1.3: Ratios

You've likely heard the word "ratio" before — it is perhaps one of the most well-known and commonly use mathematical concepts. In this section we'll define what the word ratio actually means, and get practice using ratios to answer questions in various contexts.

In this section, you will learn to:

• Define, recognize, and compute ratios in various forms
• Compare different ratios
• Use ratios to describe and answer contextualized questions
• Compute two specific real-world ratios

Introduction to Ratios

Humans find it useful to compare quantities. Given two quantities, we might want to compare them by using words like larger, smaller, much less than, much more than, and so on. These types of comparisons provide some qualitative information about how two quantities can be related.

Ratios give us a more precise way to compare two quantities. For example, if you're trying to make cookies, your recipe might call for \(2\) cups of flour and \(1\) cup of sugar. In this case, you'll use "one more" cup of flour than sugar. But let's say you wanted to make a big batch -- one that used \(3\) cups of sugar instead of \(1\) cup. In this case, would you still use one more cup of flour -- in this case, \(4\) cups of flour?

If you did this — used \(4\) cups of flour for \(3\) cups of sugar — your cookies would be very sweet and be likely to burn in the oven. The reason is that quantities used in baking are proportional. When you triple the amount of sugar, you also need to triple the amount of flour! That is, you'd need \(2 \times 3 = 6\) cups of flour if you plan to use \(1 \times 3 = 3\) cups of sugar.

The commonality here is not the difference between the two quantities. Rather, it's the fact that in either case, there is twice as much flour as sugar. This is a situation in which a ratio is useful to compare two quantities.

Definition: Ratio

A ratio is a way to describe the relative size of two quantities.

There are several common ways in which ratios are represented. We will study three of them.

Name	Notation	Example
Verbal Notation	\(a\) to \(b\)	A \(2\) to \(3\) ratio
Colon Notation	\(a \colon b\)	A \(2 \colon 3\) ratio
Fraction Notation	\(\frac{a}{b} \)	A \(\frac{2}{3}\) ratio

It is important to note that these three things are different ways of expressing the same idea. However, some of them, in particular Fraction Notation, can also have slightly different meanings in other contexts.

Each of these notations has their virtues and their drawbacks. The virtue of using Fraction Notation is that it is standard across all of mathematics. The benefit of this is that there are clear rules for how to manipulate fractions, which you can rely upon to help you solve real world problems. One drawback of Fraction Notation is that using fractions tends to make it look like you are considering part of a whole, which may not always be the case when you are using ratios.

The virtue of Verbal or Colon Notation is that they both make it clear that you are comparing two quantities. For example, we might say: "at Western Oregon University the ratio of out-of-state to in-state students is about \(1 \colon 4\)." The colon makes it clear that for every out-of-state student there are four in-state students. However, if we wrote instead that the ratio is \(\frac{1}{4}\), it is tempting to jump to the conclusion that we are making some claim about one fourth of the student population. But we are not really talking about one fourth of anything! This is why you will see all of these notations represented both in this textbook and in the broader world.

It is possible for two ratios that look different to actually represent the same ratio! That happens if the numbers in one ratio can be multiplied by the same number to get the other ratio. For example, \(\frac{2}{3}\) and \(\frac{8}{12}\) represent the same ratio, because \(2 \times 4 = 8\) and \(3 \times 4 = 12\). In other words, we can take the ratio \(\frac{2}{3}\), multiply both the top and bottom by 4, and get \(\frac{8}{12}\). Thus, they are equivalent ratios.

You may also notice that if you reduce the fraction \(\frac{8}{12}\) by cancelling the common factor of \(4\) from the top and bottom, you get \(\frac{2}{3}\). That's not a coincidence! In general, to compare or easily visualize ratios, it is often helpful to write them in a common way:

Definition: Lowest Terms and Equivalent Ratios

A ratio is in lowest terms if its fractional form has been fully reduced. That is, all common whole number factors on the top and bottom of the fraction have been cancelled out.

Two ratios are equivalent ratios if, when reduced to lowest terms, they are identical. 

Let's see an example of how we can write ratios in different ways, and how to put a ratio in lowest terms.

Example \(\PageIndex{1}\)

Write the ratio \(\frac{6}{3}\) using two other types of notation. Then write \(\frac{6}{3}\) in lowest terms. Finally, write three equivalent ratios in fractional form.

