{f1}&=V_\textrm{i1}+\text{Change}_1\\\$1,003.33&=V_\textrm{i1}-\$84.00\\\$1,087.33&=V_\textrm{i1}\end{align*}[/latex]
For the percent change in [latex]\text{Year}\;1[/latex], apply Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex]:
[latex]\begin{align*}\%C_1&=\frac{V_\textrm{f1}-V_\textrm{i1}}{V_\textrm{i1}}\times 100\\[1ex]\%C_1&=\frac{\$1,003.33-\$1,087.33}{\$1,087.33}\times 100\\[1ex]\%C_1&=-7.7253\%\end{align*}[/latex]
Calculate the price at the end of the second year:
[latex]\begin{align*}V_\textrm{f2}&=V_\textrm{i2}+\text{Change}_2\\V_\textrm{f2}&=\$1,003.33+\$113.00\\V_\textrm{f2}&=\$1,116.33\end{align*}[/latex]
For the percent change in [latex]\text{Year}\;2[/latex], apply Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex]:
[latex]\begin{align*}\%C_2&=\frac{V_\textrm{f2}-V_\textrm{i2}}{V_\textrm{i2}}\times 100\\[1ex]\%C_2&=\frac{\$1,116.33-\$1,003.33}{\$1,003.33}\times100\\[1ex]\%C_2&=11.2625\%\end{align*}[/latex]
For the overall percent change, take the old price at the beginning of the first year and compare it to the new price at the end of the second year. Apply Formula 1.9a[latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex]:
[latex]\begin{align*}\%C_\textrm{overall}&=\frac{V_\textrm{f2}-V_\textrm{i1}}{V_\textrm{i1}}\times 100\\[1ex]\%C_\textrm{overall}&=\frac{\$1,116.33-\$1,087.33}{\$1,087.33}\times100\\[1ex]\%C_\textrm{overall}&=2.6671\%\end{align*}[/latex]
Calculate the rate of change over the two years using the same two prices, but apply Formula 1.9b[latex]\begin{align*}RoC=\left(\left(\frac{V_\textrm{f}}{V_\textrm{i}}\right)^{\frac{1}{n}}\right)\times 100\end{align*}[/latex].
[latex]\begin{align*}RoC&=\left(\left(\frac{V_\textrm{f2}}{V_\textrm{i1}}\right)^{\frac{1}{n}} - 1\right)\times 100\\[1ex]RoC&=\left(\left(\frac{\$1,116.33}{\$1,087.33}\right)^{\frac{1}{2}} - 1\right)\times 100\\[1ex]RoC&=1.3248\%\end{align*}[/latex]
Step 4: Provide the information in a worded statement.
The value of the hockey card dropped [latex]7.7253\%[/latex] in the first year and rose [latex]11.2625\%[/latex] in the second year. Overall, the card rose by [latex]2.6671\%[/latex] across both years, which represents a growth of [latex]1.3248\%[/latex] in each year.
Exercises: Mechanics
For questions 1–3, solve for the unknown (?) using Formula 1.9a [latex]\begin{align*}\%C=\frac{V_\textrm{f}-V_\textrm{i}}{V_\textrm{i}}\times 100\end{align*}[/latex] (percent change).
| Table 1.9.1 | |||
| Old | New | ∆% | |
| 1) | $109.95 | $115.45 | ? |
| 2) | ? | $622.03 | 13.25% |
| 3) | 5.94% | ? | −10% |
4) If [latex]\$9.99[/latex] is changed to [latex]\$10.49[/latex], what is the percent change?
5) [latex]\$19.99[/latex] lowered by [latex]10\%[/latex] is what dollar amount?
6) What amount when increased by [latex]40\%[/latex] is[latex]\$3,500[/latex]?
7) If [latex]10,000[/latex] grows to [latex]20,000[/latex] over a period of 10 years, what is the annual rate of change?
Odd Answers
1) [latex]5.0023\%[/latex]
3) [latex]5.346\%[/latex]
5) [latex]\$17.99[/latex]
7) [latex]7.1773\%[/latex]
Exercises: Applications
8) How much, including taxes of [latex]12\%[/latex], would you pay for an item with a retail price of [latex]\$194.95[/latex]?
9) From September 8, 2007 to November 7, 2007, the Canadian dollar experienced a rapid appreciation against the US dollar, going from [latex]\$0.9482[/latex] to [latex]\$1.1024[/latex]. What was the percent increase in the Canadian dollar?
10) From 1996 to 2006, the “big three” automakers in North America (General Motors, Ford, and Chrysler) saw their market share drop from [latex]71.5\%[/latex] to [latex]52.7\%[/latex]. What is the overall change and the rate of change per year?
11) The average price of homes in Calgary fell by [latex]\$10,000[/latex] to [latex]\$357,000[/latex] from June 2009 to July 2009. The June 2009 price was [latex]49\%[/latex] higher than the June 2005 price.
a) What was the percent change from June 2009 to July 2009?
b) What was the average price of a home in June 2005?
c) What was the annual rate of change from June 2005 to June 2009?
