Basic calculations are so much a part of your everyday life that you could not escape them if you tried. While driving to school, you may hear on the radio that today’s forecasted temperature of 27°C is 3°C warmer than the historical average. In the news, consumers complain that gas prices rose 11% in three months. A commercial tells you that the Toyota Prius you are driving uses almost 50% less gas than the Honda Civic Coupe Si. After class you head to Anytime Fitness to cancel a gym membership because you are too busy to work out. You have used only four months of an annual membership for which you paid $449, and now you want to know how large a prorated refund you are eligible for.

On a daily basis you use basic calculations such as averages (the temperature), percent change (gas prices), ratios and proportions (energy efficiency), and prorating (the membership refund). To invest successfully, you must also apply these basic concepts. Or if you are an avid sports fan, you need basic calculations to understand your favourite players’ statistics.

And it’s not hard to see how these calculations would be used in the business world. Retail management examines historical average daily sales to predict future sales and to schedule the employees who will service those sales. Human resource managers continually calculate ratios between accounting and performance data to assess how efficiently labour is being used. Managers proration a company’s budget across its various departments.

This chapter covers universal business mathematics you will use whether your chosen business profession is marketing, accounting, production, human resources, economics, finance, or something else altogether. To be a successful manager you need to understand percent changes, averages, ratios, proportions, and prorating.

2.1 Percent Change

On your way to work, you notice that the price of gasoline is about 10% higher than it was last month. At the office, reports indicate that input costs are down 5.4% and sales are up 3.6% 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 40% off this week; Safeway’s ad says that Tuesday is 10% off day; and a Globe and Mail story informs you that the cost of living has risen by 3% 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 6.7% 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.

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 expresses in percentage form how much any quantity changes from a starting period to an ending period.

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 3.1 (on the next page) represents how to express the change as a percentage.

Remember two critical concepts about percent change:

1. 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 100%, not 200%. For example, if the old quantity was 25 and the new quantity is 50, note that the quantity has doubled. However, 25 of the final 50 comes from the original amount and therefore does not count toward the change. We subtract it out of the numerator through calculating 50 − 25 = 25. Therefore, the change as a percentage is The same applies to a tripling of quantity. If our new quantity is 75 (which is triple the old quantity of 25), then % The original value of 25 is once again subtracted out of the numerator. The original 100% is always removed from the formula.
2. Negative Changes. 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 a change of 15 − 20 = −5. The percent change is . Be careful in expressing a negative percent change. There are two correct ways to do this properly:
   1. “The change is −25%.”
   2. “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 −25%” would actually mean the quantity has increased by 25%.

Percent Change

*** QuickLaTeX cannot compile formula:
\frac{{

*** Error message:
File ended while scanning use of \frac .
Emergency stop.

Important Notes

To access the percent change function on your BAII Plus calculator, press 2nd ∆% located above the number 5 on your keypad. Always clear the memory of any previous question by pressing 2nd CLR Work once the function is open. Use the ↑ and ↓ 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 ∆% in its percent format without the % sign
• #PD = Number of consecutive periods for change. By default, it is set to 1. 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.

1. Rates versus Percent Changes. Sometimes you may be confused about whether questions involve rates (Section 2.3) 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, “x has increased by y%” and “x is y% more than last year” both represent a percent change.
2. 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 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%.

Paths To Success

A percent change extends the rate, portion, and base calculations introduced in Section 2.3. 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.

Give It Some Thought

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 20%. What does this situation involve?
   1. Percent change
   2. Rate, portion, base
2. 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?
   1. The 2006 price is the “New,” and the 2014 price is the “Old.”
   2. The 2006 price is the “Old,” and the 2014 price is the “New.”

Solutions:

1. a (the question involves how car prices have changed; note the key word “by”)
2. 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)

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. 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 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.

The Formula

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 Formula 3.2.

Important Notes

On your calculator, calculate the rate of change over time using the percent change (∆%) function. Previously, we had ignored the #PD variable in the function and it was always assigned a value of 1. In rate of change, this variable is the same as n in our equation. Therefore, if our question involved 10 changes, such as the annual population change of the Toronto CMA from 1996 to 2006, then this variable is set to 10.

