2.1 Percent Change
Percent Change
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 $359 futon for $215.40?
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 $215.40 futon is 60% 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 $359 futon is on sale at 40% off?
A percent change represents the change in the value of a quantity with respect to its original value, in percentage form.
Calculating the percent change
To calculate the percent change in a quantity, you need to know the original quantity and the new quantity.
The basic relationship is this: If you start with the original value and you increase it by a specific percent rate, then the new value will be:
[latex]\text{new} =\text{ original }+ \text{percentage of the original}[/latex]
or
[latex]\text{new} = \text{original} + (\text{percent increase rate})\cdot (\text{original})[/latex]
By rearranging the relationship above to solve for the percent rate of increase, we get
[latex]\text{percent increase rate}=\frac{\text{new}-\text{original}}{\text{original}}=\frac{\text{amount of increase}}{\text{original}}[/latex]
Similarly, if we look at decreasing the original value by a specific percent rate, the new value will be:
[latex]\text{new} =\text{ original}- \text{percentage of the original}[/latex]
or
[latex]\text{new} = \text{original } - (\text{percent decrease rate})\cdot (\text{original})[/latex]
By rearranging the relationship above to solve for the percent decrease rate, we get
[latex]\text{percent decrease rate}=-\frac{\text{new}-\text{original}}{\text{original}}=-\frac{\text{amount of decrease}}{\text{original}}[/latex]
If we think of percent rate as a positive number when the change is an increase in value, and negative when the change is a decrease in value, we can see from above that the fundamental relationship can be described as follows:
or
[latex]\text{new} = (\text{original})(1 + \text{percent change rate})[/latex]
We can calculate percent change rate by simply rearranging the basic relationship to solve for the percent change rate:
The Greek letter for D is Delta, or [latex]\Delta[/latex], and this symbol is used to represent difference or change. So we can also write this relationship as:
Example 2.1 A: Finding the percent rate in percent change
Suppose that last month your sales were $10,000, and they have risen to $15,000 this month. You wish to express the percent change in sales.
Answer:
Deliverable: percent change rate in sales from last month to this month = ?
In general,
[latex]\overset{\checkmark} {\text{new}}= \overset{\checkmark}{\text{original}} +\overset{?}{ (\text{percent change rate})}\cdot \overset{\checkmark}{(\text{original})}[/latex]
Rearranging for percent rate of change, we have
[latex]\begin{align*} \text{new} - \text{original} &= (\text{percent change rate})\cdot (\text{original})\\\\ \Rightarrow \text{percent change rate}&=\frac{\text{new}-\text{original}}{\text{original}}\\ \end{align*}[/latex]
Therefore,
[latex]\begin{align*} \text{percent change rate in sales}&=\frac{\text{sales this month}-\text{sales last month}}{\text{sales last month}}\\\\ &=\dfrac{15000-10000}{10000}=0.5=50\% \end{align*}[/latex]
This means that the sales increased by 50% month-over-month.
So far, in the questions regarding change in a quantity by a percentage, we figured out how to determine the new quantity and the percent change rate. What about finding the original quantity, given the new quantity and the percent rate of change?
The same strategy as above would apply. We start with the fundamental relationship, identify which quantities we know and which quantity we are looking for – the original value, and then rearrange the equation to solve for the unknown quantity:
[latex]\begin{align*} \overset{\checkmark} {\text{new}}&= \overset{?}{\text{original}} +\overset{\checkmark}{ (\text{percent change rate})}\cdot \overset{?}{(\text{original})}\\\\ &\Rightarrow \text{new} = (\text{original } )(1+\text{percent change rate })\\\\ &\Rightarrow \frac{\text{new}}{1+\text{percent change rate}}=\text{original } \end{align*}[/latex]
Therefore, to calculate the original amount, given the final amount and the percent change rate, we have:
Example 2.1B: Finding the original amount in percent change
Suppose that you borrowed some dollar amount and had to pay 2.55% interest. If the final amount you owed was $2510, what was the amount you borrowed?
