2.2: Rates

We've now studied ratios and proportions, and we've seen that proportions give us a way to solve problems involving ratios. We turn now to a specific type of ratio, called a rate, and investigate some interesting applications of rates.

In this section, you will learn to:

• Recognize and compute rates and unit rates in context
• Use currency exchange rates to calculate the relative values of different types of currency
• Calculate cost per unit in context, and use cost per unit to determine which of two purchasing options is a better deal

Introduction to Rates

We'll start with a simple example that might be relevant to your daily life.

Example \(\PageIndex{1}\)

Let's say that your favorite brand of shampoo is Herbal Essences. Bi-Mart sells \(11.7\) ounces of this shampoo for \(\$5.39\). On the other hand, you can get \(29.2\) ounces of this shampoo for \(\$7.97\) on Amazon. What shampoo would give you the better deal?

Sure, the \(29.2\) ounce shampoo in the above example is more expensive than the \(11.7\) ounce bottle, but does that mean that the more expensive option gives you less value? How can you compare these two quantities when they have different prices and different volumes? Rates can help us answer this question.

Definition: Rate

A rate is a ratio in which the units of the two quantities being compared are different.

In fact, many of the ratios we have seen already are rates. For example, when we computed our hourly pay, we were actually finding a rate, because we compared dollars to hours worked. Other examples of rates include:

• Miles per gallon
• Rotations per minute
• Kilometers per hour

Do you see a pattern with all of these rates? They all have one word in common: the word per. This word, which is often ignored, means "for each" or "for every one" in this context. If you see the word "per" between two units, there's a good chance you are dealing with a rate.

In fact, most of the rates we see in our every day lives are a particular type of rate.

Definition: Unit Rate

A unit rate is a rate expressed so that the denominator of the corresponding ratio is \(1\). (Recall that the "denominator" is the bottom number of the ratio when it is expressed in fraction form.) 

Think about the "miles per gallon" example above. If we hear, "this car gets 30 miles per gallon," we know that for every 30 miles driven, the car uses 1 gallon of gas. The number 1 is implicit in the way we express most rates. Even if we are given a rate that is not a unit rate, we can convert it to a unit rate using proportions, as in the example below.

Example \(\PageIndex{2}\)

Which of the following are unit rates? For those that are not, convert them to unit rates

• \(5\) miles per hour
• \(7\) feet every \(2\) seconds

Solution

• The expression "\(5\) miles per hour" is already a unit rate, since it means "5 miles for every one hour." We can express this ratio in the following way: \[\frac{5 \text{ miles}}{1 \text{ hour}}\] The denominator of this fraction is \(1\), so this is a unit rate.
• The expression "\(7\) feet every 2 seconds" is not a unit rate. As a fraction, it is represented as follows: \[\frac{7 \text{ feet}}{2 \text{ seconds}}\] The fraction above does not have a \(1\) in the denominator, so it is not a unit rate. To convert it into a unit rate, we will set up a proportion involving the ratio above, and an equivalent ratio that does have a 1 in the denominator: \[\frac{7 \text{ feet}}{2 \text{ seconds}} = \frac{x \text{ feet}}{1 \text{ seconds}} \] \[\frac{7 \text{ feet}}{2 \text{ seconds}}\] Now we'll use Cross Multiplication and Division undoes Multiplication to solve for \(x\): \[\begin{align*} 2x &= 7 \\ x &= \frac{7}{2} \\ x & = 3.5 \end{align*} \] That means that a unit rate equivalent to "\(7\) feet every \(2\) seconds" is "\(3.5\) feet every one second." This can be rephrased as "\(3.5\) feet per second," which is more easily understood.

An alternative way to solve the second example would simply to divide \(7 \div 2 = 3.5\). The fact that these alternative ways give the same answer serves to illustrate how useful our method of solving proportions is in different contexts.

Currency Exchange Rates

We'll start by discussing a bit of background about world currencies and how they are used.

There are over \(160\) national currencies in the world. When you travel between two countries that use different currencies, it is necessary to exchange some of one type of currency for the other. This is particularly true in countries where cash is used more frequently that credit/debit cards. It's always a good idea when traveling internationally to carry a small amount of that country's currency in either coins or bills. You can get currency from other countries by ordering it through a bank ahead of your travel. You can also exchange currency at a currency exchange business in the airport, port, or border crossing when you are traveling, though these methods usually have fees associated with them and can be exploitative. In some places, ATMs can be used to dispense currency as well.