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:

1. If you are looking for the overall rate of change from beginning to end, you need to calculate the percent change.
2. 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 n = 1. Thus you can solve any percent change question using Formula 3.2 instead of Formula 3.1.

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.

1. 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?
2. 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?
3. 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?

Solutions:

1. Percent change; looking for overall change
2. Rate of change over time; looking for change per year
3. Rate of change over time; looking for change per year

2.2 Averages

No matter where you go or what you do, averages are everywhere. Let’s look at some examples:

• Three-quarters 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 125 mL size for $1.99 while a comparable brand sells for $1.89 for 110 mL. Which is the better deal? A fair comparison requires you to calculate the average price per millilitre.
• Your local transit system charges $2.25 for an adult fare, $1.75 for students and seniors, and $1.25 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 $8,000 in Roller Coasters Inc. The stock value has changed by 9%, −7%, 13%, 4%, and −2% 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 year-over-year sales? One approach would involve taking company sales for each of the 52 weeks in the current year and comparing these with the sales of all 52 weeks from last year. This involves 104 weekly sales figures with 52 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:

1. The data itself—you need the value for each piece of data.
2. 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.

Simple Average

As expressed in Formula 3.3, you calculate a simple average by adding together all of the pieces of data then taking that total and dividing it by the quantity.

Important Notes

Although mentioned earlier, it is critical to stress that a simple average is calculated only when all of the following conditions are met:

1. All of the data shares the same level of importance toward the calculation.
2. All of the data appear the same number of times.
3. 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.

Give It Some Thought

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 15, 30, 40, and 45 is: a. lower than 20. b. between 20 and 40, inclusive. c. higher than 40.
2. If the simple average of three pieces of data is 20, which of the following data do not belong in the data set? Data set: 10, 20, 30, 40 a. 10 b. 20 c. 30 d. 40

Solutions:

1. b (a simple average should fall in the middle of the data set, which appears spread out between 15 and 45, so the middle would be around 30)
2. d (if the number 40 is included in any average calculation involving the other numbers, it is impossible to get a low average of 20)

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 one-hour 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:

1. The data itself—you need the value for each piece of data.
2. 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).

The Formula

As expressed in Formula 3.4, 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.

Weighted Average

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.

Give It Some Thought

In each of the following, determine which information is the data and which is the weight.

1. Rafiki operates a lemonade stand during his garage sale today. He has sold 13 small drinks for $0.50, 29 medium drinks for $0.90, and 21 large drinks for $1.25. What is the average price of the lemonade sold?
2. Natalie received the results of a market research study. In the study, respondents identified how many times per week they purchased a bottle of Coca-Cola. Calculate the average number of purchases made per week.

How do you average a percent change? If sales increase 100% this year and decrease 50% next year, is the average change in sales an increase of per year? The answer is clearly “no.” If sales last year were $100 and they increased by 100%, that results in a $100 increase. The total sales are now $200. If sales then decreased by 50%, you have a $100 decrease. The total sales are now $100 once again. In other words, you started with $100 and finished with $100. That is an average change of nothing, or 0% 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.

The Formula

In business mathematics, you most commonly use a geometric average to average a series of percent changes. Formula 3.5 is specifically written to address this situation.

Geometric Average

Important Notes

A critical requirement of the geometric average formula is that every (1 + Δ%) expression must result in a number that is positive. This means that the Δ% cannot be a value less than -100% else Formula 3.5 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 +100% and -50% is 25%, and the geometric average is 0%. This characteristic can be used as an error check when you perform these types of calculations.

Give It Some Thought

For the first three questions, determine whether you should calculate a simple, weighted, or geometric average.

1. Randall bowled 213, 245, and 187 in his Thursday night bowling league and wants to know his average.
2. Cindy invested in a stock that increased in value annually by 5%, 6%, 3%, and 5%. She wants to know her average increase.
3. A retail store sold 150 bicycles at the regular price of $300 and 50 bicycles at a sale price of $200. The manager wants to know the average selling price.
4. Gonzalez has calculated a simple average of 50% and a geometric average of 60%. He believes his numbers are correct. What do you think?