Answer:
Deliverable: the amount borrowed
Since the amount borrowed increased to the final amount by a percentage, we have:
[latex]\begin{align*} \overset{\checkmark} {\text{final amount}}&= \overset{?}{\text{amount borrowed}} \\ &+\overset{\checkmark}{ (\text{percent increase rate})}\cdot \overset{?}{(\text{amount borrowed})}\\\\ \end{align*}[/latex]
Hence, the original amount you borrowed was $2,447.59.
To recognize problems that represent change by a percentage of the original, look for words such as:
- “percent more (than)” or “increase (from) by percent” or “grow by percent”
- “percent less (than)” or “decrease (from) by percent” or “reduce by percent” or “percent off“
Beware: A common error arises from incorrect interpretations of the word “of” versus “off”. For example, 30% of 70 is [latex]0.3\cdot 70[/latex]. However, 30% off 70 means that 70 is being reduced by 30%. In other words this is a percent change problem where 70 is the original amount, 30% is the percent change rate, and the new amount is [latex](1+0.3)\cdot 70[/latex].
Remember four critical concepts about percent change:
Be very clear on what the original amount is and what the new amount is. When solving percent change problems, take a moment to rephrase the problem, using the words “from” and “to” to ensure you know where you are starting, and where you are ending.
For example, if we know that the population of Oshawa in 2016 was 159,455 and that in 2011 it was 149,607, and we wanted to know the percent change in the population during that period, then the original amount would be the population size in 2011 and the new amount would be the population size in 2016.
Correct:
[latex]\begin{align*} \text{percent change}&=\dfrac{\text{new}-\text{original}}{\text{original}}=\dfrac{\text{pop}_{2016}-\text{pop}_{2011}}{\text{pop}_{2011}}\\\\ &=\frac{159455-149607}{149607}\\\\ &\approx 0.0658=6.58\% \end{align*}[/latex]
Incorrect:
[latex]\begin{align*} \text{percent change}&=\frac{\text{new}-\text{original}}{\text{original}}=\dfrac{\text{pop}_{2016}-\text{pop}_{2011}}{\text{pop}_{2016}}\\\\ &=\frac{159455-149607}{159455}\\\\ &\approx 0.0618=6.18\% \end{align*}[/latex]
Be very clear on whether you are talking about change in general or specifically increase or decrease. A negative change must be expressed with a negative sign or equivalent wording.
For example, if the old quantity was 20 and the new quantity is 15, this is a decrease of 5, or an amount change of [latex]15 − 20 = −5[/latex]. The percent change in this case is then [latex]\frac{{15-20}}{{20}} = -25\%[/latex].Be careful in expressing a negative percent change in words and symbols. There are two correct ways to do this properly:
-
- “The change is −25%.”
- “The amount has decreased by 25%.”
Note in the second statement that the word “decreased” replaces the negative sign. Avoid combining a negative number with the word “decreasing” or similar, as this might cause some confusion – recall that two negatives make a positive, so “the amount has decreased by −25%” would actually mean the quantity has increased by 25%.
Always clarify whether you are talking about percent change rate or percent change amount. The change amount refers to the difference of new and original amount, i.e., [latex]\text{new}-\text{original}[/latex], whereas the change rate refers to the ratio of the change amount with respect to the original amount.
Often times we refer to both as “percent change” and you are expected to determine from the context of the problem if this refers to the change rate or the change amount.
In sequential percent changes, do not simply add up the percent change rates. If the original quantity changes by a percentage and then the result changes by another percentage, you must first calculate the new amount from the first change, which then becomes the original amount in the second change.
For example, if an investment increases in value in the first year by 10% and then declines in the second year by 6%, this is not an overall increase of 10% − 6% = 4%. Why? If you originally had $100, an increase of 10% (which is $100 × 10% = $10) results in $110 at the end of the first year. You must calculate the 6% decline in the second year using the $110 balance, not the original $100. This is a decline of $110 × (−6%) = -$6.60, resulting in a final balance of $103.40. Overall, the percent change is 3.4%.
