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 represent a mathematical concept that is 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, also called allocation, 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 similar or different 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 liters 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. Units of measurement for each quantity in the ratio must be clear and remain consistent. Ideally, the quantities of the same type should use the same unit of measurement. If the units of measurement are the same for all terms in the ratio, we don’t have to state them. However, if they are different, say 2 cups of flour to 1.5 teaspoons of baking powder, we must state that 2 refers to cups, and 1.5 refers to teaspoons.
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. This is similar to fractions being expressed in higher and lower terms. We now apply the same knowledge to ratios to make the relationship as clear as possible.

When you simplify ratios to lower terms, remember two important rules when working with rations:

1. What you do to one term in the ratio, you must do to the other. Recall that ratios of two terms can be represented as fractions. Also recall that, when rewriting fractions into different but equivalent forms, the operation applied to the numerator must also be applied to the denominator, and vice versa. The same applies to ratios.  In other words, if a mathematical operation is performed on a term in a ratio, the same operation must also be performed on every other term in the ratio. If this rule is violated then the relationship between the terms is broken, i.e., the resulting ratio is not the same as the original ratio.
2. Integer terms are ideal. Integers are easier to understand than decimals and fractions. When rewriting ratios into equivalent forms, aim to apply the operations that will result in every term being an integer.

Important Notes

To reduce the ratio to lowest terms, one must be able to factor each term, and for that one must know the multiplication tables. However, there are a few simple tricks that we can use to check if a number is divisible by 2, 3, 4, 5, 6, 9, or 10:

• A number is divisible by 2 if it is even, i.e., it ends in 0, 2, 4, 6 or 8. (example: 2137886 is divisible by 2 because it ends with an even number, and 2137887 is not divisible by 2 because it does not end with an even number)
• A number is divisible by 3 if the sum of its digits is divisible by 3. (example: 2137887 is divisible by 3 because 2+1+3+7+8+8+7=36 is divisible by 3, but 2137886 is not divisible by 3 because 2+1+3+7+8+8+6=35 is not divisible by 3)
• A number is divisible by 4 if the last two digits of the number make a number that is divisible by 4. (example: 2137884 is divisible by 4 because 84 is divisible by 4, but 2137885 is not divisible by 4 because 85 is not divisible by 4)
• A number is divisible by 5 if it ends in 0 or 5.
• A number is divisible by 6 if it is divisible by 2 and 3, i.e., it is even and its digits add up to a number divisible by 3.
• A number is divisible by 9 if its digits add up to a number divisible by 9.
• A number is divisible by 10 if it ends in 0.

Proportions

Knowing the relationship between specific quantities is helpful, but what if the quantity you are considering does not match the specific quantity expressed in the existing ratio? For example, you know you have to maintain a ratio of 2 administrators to 5 clerks, and you are looking at hiring 30 clerks. How does that reflect on the number of administrators you require to maintain the given ratio?

A proportion is a statement of equality between two ratios. For example, the following statement is a proportion because it is a statement of equality between two equal ratios:

[latex]2:3=200:300[/latex]

Just as we have both algebraic expressions and algebraic equations, there are ratios and proportions. And just as in algebraic expressions and algebraic equations, we can have terms that involve unknowns, or variables. Because proportions are equations, if they involve an unknown variable, we may be able to solve for that variable using the algebraic operations on equations as we have before.

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.

Let’s first work with algebraic proportions of two-term ratios and involving one unknown, without a context behind the underlying ratios.

Let’s consider a general proportion statement, where three quantities are known and one is unknown:

[latex]a:b=m:n[/latex]

We can rewrite this as

[latex]\frac{a}{b}=\frac{m}{n}[/latex]

By multiplying both sides with [latex]bn[/latex], we get

[latex]a\cdot n=b\cdot m[/latex]

We can now solve this equation for whichever quantity is the unknown.

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, follow these guidelines:

If there are only two terms on each side of the proportion and only one unknown, rewrite the proportion as a linear equation and solve for the unknown.

Note that if there is more than one unknown, the proportion is not going to be uniquely solvable, i.e., there will be infinitely many possibilities for the values of the unknown, described through a relationship arrived to by solving for one of them in terms of the other.

For example, [latex]x : 5 = y : 10[/latex] is not a uniquely solvable proportion since the proportion ratios have only two terms but there are two unknowns, [latex]x[/latex] and [latex]y[/latex]. If we attempt to solve it we will get [latex]10x=5y[/latex], which can be simplified as [latex]2x=y[/latex]. This will be true for any pair of values [latex]x[/latex] and [latex]y[/latex] where [latex]y[/latex] is twice as large as [latex]x[/latex].

If there are more than two terms in the ratios in the proportion, extract a two-term ratio proportion that has only one unknown.

For example, if you have [latex]3 : x : 6 = 9 : 4 : z[/latex], you can use [latex]3 : x  = 9 : 4[/latex] to solve for [latex]x[/latex] and [latex]3 : 6 = 9 : z[/latex] to solve for [latex]z[/latex].

Things To Watch Out For

When working with proportions that have more than two terms in the ratios and extracting a two-term ratio proportion, 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. For example, consider the proportion [latex]6 : 5 : 4 = 18 : 15 : y[/latex]. You cannot select the first and third terms on the left side while selecting the second and third terms on the right side. In other words, [latex]6 : 4 \neq 15 : y[/latex] since 1st term : 3rd term ≠ 2nd term : 3rd term.