Solutions:

1. Simple; each item has equal importance and frequency.
2. Geometric; these are a series of percent changes on the price of stock.
3. Weighted; each item has a different frequency.
4. At least one of the numbers is wrong since a geometric average is always smaller than or equal to the simple average

2.3 Ratios, Proportions, and Prorating

You and your business partner have a good problem: Consumers are snapping up packets of your new eucalyptus loganberry facial scrub as fast as you can produce them. Each packet of the scrub contains 600 mg of loganberry extract and 80 mg of eucalyptus oil, as well as water and clay and other ingredients. Ratios are invaluable in understanding the relationship among different quantities, such as how much of each ingredient you need.

You have no trouble obtaining the water and clay, but the loganberry extract and eucalyptus oil are in short supply because of poor weather. In any time period, your suppliers can provide seven times as much loganberry extract as eucalyptus oil. To figure out which ingredient is limiting your production, you need proportions.

Sometimes you need to relate a proportion to the total of the quantities. This requires prorating skills. For example, once your business has grown, you start using a production line, which follows the common practice of producing more than one product. The papaya facial scrub you have introduced recently and the eucalyptus loganberry scrub require the same amount of production time. Your equipment capacity is 1,000 units per day. How many units of each type of scrub must you produce to meet market demand in the ratio of nine to two?

Years later, you are splitting the profits of your business partnership in proportion to each partner’s total investment. You invested $73,000, while your partner invested $46,000. With total profits of $47,500, what is your share?

Understanding the relationships among various quantities and how the components relate to an overall total emphasizes the need for understanding ratios, proportions, and prorating.

What Is a Ratio?

A ratio is a fixed relationship between two or more quantities, amounts, or sizes of a similar nature. For a ratio to exist, all terms involved in the ratio must be nonzero. Examine the criteria of this definition more closely:

1. There Must Be Two or More Quantities. A ratio does not exist if only one quantity is involved. For example, the fuel tank on your Mustang takes 60 litres of gasoline. This is not a ratio, as there is no relationship to any other quantity, amount, or size. On the other hand, if you compare your fuel tank to the fuel tank of your friend’s Hummer, you now have two quantities involved and could say that her fuel tank has twice the capacity of yours.
2. All Terms Must Be of a Similar Nature. In all of the examples provided note that all quantities, amounts, or sizes are based on the same unit. In the cosmetics formulation it was milligrams to milligrams; in the production line, it was units to units; in the investment scenario, it was dollars to dollars. For a ratio to have meaning and to be properly interpreted, all terms of the ratio must be expressed in a similar nature. When you place different units such as kilometres and metres into the same ratio, the result is confusing and will lead to misinterpretation of the relationship.
3. All Terms Must Be Nonzero. The numbers that appear in a ratio are called the terms of the ratio. If we have a recipe with four cups of flour to one cup of sugar, there are two terms: four and one. If any term is zero, then the quantity, amount, or size does not exist. For example, if the recipe called for four cups of flour to zero cups of sugar, there is no sugar! Therefore, every term must have some value other than zero.

Let’s continue using the example of four cups of flour to one cup of sugar. Business ratios are expressed in five common formats, as illustrated in the table below.

All of these formats work well when there are only two terms in the ratio. If there are three or more terms, ratios are best expressed in the colon format. For example, if the recipe called for four parts of flour to one part of sugar to two parts of chocolate chips, the ratio is 4 : 1 : 2. The fraction, decimal, and percent forms do not work with three or more terms.

Simplification and Reduction of Ratios

When a ratio is used to express a relationship between different variables, it must be easy to understand and interpret. Sometimes when you set up a ratio initially, the terms are difficult to comprehend. For example, what if the recipe called for 62½ parts flour to 25 parts sugar? That is not very clear. Expressing the same ratio another way, you can say the recipe requires 5 parts flour to 2 parts sugar. Note how the relationship is clearer in the latter expression. Either way, though, both ratios mean the same thing; in decimal format this ratio is expressed as a value of 2.5. Recall that Section 2.2 discussed how fractions are expressed in higher and lower terms. We now apply the same knowledge to ratios to make the relationship as clear as possible.