Strategy summary
To solve any question about percent change, follow these steps:
Step 1: Verify that the question is about percent change, i.e., that it involves two quantities and a percentage rate that changes one quantity to another, and identify the task.
Step 2: Set up the fundamental relationship of percent change and identify which quantity you are asked to determine and which quantities you are given.
Step 3: Rearrange the fundamental relationship equation to solve for the unknown quantity.
Step 4: Substitute the known values into the new equation to determine the value of the unknown quantity.
Example 2.1 C: Finding the new amount in percent change
Due to increased efficiencies, a company is expecting their manufacturing costs to drop over the next year from current $56,915,000 by 2.71%. What are the predicted manufacturing costs a year from now?
Answer:
Deliverable: new cost amount?
Since the new cost amount is a decrease from the current cost by a percentage,
[latex]\begin{align*} \overset{?} {\text{new cost}}&= \overset{\checkmark}{\text{current cost}} -\overset{\checkmark}{ (\text{percent decrease rate})}\cdot \overset{\checkmark}{(\text{current cost})}\\\\ &=( \text{current cost})\cdot(1- \text{percent decrease rate})\\\\ &=(56915000)(1-0.0271)\\\\ &=55372603.5 \end{align*}[/latex]
and so they are expecting their manufacturing costs a year from now to drop to $55,372,603.50.
Give It Some Thought
- 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 20%. What does this situation involve?
- Percent change
- Rate, portion, base
- Manuel had his home custom built in 2006 for $300,000. In 2014 he had it professionally appraised at $440,000. He wants to figure out how much his home has appreciated. How would he do so?
- The 2006 price is the “New,” and the 2014 price is the “Old.”
- The 2006 price is the “Old,” and the 2014 price is the “New.”
Solutions:
- (a) (the question involves how car prices have changed; note the key word “by”)
- (b) (the 2006 price is what the house used to be worth, which is the old quantity; the 2014 price represents the new value of the home)
Example 2.1 D: Price of New Cars in Canada
In 1982, the average price of a new car sold in Canada was $10,668. By 2009, the average price of a new car had increased to $25,683. By what percentage has the price of a new car changed over these years?
Answer:
Deliverable: percent change rate in the price of the new car from 1982 to 2009= ? (∆% = ?)
Since the price increased by a percentage,
[latex]\overset{\checkmark} {\text{new}}= \overset{\checkmark}{\text{original}} +\overset{?}{ (\text{percent change rate})}\cdot \overset{\checkmark}{(\text{original})}[/latex]
Rearranging for percent change rate, we have
[latex]\begin{align*} \text{new} - \text{original} &= (\text{percent change rate})\cdot (\text{original})\\ &\Rightarrow \text{percent change rate}=\frac{\text{new}-\text{original}}{\text{original}} \end{align*}[/latex]
and so
[latex]\begin{align*} \text{percent change rate in price}&=\dfrac{\text{price}_{2009}-\text{price}_{1987}}{\text{price}_{1987}}\\\\ &=\dfrac{25683-10668}{10668}\approx 1.4075=140.75\% \end{align*}[/latex]
This means that the average price of a new car sold in Canada increased by approximately 140.75% from 19912 to 2009.
Example 2.1 E: Price Changes
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 $5.99 for 240 mL. To account for cost changes, the manufacturer decided to change the price to $5.79, but also reduce the bottle size to 220 mL. What was the percent change in the price per millilitre?