Solution

First, we will write \(\frac{6}{3}\) in the two other types of notations. They are:

• \(6\) to \(3\)
• \(6 \colon 3\)

Next, we will reduce \(\frac{6}{3}\) to lowest terms. There is a common factor of \(3\) in both the top and bottom of the fraction, so when we cancel those factors out, \(\frac{6}{3}\) reduces to \(\frac{2}{1}\). Now there are no more common factors (besides \(1\), but canceling that does not change the answer). So, this ratio in lowest terms is \(\frac{2}{1}\). Note that this is probably more easily understood than \(\frac{6}{3}\) in most contexts.

Last, we'll find several equivalent ratios. To do that, we'll multiply both the top and bottom of the fraction \(\frac{6}{3}\) by the same numbers. For example, if we multiply both by \(2\), we get \(\frac{12}{6}\). This is equivalent to \(\frac{6}{3}\). We also found that \(\frac{2}{1}\) is equivalent to \(\frac{6}{3}\), since we can multiply both the top and bottom by \(\frac{1}{3}\). Said another way, you can see this by dividing both by the top and bottom numbers by \(3\). 

There are many other equivalent ratios: \(\frac{4}{2}\), \(\frac{10}{5}, \frac{18}{9}, \frac{200}{100}\) — any fraction where the top number is twice the bottom number. This is the essence of equivalent ratios: even though they have different numbers, those two numbers have the same relationship to one another.

Next, we need to understand ratios in context. See if you can answer the question below before revealing the answer!

Example \(\PageIndex{2}\)

The director of an animal shelter is writing a newsletter about the current operations at the shelter. In the animal shelter, there are \(18\) dogs and \(24\) cats. In her newsletter, she wants to list the ratio of cats to dogs in an easy-to-understand way. What sentence should she write?

Solution

Here's one possibility: the ratio of cats to dogs in the shelter is \(24 \colon 18\). (Notice that the order matters here!)

That ratio gives an accurate description, but it doesn't lead to an easily visualized comparison for the newsletter readers. So, she should put it in lowest terms. If we rewrite our ratio as \(\frac{24}{18}\), we can reduce this to lowest terms by eliminating a factor of \(6\) from both top and bottom. Then the ratio becomes \(\frac{4}{3}\), which has no more common factors to cancel.

The sentence she writes will be: "There's a \(4\) to \(3\) ratio of cats to dogs in the shelter."

Now, of course, there are many possible good answers to the question above. In this class, grading will be much more fluid than in other math classes, because often there is more than one good answer when we are asked to contextualize mathematics. You should focus on making the mathematics understandable in context. Throughout these notes you will see many more examples of how to do that.

In some contexts, ratios are rounded slightly so they are easier to understand. This happens particularly when the actual quantities being compared are large, but it's helpful to represent the ratio using smaller numbers.

Example \(\PageIndex{3}\)

According to the WOU website, as of 2023, the total enrollment is listed as 3752 students, and the student-to-faculty ratio is listed as \(13 \colon 1\). You want to know how many faculty there are at WOU. How might you find this out, and how do you explain your answer?

Solution

To figure this out, we need to find a student-to-faculty ratio that is equivalent to \(13 \colon 1\), but where the number of students is 3752. We will use the a letter, \(x\), to represent the number of faculty. That is, we want to find a value for \(x\) so that

\[3752:x \quad \text{and} \quad 13:1\]

are equivalent ratios. If these are equivalent ratios, then we must be able to multiply 13 by some number to get \(3752\). To find this number, we simply divide \(3752\div 13\), since multiplication and division are inverse operations. This gives us

\[3752 \div 13 = \frac{3752}{13} \approx 288.6\]

using a calculator, rounding to the nearest tenth. That is, \(13 \times 288.6 \approx 3752\). You can check this using a calculator.

In order for these ratios to be equivalent, we have \(1 \times 288.6 = x\). Therefore, it must be \(288.6 = x\), and so there are \(288.6\) faculty at WOU. But this answer is not a whole number, even though it is referring to a number of people. So, what does it mean to have \(288.6\) faculty?

It could mean several things. First, it could mean that some faculty are part-time, which means that they are counted as only part of a whole faculty member. But it could also mean that the ratio \(13 \colon 1\) was just slightly rounded, which is also likely. For example, say there are \(289\) faculty — then the real student to faculty ratio would be \(3752 \colon 289\), which cannot be reduced. This ratio, \(3752\) to \(289\), would not be a good ratio to put on the website, because it's hard to understand. The ratio \(13 \colon 1\), while it may have been rounded just slightly, provides a better picture of what class sizes at Western look like.

Now, in the previous example, you may have seen a quicker way to solve this problem using proportions. In general, if you find a way to solve a problem that works for you, as long as you get the same numerical answer, you can use that method. We will cover proportions in the next section. In this section, we are choosing to focus on equivalent ratios as a method for problem-solving.