When you reduce ratios to lower terms, remember two important characteristics involving the cardinal rule and integers:

1. The Cardinal Rule. Recall from Section 2.5 that the cardinal rule of algebra states, “What you do to one you must do to the other.” In other words, whatever mathematical operation is performed on a term in a ratio must be equally performed on every other term in the ratio. If this rule is violated then the relationship between the terms is broken.
2. Maintaining Integers. Integers are easier to understand than decimals and fractions. In reducing a ratio to lower terms, aim to maintain every term as an integer (much as in Section 2.2).

Important Notes

To factor any term, recall your multiplication tables. Assume the ratio is 36 : 24. It is always best to pick the lowest term in the ratio when factoring. Looking at the 24 and recall what multiplies together to arrive at 24. In this case, your factors are 1 × 24, 2 × 12, 3 × 8, and 4 × 6. The goal is to find the largest value among these factors that also divides evenly into the other term of 36. The largest factor that makes this true is 12. Therefore, perform step 4 in our reduction process by dividing every term by a factor of 12, resulting in a ratio of 3 : 2.

Things To Watch Out For

Always ensure that before you apply any of the reduction steps your relationship meets the requirements of being a ratio in the first place. For example, the expression of 10 km : 500 m is not a 1 : 50 ratio since it violates the “similar nature” characteristic of the ratio definition. You need to convert the metres into kilometres, producing 10 km : 0.5 km. Now that you have a ratio, the reduction to lowest terms results in a 20 : 1 ratio. This is very different than 1 : 50! The lesson learned is to make sure you are working with a proper ratio before you manipulate it.

Paths To Success

In the fourth step of the procedure, do not feel compelled to find what is called the “magic factor.” This is the single factor that reduces the ratio to its lowest terms in a single calculation. Although this is nice if it happens, you may as well find any factor that will make the ratio smaller and easier to work with.

• For example, assume a ratio of 144 : 72 : 96. It may not be very apparent what factor goes into each of these terms on first glance. However, they are all even numbers, meaning they can all be divided by 2. This produces 72 : 36 : 48.
• Once again, the “magic factor” may not be clear, but every term is still even. Divide by 2 again, producing 36 : 18 : 24.
• If no magic factor is apparent, again note that all numbers are even. Divide by 2 yet again, producing 18 : 9 : 12.
• The common factor for these terms is 3. Divided into every term you have 6 : 3 : 4. There is no common factor for these terms, and the ratio is now in its lowest terms.

In the above example it took four steps to arrive at the answer—and that is all right. Did you notice the “magic factor” that could have solved this in one step? You can find it if you multiply all of the factors you have: 2 × 2 × 2 × 3 = 24. Dividing every term in the original ratio by 24 produces the solution in one step. If you did not notice this “magic factor,” there is nothing wrong with taking four steps (or more!) to get the answer. In the end, both methods produce the same solution.

Reducing a Ratio to the Smallest Term of One

Your goal in reducing a ratio is to make it easier to understand. Sometimes, even after you have applied the techniques for reducing the ratio, the end result is still hard to grasp. Look at part (d) of Example 3.3A. You arrived at a final solution of 41 : 55, with no further reduction possible. Think of this as 41 cups of flour to 55 cups of sugar. The relationship is not very clear.

In these circumstances, although integers are preferable in general, you must reduce the ratio further, reintroducing decimals to make the relationship more comprehensible. This means you apply a technique called “reducing the ratio to the smallest term of one.” In this technique, the smallest term in the ratio will have a value of 1 once you perform the ratio reduction and simplification.

Important Notes

You may have to make a judgment call when you decide whether to leave a reduced ratio alone or to reduce it to a ratio where the smallest term is one. There is no clear definitive rule; however, keep the following two thoughts in mind:

1. For purposes of this textbook, you are provided with clear instructions on how to handle the ratio, allowing everyone to arrive at the same solution. Your instructions will read either “Reduce the ratio to its smallest terms” or “Reduce the ratio to the smallest term of one.”
2. In the real world, no such instructions exist. Therefore you should always base your decision on which format is easier for your audience to understand. If the reduced fraction leaves your audience unable to understand the relationship, then use the reduction to a ratio with the smallest term of one instead.

What Are Proportions?