Answer:
Task: percent change rate in price/ml
In general we have:
[latex]\overset{\checkmark} {\text{new}}= \overset{\checkmark}{\text{original}} +\overset{?}{ (\text{percent change rate})}\cdot \overset{\checkmark}{(\text{original})}[/latex]
Rearranging for percent change rate, we have
[latex]\begin{align*} \text{new} - \text{original} &= (\text{percent change rate})\cdot (\text{original})\\ &\Rightarrow \text{percent change rate}=\dfrac{\text{new}-\text{original}}{\text{original}} \end{align*}[/latex]
and so
[latex]\begin{align*} \text{% change rate in price/ml}&=\dfrac{\text{adjusted price/ml}-\text{reg. price per ml}}{\text{reg. price/ml}}\\\\ &=\dfrac{\dfrac{5.79}{220}-\dfrac{5.99}{240}}{\dfrac{5.99}{240}}\approx 0.054485=5.4485\% \end{align*}[/latex]
Therefore the price per mL has actually increased by 5.4485%. Note that, to the consumer, it would appear as if the price has been lowered from $5.99 to $5.79, as most consumers would not notice the change in the bottle size.
Concept Check
Try out your understanding of the relationships related to percent change
MathMatize: Percent change relationships
Rate of Change over Time
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. But 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 4,263,759 in 1996 to 5,113,149 in 2006. What annual percentage growth in population does this reflect? Notice that, by asking this question, we are not interested in calculating the change in population over the 10 years; instead we want to determine the percentage change in each of the 10 years. The rate of change over time measures the percent change in a variable per time period.
Calculating the rate of change over time
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 1996 to 1997 is based on the original population size of 4,263,759. However, the percent change from 1997 to 1998 is based on the new population figure for 1997. 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 the following the rate of change over time formula:
Rate of change per period over [latex]n[/latex] periods
[latex]\text{periodic RoC}=\sqrt[n]{\frac{\text{new}}{\text{original}}}-1[/latex]
How It Works
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 the rate of change over time formula.
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 the rate of change over time formula .
Give It Some Thought
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.
- When Peewee started five-pin bowling with the Youth Bowling Canada (YBC) in 1997, his average was 53. In 2011, he finished his last year of the YBC with an average of 248. How did his average change from 1997 to 2011?
- A stock was priced at $4.34 per share in 2006 and reached $7.15 per share in 2012. What annual return did a shareholder realize?
- In 2004, total sales reached $1.2 million. By 2010, sales had climbed to $4.25 million. What is the growth in sales per year?
Give it some Thought Answers
- Percent change; looking for overall change
- Rate of change over time; looking for change per year
- Rate of change over time; looking for change per year
Concept Check
Test your understanding of the concept of rate of change over time
MathMatize: Rate of change over time
Example 2.1 F: Annual Population Growth
Using the Toronto CMA information, where the population grew from 4,263,759 in 1996 to 5,113,149 in 2006, calculate the annual percent growth in the population.
Answer:
Task: annual percent growth = ? [latex]\rightarrow[/latex] percent change per year[latex]\rightarrow[/latex] RoC=?
In general, the periodic percent change is calculated by:
[latex]\begin{align*} \text{periodic percent change}&=\sqrt[\text{# of periods}]{\frac{\text{new amount}}{\text{original amount}}}-1\\\\ &=\left({\frac{\text{new amount}}{\text{original amount}}}\right)^{1/(\text{# of periods})}-1 \end{align*}[/latex]
Therefore
[latex]\begin{align*} \text{annual % percent change in population} &=\left({\frac{\text{pop}_{2006}}{\text{pop}_{1996}}}\right)^{\frac{1}{\text{# of years}}}-1\\\\ &=\left(\frac{5113149}{4263759}\right)^{\frac{1}{2006-1996}}-1\\ &\approx 0.018332=1.88332\% \end{align*}[/latex]
Therefore, over the 10 year span from 1996 to 2006, the CMA of Toronto grew by an average of 1.8332% per year.
Example 2.1 G: Percent Changes and Rate of Change Together
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 $113 in the second year, determine the following:
a. What is the percent change in each year if the card is valued at $1,003.33 at the end of the first year?
b. Over the course of the two years, what was the overall percent change in the value of the card?
c. What was the rate of change per year?