Let's see one more example.

Example \(\PageIndex{4}\)

In a given afternoon, a librarian checks out \(52\) total books. Of those, \(24\) are fiction books. What is the ratio, in lowest terms, of non-fiction books to fiction books checked out of the library that afternoon?

Solution

In a given afternoon, a librarian checks out 52 total books. Of those, 24 are fiction books. What is the ratio, in lowest terms, of non-fiction books to fiction books checked out of the library that afternoon?

In this problem we need to know that library books are either fiction or non-fiction books. Therefore, the number of non-fiction books can be found by calculating \(\underset{\text{total books}}{52} - \underset{\text{fiction books}}{24} = \underset{\text{non-fiction books}}{28}\)

Thus, the ratio of non-fiction books to fiction books is \(28 \colon 24\). We see that both \(28\) and \(24\) have a factor of \(4\), so when we divide both parts of the ratio by \(4\), we get a ratio of \(7 \colon 6\). That ratio is in lowest terms, so the answer is: There was a ratio of \(7 \colon 6\) non-fiction books to fiction books checked out of the library that afternoon. 

Two Important Real-World Ratios

The remainder of this section is dedicated to two real-world ratios that you will likely encounter at some point in your life. They are:

• Your cholesterol ratio
• Your debt-to-income ratio

Before we proceed, you should know: the person writing these notes is neither a medical doctor nor a financial professional. The information in this section should not replace the advice of an actual physician or financial planner! The guidelines outlined below are based on publicly available information, and sources are cited where appropriate. In some cases some simplifications have been made for the sake of clarity.

First, we'll study the cholesterol ratio, which is used to indicate risk of heart attack and coronary disease by doctors. Cholesterol is a fat-like substance that's found in many common foods. There are two main types of cholesterol:

• HDL, or high-density lipoprotein, which is sometimes called "good cholesterol." HDL carries other cholesterol away from your arteries and to your liver, thereby reducing the risk of clogged arteries. HDL is found in foods like avocado, nuts, and vegetable-based oils. Smoking can also decrease HDL levels.
• LDL, or low-density lipoprotein, which is sometimes called "bad cholesterol." LDL is the type of cholesterol that sticks to the sides of your arteries, thereby increasing the risk of clogged arteries. LDL is found in foods with lots of saturated and trans fats, like butter, red meat, and fried foods. Smoking can also increase LDL levels.

You can get your own cholesterol tested by requesting a lipoprotein panel from a physician. In general, it is ideal to have an LDL level below 100 milligrams per deciliter (mg/dL) and an HDL level above 45 mg/dL. Additionally, your total cholesterol should be below 200 mg/dL. Various demographic factors affect your personal ideal levels.

However, since the two types of cholesterol interact, doctors will often monitor a number known as your cholesterol ratio. This is calculated in the following way:

\[\underset{\text{HDL}}{45} + \underset{\text{LDL}}{110} = \underset{\text{total cholesterol}}{155}\]

where the "total cholesterol" is calculated as

\[\text{cholesterol ratio} = \frac{\text{total cholesterol}}{\text{HDL}} = \frac{155}{45} \approx 3.44\]

Since HDL is good cholesterol, and LDL is bad cholesterol, a low cholesterol ratio is an indicator of better health and lower heart attack risk. Mathematically, we want the top of the fraction, LDL + HDL, to be relatively low compared to the bottom, which is just HDL. In general, the ideal ratio is 3.5 or lower, and should be kept below 5.

Note that the ratio has no units — it is simply a number. As with LDL and HDL recommendations, various demographic factors affect the exact recommended levels. You can find more information about this here.

Example \(\PageIndex{5}\)

You receive the results of a lipoprotein panel from your doctor. Your LDL is 110 mg/dL, and your HDL is 45 mg/dL. What is your cholesterol ratio, and what does this indicate about your risk for heart disease? Round numerical answers to two decimal places.

Solution

To calculate your cholesterol ratio, we need to know the total cholesterol. We calculate this as follows: \[\underset{\text{HDL}}{45} + \underset{\text{LDL}}{110} = \underset{\text{total cholesterol}}{155}\]

Now we calculate your cholesterol ratio as follows \[\text{cholesterol ratio} = \frac{\text{total cholesterol}}{\text{HDL}} = \frac{155}{45} \approx 3.44\]

This ratio represents a fairly low risk for heart disease, since it is below 3.5. Additionally, your total cholesterol is below 200 mg/dL, which is good. However, the LDL number is a bit high — ideally, it should be below 100 mg/dL. 