But there is a big question here: how much of the other type of currency will you get when you exchange? Most countries do not use the United States Dollar or an equivalent, so this question can be hard to answer. For example, Mexico uses Mexican Pesos as its national currency. At the time this book was written, \(1\) Mexican Peso was worth about \(6\) US cents according to a reliable currency exchange website. That doesn't meant that product cost significantly more or less in either place. The costs of items are scaled to reflect the fact that one Mexican Peso has a relatively smaller value than one US Dollar. For example, a taco plate in Mexico might cost \(90\) Mexican pesos, which is equivalent to \(15\) US Dollars. When the individual units of currency have relatively smaller value, the sticker price on the item goes up. The opposite is also true -- if a currency has a higher value, the items priced in that currency tend to have lower prices. This can be very confusing for travelers.

What's more, these exchange rates change from day to day, depending on international economic factors. At any time, it is possible to look up the exchange rate using the internet, or by going through a bank and asking what exchange rate they are using. There may even be slight discrepancies between these numbers. In this text, we'll use exchange rates that are current at the time of writing according to Google's currency exchange rate lookup feature.

Since currency exchange rates are, at their most basic mathematical level, ratios, we can use our techniques from earlier chapters to solve currency exchange problems. In this section we will avoid using the \(\$\) sign and instead write out "USD" for US Dollar amounts. This helps avoid confusion with symbols, since other countries also use the \(\$\) symbol to mean the currency of that country.

Let's look at a simple example. Please note that example values in this section may not be current as this book is only updated periodically.

Example \(\PageIndex{3}\)

Each Japanese Yen is worth \(.0093\) USD. Prior to a trip to Japan, you want to exchange \(250\) US Dollars for Yen. How many Japanese Yen will you get in this exchange? Round to the nearest whole.

Solution

We are given the exchange rate of \(1\) Japanese Yen to \(.0093\) US Dollars. Before we start, let's think for a second about this rate: it means that each Yen is equivalent to less than \(1\) US cent, because \(1\) US cent is \(.01\) US dollars, and\(.0093\) is just slightly less than \(.01\). In forming this observation, we can conclude that \(250\) USD will give us a very large amount of Yen, since each USD will be equivalent to over \(100\) Yen.

To actually solve this problem, we set up a proportion that compares Yen to USD, using \(x\) to stand for the quantity we want to find:

\[\frac{1 \text{ Yen}}{.0093 \text{ USD}} = \frac{x \text{ Yen}}{250 \text{ USD}}\]

Notice that we've got the exchange rate we are given on the left, and on the right we've got what we're trying to find. The units are labeled so that we know everything is in the correct place. Now we simply solve this proportion:

\[\begin{align*} \frac{1}{.0093} &= \frac{x}{250} \\ .0093x &= 250 \\ x & = \frac{250}{.0093} = 26,881.72 \end{align*}\]

Rounding to the nearest whole, we find that we get \[26,882\] Japanese Yen. This may seem like an enormous amount, but based on our observation before starting this problem, it makes sense that the number of Yen is large.

Currency exchange rate problems can always be solved using a method similar to the one above. Depending on the information you are given, your unknown may occur in a different place, but our techniques for solving proportions allow us to find any unknown value, so long as we have all of the other information.

What's the Better Deal?

To introduce this last application of rates, we'll start with an example. Note that throughout this section, we will use the \(\$\) sign again, and all prices are assumed to be in US Dollars. The next example should look familiar to you.

Example \(\PageIndex{4}\)

Let's say that your favorite brand of shampoo is Herbal Essences. Bi-Mart sells \(11.7\) ounces of this shampoo for \(\$5.39\). On the other hand, you can get \(29.2\) ounces of this shampoo for \(\$7.97\) on Amazon. What shampoo would give you the better deal?

Solution

Before attempting to solve this problem, let's understand what is being asked. For our purposes, we consider a choice to be the "best deal" if the cost per unit of the product is lower. In this case, the unit that is used to measure the shampoo is ounces. Therefore, this question could be rephrased as: which of these two purchasing options has the lowest cost per ounce?

In order to solve this question, we want to determine the cost per ounce of each possible purchase. In other words, we want to find the unit rate of cost for each purchase, and the lower unit rate will correspond to the better deal.

For the \(11.7\) ounce shampoo you can buy at Bi-Mart, we can calculate the cost per ounce by dividing the cost by the number of ounces:

\[\frac{\$5.39}{11.7 \text{ ounces}} \approx \$0.46 \text{ per ounce}\]

For the \(\$7.97\) ounce shampoo available on Amazon, we calculate the cost per ounce in the same way:

\[\frac{\$7.97}{29.2 \text{ ounces}} \approx \$0.27 \text{ per ounce}\]

In summary, the \(11.7\) ounce bottle costs \(\$0.46\) per ounce, and the \(29.2\) ounce bottle cost \(\$0.27\) per ounce. That means that the \(29.2\) ounce bottle is a better deal, since it costs less per ounce.