Answer:
a. % change year 0 [latex]\rightarrow[/latex] year 1 = ?; % change year 1 [latex]\rightarrow[/latex] year 2 =?
Conditions: year 0 to year 1 change = -84, year 1 to year 2 change = 113, value at year 1 = 1003.33
year 0 [latex]\rightarrow[/latex] year 1:
[latex]\begin{align*} \text{% change year 0 to year 1}&=\frac{\text{amount of change}}{\text{original}}=\frac{\overset{\checkmark}{\text{amount of change}}}{\underset{?}{\text{value in year 0}}} \end{align*}[/latex]
Since
[latex]\text{value in year 1}=(\text{value in year 0})+(\text{year 0 to year 1 change})[/latex]
we have that
[latex]\text{value in year 0}=(\text{value in year 1})-(\text{year 0 to year 1 change})[/latex]
Therefore,
[latex]\begin{align*} \text{% change year 0 to year 1}&=\frac{\overset{\checkmark}{\text{amount of change}}}{\underset{?}{\text{value in year 0}}}\\ &=\frac{\overset{\checkmark}{\text{amount of change}}}{\underset{\checkmark}{(\text{value in year 1})}-\underset{\checkmark}{(\text{year 0 to year 1 change})}}\\ &=\frac{-84}{1003.33-(-84)}\\ &\approx -0.077253=-7.7253\% \end{align*}[/latex]
year 1 [latex]\rightarrow[/latex] year 2:
The process is the same as above, so:
[latex]\begin{align*} \text{% change year 1 to year 2}&=\frac{\text{amount of change}}{\text{original}}\\ &=\frac{\overset{\checkmark}{\text{amount of change}}}{\underset{\checkmark}{(\text{value in year 1})}}\\ &=\frac{113}{1003.33}\approx 0.112625=11.2625\% \end{align*}[/latex]
Hence, the value of the card decreased by approximately 7.7253% over the course of first year and increased by approximately 11.2625% over the second year.
b. overall percent change rate from year 0 to year 2?
[latex]\begin{align*} \Delta_{overall}\%&=\frac{\text{amount of change}}{\text{original amount}}\\ &=\frac{\overset{\checkmark}{(\text{change from year 0 to year 1})}+\overset{\checkmark}{(\text{change from year 1 to year 2})}}{\underset{?}{\text{value in year 0}}}\\ &=\frac{\overset{\checkmark}{(\text{change from year 0 to year 1})}+\overset{\checkmark}{(\text{change from year 1 to year 2})}}{\underset{\checkmark}{(\text{value in year 1})}-\underset{\checkmark}{(\text{year 0 to year 1 change})}}\\ &=\frac{-84+113}{1003.33-(-84)}\approx 0.026671=2.6672\% \end{align*}[/latex]
c. annual RoC in value from year 0 to year 2 = ?
We have
[latex]\begin{align*} \text{RoC}&=\left(\frac{\text{value in year 2}}{\text{value in year 0}}\right)^{1/(\text{# of years})}-1\\\\ &=\left(\frac{\text{value in year 1 + change from year 1 to year 2}}{\text{value in year 1 - change from year 0 to year 1}}\right)^{1/(\text{# of years})}-1\\\\ &=\left(\frac{1003.33+113}{ 1003.33-(-84)}\right)^{1/2}-1\\\\ &\approx 0.013248=1.3248\% \end{align*}[/latex]
The value of the hockey card dropped 7.7253% in the first year and rose 11.2625% in the second year. Overall, the card rose by 2.6672% over the two years, which represents a growth of 1.3248% per year.
Section Exercises
Work on section 2.1 exercises in Fundamentals of Business Math Exercises. Discuss your solutions with your peers and/or course instructor.
You may consult answers to select exercises: Fundamentals of Business Math Exercises – Select Answers