Knowing the relationship between specific quantities is helpful, but what if your quantity happens not to match the specific quantity expressed in the existing ratio? Example 3.3B illustrated that you need to learn how to apply the ratio to meet current conditions. The ratio in the inventory system was expressed in terms of 27 Nintendo Wiis; however, the shelf displayed 38 units. How can we relate an existing ratio to a needed ratio?

A proportion is a statement of equality between two ratios. Just as we have both algebraic expressions and algebraic equations, there are ratios and proportions. With algebraic expressions, only simplification was possible. When the expression was incorporated into an algebraic equation, you solved for an unknown. The same is true for ratios and proportions. With ratios, only simplification is possible. Proportions allow you to solve for any unknown variable. The Wii and PlayStation example could have been set up as follows:

Proportions

A proportion must adhere to three characteristics, including ratio criteria, order of terms, and number of terms.

• Characteristic #1: Ratio Criteria Must Be Met. By definition, a proportion is the equality between two ratios. If either the left side or the right side of the proportion fails to meet the criteria for being a ratio, then a proportion cannot exist.
• Characteristic #2: Same Order of Terms. The order of the terms on the left side of the proportion must be in the exact same order of terms on the right side of the proportion. For example, if your ratio is the number of MP3s to CDs to DVDs, then your proportion is set up as follows:

MP3 : CD : DVD = MP3 : CD : DVD

• Characteristic #3: Same Number of Terms. The ratios on each side must have the same number of terms such that every term on the left side has a corresponding term on the right side. A proportion of MP3 : CD : DVD = MP3 : CD is not valid since the DVD term on the left side does not have a corresponding term on the right side.

When you work with proportions, the mathematical goal is to solve for an unknown quantity or quantities. In order to solve any proportion, always obey the following four rules:

1. Rule #1: At Least One Value for Any Term Is Known. At least one of the left side or right side values for each term must be known. For example, x : 5 = y : 10 is not a solvable proportion since the corresponding first terms on both sides are unknown. However, 15 : 5 = y : 10 is solvable since at least one of the first corresponding terms (the 15 and the y) is known.
2. Rule #2: One Pair of Corresponding Terms Must Be Known. At least one pair of corresponding terms on the left side and right side must have both quantities known. For example, 3 : x : 6 = y : 4 : z is not a solvable proportion since there is no pair of first terms (3 and y), second terms (x and 4), or third terms (6 and z) that produces a pair of known values. However, 3 : x : 6 = 9 : 4 : z is a solvable proportion since the first terms on both sides (the 3 and 9) are known.
3. Rule #3: Obey BEDMAS and Perform Proper Algebraic Manipulation. To manipulate a proportion, you must satisfy the rules of BEDMAS (Section 2.1) and all of the rules of algebra (Sections 2.4 and 2.5). Violating any of these rules breaks the equality of the ratios and produces an incorrect proportion.
4. Rule #4: Use the Fractional Format. The fractional format for ratios is recommended for solving a proportion. The other four formats generally make solving the proportion much more difficult, and the mathematical operations required become unclear.

Things To Watch Out For

You must always pick terms from the same positions on both sides of the proportion. Otherwise, you will violate the equality of the proportion, since the terms are no longer in the same order on both sides and Characteristic #2 is not satisfied. For example, in working with the proportion 6 : 5 : 4 = 18 : 15 : y from above, you cannot select the first and third terms on the left side and also select the second and third terms on the right side. In other words, 6 : 4 ≠ 15 : y since 1st term : 3rd term ≠ 2nd term : 3rd term.

Paths To Success

It is always easiest to solve a proportion when the unknown variable is in the numerator. This characteristic requires minimal algebra and calculations to isolate the variable. If you find yourself with an unknown variable in the denominator, you can mathematically invert the fraction on both sides, since this obeys the cardinal rule of “what you do to one, you must do to the other.” When you invert, the numerator becomes the denominator and vice versa. For example, if the proportion is , inversion then produces a proportion of 2.09 and later!

Notice that isolating the unknown variable in the inverted proportion requires only a multiplication of 12 on both sides. This is a lot less work!

Give It Some Thought

1. Some of the following proportions violate the characteristics or rules of proportions. Examine each and determine if all the rules and characteristics are met. If not, identify the problem.

2. In the following solved proportions, which person properly executed step 2 of the proportion steps?