You may have approached the problem above in a different way, by noticing that if you bought two of the \(11.7\) ounce bottles, you'd only have \(23.4\) ounces of shampoo, but have paid \(2 \times \$5.39 = \$10.78\), whereas you could have paid \(\$7.97\) for more shampoo, so Amazon is a better deal. That would be a reasonable way to approach this problem, but not all situations are as easy to calculuate in your head.

These sorts of questions can get trickier if the units involved are not immediately obvious. Let's see another example.

Example \(\PageIndex{5}\)

A pizza parlor sells circular pizzas. A large pizza costs \(\$12\), and a family size pizza costs \(\$16\). A large pizza has a \(12\) inch diameter, and a family size pizza has a \(14\) inch diameter. Which pizza is the better deal?

Solution

This problem looks almost too easy at first — certainly the \(12\)-inch pizza is better because it costs a dollar per inch, and the other one costs more per inch!

But wait: what is an "inch" of pizza? That's not a relevant measurement here. The more important unit to consider, when thinking about pizza, is the number of square inches of pizza you get. The area of the pizza is what matters most, not the diameter.

Our first step will therefore be to calculate the areas of the pizza involved. In order to do that, you need to know the area of a circle. The area \(A\) of a circle with radius \(r\) is given by \[A = \pi r^2\] where \(\pi\) is a special constant number related to circles. While \(\pi\) is irrational, meaning a decimal that goes on forever without repeating, it is usually safe to use \(\pi \approx 3.14\) in calculations.

So we proceed to calculate the areas of the two pizzas. The large pizza has a \(12\) inch diameter. The diameter is twice the radius, so this pizza has a \(6\) inch radius. We can find area of the large pizza by calculating \[A_{\text{large}} = \pi r^2 = \pi \cdot (6^2) = 36 \pi \approx 113 \text{ square inches}\]

Similarly, the diameter of the family-size pizza is \(14\) inches, so its radius is \(7\) inches. Therefore, the area of the family-size pizza is given by \[A_{\text{family}} = \pi r^2 = \pi \cdot (7^2) = 49 \pi \approx 154 \text{ square inches}\]

Now that we've got the areas of both pizzas, we can ask: what is the cost per square inch of each pizza? The pizza with the lower cost per square inch will be a better deal.

We find the cost per square inch by dividing the cost of the pizza by the number of square inches. For the large pizza, the cost per square inch is \[\frac{\$12}{113 \text{ square inches}} = \$.106 \text{ per square inch}\]

For the family size pizza, the cost per square inch is \[\frac{\$16}{154 \text{ square inches}} = \$.104 \text{ per square inch}\]

Comparing these two numbers, we see that the \(\$16\) pizza, which was the \(14\) inch family size pizza, has a slightly lower cost per square inch. Therefore, the family size \(14 \)inch pizza is the better deal.

This example illustrates two things:

• It's important to think critically about the question to find the right interpretation. In this case we needed to first find the area of each pizza to make sense of the problem.
• Sometimes cost per unit amounts can be very close, and it's necessary to include many digits following the decimal point. Here it was necessary to go to the third decimal place to see which number was larger.

Keep these things in mind as you answer questions that ask: what is the better deal?

Exercises

1. Write down at least three other examples of rates that were not discussed in this particular chapter.
2. Is 8 meters every 5 seconds a unit rate? If not, convert it to a unit rate.
3. Is 6 beats per minute a unit rate? If not, convert it to a unit rate.
4. Suppose that the exchange rate for US Dollars to Moroccan Dirhams is \(1\) to \(9.67\).
   1. Before a trip to Morocco you exchange 300 US Dollars for Moroccan Dirhams. How many Dirhams do you get?
   2. At the end of the trip, you have 45 Moroccan Dirhams left. If you exchange these for US Dollars, how many Dollars will you get back?
5. You are told that \(1\) Swiss Franc is worth \(1.04\) US Dollars, and that \(1\) Russian Ruble is worth \(.014\) US Dollars. On a trip from Switzerland to Russia, you exchange \(50\) Swiss Francs for Russian Rubles. How many Rubles do you get? Round to the nearest whole. [Hint: you can do two separate currency exchange calculations.]
6. Suppose that \(32\) ounces of soda costs \(\$1.69\), and \(54\) ounces of the same soda costs \(\$3.47\). Which is the better deal and why?
7. You are purchasing land on which to build a home. There are two rectangular lots to choose from. Lot A is \(80\) feet by \(110\) feet, and costs \(\$94,000\). Lot B is \(70\) feet by \(120\) feet, and costs \(\$87,000\). Assuming that all other characteristics of the properties are equally desirable, and you are willing to accept either size lot, which is the better deal and why? [Hint: a rectangle's area is calculated by its length times its width.]