12) On October 28, 2006, Saskatchewan lowered its provincial sales tax (PST) from [latex]7\%[/latex] to [latex]5\%[/latex]. What percent reduction does this represent?
13) A local Superstore sold [latex]21,983[/latex] cases of its President’s Choice cola at [latex]\$2.50[/latex] per case. In the following year, it sold [latex]19,877[/latex] cases at [latex]\$2.75[/latex] per case.
a) What is the percent change in price year-over-year?
b) What is the percent change in quantity year-over-year?
c) What is the percent change in total revenue year-over-year? (Hint: revenue = price × quantity)
14) A bottle of liquid laundry detergent priced at [latex]\$16.99[/latex] for a [latex]52[/latex]-load bottle has been changed to [latex]\$16.49[/latex] for a [latex]48[/latex]-load bottle. By what percentage has the price per load changed?
Odd Answers
9) [latex]16.2624\%[/latex]
11a) [latex]−2.7248\%[/latex]
11b) [latex]\$246,308.73[/latex]
11c) [latex]10.4833\%[/latex]
13a) [latex]10\%[/latex]
13b) [latex]−9.5801\%[/latex]
13c) [latex]−0.5381\%[/latex]
Exercises: Challenge, Critical Thinking, & Other Applications
15) At a boardroom meeting, the sales manager is happy to announce that sales have risen from [latex]\$850,000[/latex] to [latex]\$1,750,000[/latex] at a rate of [latex]4.931998\%[/latex] per year. How many years did it take for the sales to reach [latex]\$1,750,000[/latex]?
16) The Nova Scotia Pension Agency needs to determine the annual cost of living adjustment (COLA) for the pension payments made to its members. To do this, it averages the consumer price index (CPI) for both the previous fiscal year and the current fiscal year. It then calculates the percent change between the two years to arrive at the COLA. If CPI information is as follows, determine the COLA that the pensioners will receive.
| Table 1.9.2 | |||||||
| Previous Fiscal Year | Current Fiscal Year | ||||||
| Nov. | 109.2 | May | 112.1 | Nov. | 111.9 | May | 114.6 |
| Dec. | 109.4 | June | 111.9 | Dec. | 112.0 | June | 115.4 |
| Jan. | 109.4 | July | 112.0 | Jan. | 111.8 | July | 115.8 |
| Feb. | 110.2 | Aug. | 111.7 | Feb. | 112.2 | Aug. | 115.6 |
| Mar. | 111.1 | Sep. | 111.9 | Mar. | 112.6 | Sep. | 115.7 |
| Apr. | 111.6 | Oct. | 111.6 | Apr. | 113.5 | Oct. | 114.5 |
17) During The Bay’s warehouse clearance days, it has reduced merchandise by [latex]60\%[/latex]. As a bonus, today is Scratch ’n’ Save day, where you can receive up to an additional [latex]25\%[/latex] off the reduced price. If you scratched the maximum of [latex]25\%[/latex] off, how many dollars would you save off an item that is regularly priced at [latex]\$275.97[/latex]? What percent savings does this represent?
18) Federal Canadian tax rates for 2010 and 2011 are listed below. For example, you pay no tax on income within the first bracket, [latex]15\%[/latex] on income within the next bracket, and so on. If you earned [latex]\$130,000[/latex] in each year, by what percentage did your federal tax rate change? In dollars, what was the difference?
| Table 1.9.3 | ||
| 2010 Tax Brackets | Taxed at | 2011 Tax Brackets |
| $0–$10,382 | 0% | $0–$10,527 |
| $10,383–$40,970 | 15% | $10,528–$41,544 |
| $40,971–$81,941 | 22% | $41,545–$83,088 |
| $81,942–$127,021 | 26% | $83,089–$128,800 |
| $127,022+ | 29% | $128,801+ |
19) Melina is evaluating two colour laser printers for her small business. A Brother model is capable of printing [latex]21[/latex] colour pages per minute and operates [latex]162.5\%[/latex] faster than a similar Hewlett-Packard model. She needs to print [latex]15,000[/latex] pages for a promotion. How much less time (stated as a percentage) will it take on the Brother model?
20) A chocolate bar has been priced at [latex]\$1.25[/latex] for a [latex]52[/latex] gram bar. Due to vending machine restrictions, the manufacturer needs to keep the price the same. To adjust for rising costs, it lowers the weight of the bar to [latex]48[/latex] grams.
a) By what percentage has the price per gram changed?
b) If this plan is implemented over two periods, what rate of change occurs in each period?
Odd Answers
15) [latex]15\;\text{years}[/latex]
17) [latex]\text{Amount saved}=\$193.18[/latex]; [latex]\%C=70.0004\%[/latex]
19) [latex]62.5\%\;\text{less time}[/latex]