Give It Some Thought

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

2. In the following problem, which person properly executed the first step in solving proportions with more than two ratio terms?

Prorating (or Allocations)

Ratios and proportions are commonly used in various business applications. But there will be numerous situations where your business must allocate limited funds across various divisions, departments, budgets, individuals, and so on.

In the opener to this section, one example discussed the splitting of profits with your business partner, where you must distribute profits in proportion to each partner’s total investment. You invested $73,000 while your partner invested $46,000. How much of the total profits of $47,500 should you receive?

The process of prorating or allocating is the process of taking of a total quantity and allocating or distributing it proportionally. In the above example, you must take the total profits of $47,500 and distribute it proportionally with your business partner based on the investment of each partner. The proportion is:

your investment : your partner’s investment = your profit share : your partner’s profit share

This proportion has two major concerns:

1. You don’t know either of the terms on the right side. As per the rules of proportions, this makes the proportion not uniquely solvable.
2. There is a piece of information from the situation that you didn’t use at all! What happened to the total profits of $47,500?

Every prorating situation involves a hidden term. This hidden term is usually the sum of all the other terms on the same side of the proportion and represents a total. In our case, it is the $47,500 of total profits. This quantity must be placed as an extra term on both sides of the proportion to create a proportion that can actually be solved. In other words, we can rewrite the above proportion as follows:

[latex]\begin{align*} &\text{your investment} : \text{your partner's investment }:\text{total investment}\\ &=\text{your profit share} : \text{your partner's profit share}:\text{ total profit} \end{align*}[/latex]

This then allows us to calculate both of the profit shares:

[latex]\begin{align*} &\overset{\checkmark}{\text{your investment}} :\overset{?}{\text{total investment}}=\overset{?}{\text{your profit share}} : \overset{\checkmark}{\text{ total profit}}\\ &\\ \Rightarrow &\overset{\checkmark}{\text{your investment}} :(\overset{\checkmark}{\text{your investment}}+\overset{\checkmark}{\text{your partner's investment}})=\overset{?}{\text{your profit share}} : \overset{\checkmark}{\text{ total profit}}\\ &\\ \Rightarrow &(\overset{\checkmark}{\text{your investment}}+\overset{\checkmark}{\text{your partner's investment}})\cdot (\overset{?}{\text{your profit share}})=(\overset{\checkmark}{\text{your investment}})\cdot(\overset{\checkmark}{\text{ total profit}})\\ &\\ \Rightarrow & \text{your profit share}=\frac{(\overset{\checkmark}{\text{your investment}})\cdot(\overset{\checkmark}{\text{ total profit}})}{(\underset{\checkmark}{\text{your investment}}+\underset{\checkmark}{\text{your partner's investment}})} \end{align*}[/latex]

Therefore,

[latex]\begin{align*} \text{your}&\text{ profit share}\\ &=\frac{(\text{your investment})}{(\text{your investment}+\text{your partner's investment})}\cdot(\text{ total profit})\\ &\\ & =(\text{your share in the investment})\cdot({\text{ total profit}})\\ \end{align*}[/latex]

Similarly,

[latex]\begin{align*} \text{your} &\text{ partner's profit share}\\ &=\frac{({\text{your partner's investment}})\cdot({\text{ total profit}})}{({\text{your investment}}+{\text{your partner's investment}})}\\ &\\ & =(\text{your partner's share in the investment})\cdot({\text{ total profit}})\\ \end{align*}[/latex]

Do you notice a potential strategy? We started with individual investment amounts and the total profit amount and determined that each partner’s share in the profit is equal to the product of the partner’s share in the investment and the total profit. This was not a coincidence. In fact, this can be generalized to any situation where an initial ratio is given and an amount is to be shared according to that ratio.

Here is the general approach:

For example, to solve the profit-splitting scenario discussed above, let [latex]y[/latex] represent your profit share and [latex]p[/latex] represent your partner’s share. Recall that the profits of $47,500 were to be shared according to the investment ratio of:

[latex]\text{your investment} : \text{your partner’s investment}=73000 : 46000[/latex]

This means that the total investment is [latex]73000 + 46000=\$119,000[/latex]

[latex]\begin{align*} \text{your share in the investment}&=\frac{\text{your investment}}{\text{total investment}}\\ &=\frac{73000}{119000}=\frac{73}{119} \end{align*}[/latex]

[latex]\begin{align*} \text{your partner's share in the investment}&=\frac{\text{your partner's investment}}{\text{total investment}}\\ &=\frac{46000}{119000}=\frac{46}{119} \end{align*}[/latex]

Therefore,

[latex]\begin{align*} y&=(\text{your share in the investment})\cdot(\text{profit})\\ &=\frac{73}{119}\cdot 47500=29138.66 \end{align*}[/latex]

and

[latex]\begin{align*} p&=(\text{your partner's share in the investment})\cdot(\text{profit})\\ &=\frac{46}{119}\cdot 47500=18361.34 \end{align*}[/latex]

So you will receive $29,138.66 of the total profits and your partner will receive $18,361.34.