The last ratio we'll study is the debt-to-income ratio, also called DTI. This ratio is used frequently by lenders to determine if someone is eligible to borrow money for a large purchase or credit line. This is not be confused with a person's credit score. 

To calculate a person's debt-to-income ratio, we use the following formula:

\[\text{DTI }= \text{debt-to-income ratio} = \frac{\text{total monthly debt payments}}{\text{monthly gross income}}\]

Let's examine the two parts of the fraction. A person's monthly gross income is the amount of money they make in a month before any taxes or deductions, such as federal tax, state tax, or Social Security tax are taken out. This amount can be found on a pay stub, and it's likely higher than what you actually take home in your paycheck.

A person's total monthly debt payments can be calculated by adding up the debt payments they make each month to debts they have. Debt payments include things like:

• House payment (mortgage) or rent payments
• Student loan payments
• Auto loan payments
• Credit card payments
• Child support or alimony payments
• Any other type of debt: personal loan, medical loan, etc.

Other monthly payments that you might make are typically not considered debt. For example, groceries, utilities, subscriptions, and gas do not count towards your total monthly debt payments

It's important to note that different lenders use slightly different definitions of what counts as debt. The list above is intended to be as general as possible, and is what we will use in this class; however, you may encounter slightly different criteria when you actually apply for a loan. For more information, visit a financial institution's website such as this one.

Once the debt-to-income ratio is calculated, it is compared to a benchmark number. In practice, that benchmark number varies, but is typically around .36, which is what we will use in this class. If your DTI is less than .36, that means you are a good candidate for lending — your debts are relatively low compared to your income. If your DTI is more than .36, that means you are a riskier candidate for lending, because your debts are already somewhat high compared to your income.

In general, you want to keep your debt-to-income ratio as low as possible. However, some lenders accept a DTI of up to 50%, depending on the type of loan.

Example \(\PageIndex{6}\)

You currently make \(\$2800\) gross income per month. Your rent costs \(\$550\) per month, your monthly minimum credit card payments are \(\$300\), you spend \(\$350\) per month on groceries, and your car payment is \(\$150\) per month. What is your debt-to-income ratio? If you applied for a home loan, do you think you would be approved? Why or why not? 

Solution

First, we add up all of the monthly debt payments. Be careful -- not every monthly expense is a debt payment! You will need to read carefully and consult the list of debts above to tell what expenses are and aren't debts. The debts in this case are the rent, credit card payments, and car payments. The groceries are not a debt payment, so we don't include them. We add the debt payments to find the total monthly debt payments: \[\underset{\text{rent}}{\$550} + \underset{\text{credit cards}}{\$300} + \underset{\text{car loan}}{\$150} = 1000 = \text{ total monthly debt payments}\]

Now, to find the DTI, we calculate \[{DTI }= \frac{\text{total monthly debt payments}}{\text{Monthly gross income}} = \frac{\$1000}{\$2800} \approx .357\]

We see that this DTI is just slightly less than the ideal number, .36. That means that you're likely a good candidate for a home loan, but close to being a risky candidate. You are fairly likely to be approved for a home loan.

Exercises

Make sure to include supporting work for all answers

1. Consider the ratio \(6 \colon 11\).
   1. Write \(6 \colon 11\) in two other ways.
   2. Write three equivalent ratios.
   3. Is the ratio \(6\colon 11\) in lowest terms? Explain why or why not in a complete sentence.
2. In a given pond, there are 22 rock fish for every 121 minnows. Express the ratio of minnows to rock fish in lowest terms.
3. The ratio of the amount of food that my cat Jade eats to the amount of food that my cat Poe eats is \(3 \colon 2\). If Jade eats 6 ounces of food, how many ounces of food will Poe eat? (Your answer should be a whole number.)
4. Sourdough bread uses a 5 to 4 ratio of flour to sourdough starter. If you have 10 cups of flour, how much sourdough starter should you use?
5. Elizabeth's total cholesterol is 182 and her LDL is 128. What is her cholesterol ratio? How might you evaluate her overall heart health in 1-2 sentences?
6. Mark's gross monthly income is $2300. His rent is $575, his monthly car payment is $120, his utilities cost $300 per month, his student loan payment is $63 per month, and he spends $300 per month on groceries. What is his debt-to-income ratio? If he applied for a mortgage, do you think he would be approved? Why or why not?
7. Do a bit of research and write a paragraph (4-5 sentences) about what a person's credit score is, including how it is calculated and how it affects one's chances of being approved to borrow money.