The best examples of rational expressions: 3 practical examples you’ll actually use

Starting with real examples of rational expressions

Before we talk about rules, let’s jump straight into situations. A rational expression is just a fraction where the top and bottom are polynomials (no square roots of x, no trig, just regular algebraic expressions). For instance, something like \(\frac{2x+1}{x-3}\) is a classic example of a rational expression.

But that’s the textbook version. Let’s anchor this in real life first. Here are some quick examples of rational expressions you might bump into without realizing it:

• The formula \(\frac{\text{distance}}{\text{time}}\) becomes \(\frac{50x}{x-2}\) if distance and time depend on some variable \(x\).
• A phone plan cost like \(\frac{40 + 0.05n}{n}\), where \(n\) is the number of gigabytes you use.
• A medical dosage per pound of body weight might look like \(\frac{75m}{w}\), where \(m\) is milligrams and \(w\) is weight in pounds.

These are all examples of rational expressions because they are “polynomial over polynomial.” Now let’s slow down and walk through 3 practical examples in detail, then build out more variations.

Example of a rational expression in speed and travel

Imagine you’re planning a road trip. You drive at a speed that depends on traffic conditions, which we’ll call \(v\) miles per hour. Your time to travel a fixed distance, say 180 miles, can be written as the rational expression

[
T(v) = \frac{180}{v}.
]

Here:

• The numerator 180 is a constant polynomial.
• The denominator \(v\) is a first-degree polynomial.

So \(T(v)\) is a simple but powerful example of a rational expression. It tells you how your travel time changes as your speed changes.

Now let’s make it more realistic.

Adding a delay: a more practical version

Suppose there’s always a 30-minute (0.5 hour) delay for traffic and rest stops. Then your total time is

[
T(v) = \frac{180}{v} + 0.5.
]

This is still built around a rational expression, and it’s exactly the kind of formula transportation planners and mapping apps use behind the scenes.

If you want examples of rational expressions: 3 practical examples, this is our first big one:

• Travel time as a function of speed: \(T(v) = \frac{180}{v}\).
• Travel time with a fixed delay: \(T(v) = \frac{180}{v} + 0.5\).

You can even compare two drivers. Suppose one driver averages \(v\) mph and another averages \(v+10\) mph. The time saved is

[
\text{Time saved} = \frac{180}{v} - \frac{180}{v+10},
]

another excellent example of a rational expression that has a very intuitive meaning: “how much faster is it if I drive a bit quicker?”

Money and budgeting: examples of rational expressions in everyday costs

Let’s move to money, because nothing makes math feel more real than seeing it affect your wallet.

Streaming subscription example

Say you and your friends share a streaming service that charges \(20 per month plus \)3 per user for extra profiles. If \(n\) is the number of users, then the cost per person is

[
C(n) = \frac{20 + 3n}{n}.
]

This is a very natural, very real example of a rational expression:

• The numerator \(20 + 3n\) is a polynomial.
• The denominator \(n\) is a polynomial.

We can simplify this:

[
C(n) = \frac{20}{n} + 3.
]

Now the meaning is clear:

• Everyone pays a fixed $3 for their own profile.
• The $20 base fee is split as \(\frac{20}{n}\) among all users.

As \(n\) gets larger, \(\frac{20}{n}\) gets smaller, which is a nice way to see how sharing reduces cost.

Phone data plan example

Here’s another money-based example of a rational expression. Suppose a phone plan charges a \(15 monthly access fee plus \)0.10 per gigabyte of data, and you want the cost per gigabyte. If \(g\) is the number of gigabytes used, then

[
P(g) = \frac{15 + 0.10g}{g}.
]

Again, this is a rational expression, and simplifying it gives

[
P(g) = \frac{15}{g} + 0.10.
]

This tells you:

• There’s a baseline 10 cents per gigabyte.
• The access fee spreads out over your usage, decreasing per gigabyte as \(g\) increases.

Both of these are strong examples of rational expressions: 3 practical examples type problems that teachers love to assign, because they’re easy to picture and they build good algebra habits.

Health and science: rational expressions that show up in data

Rational expressions aren’t just for classroom word problems. They show up in real scientific work, especially when you’re looking at rates.

Dosage per body weight

Health professionals often think in terms of “milligrams per kilogram” or “milligrams per pound.” While you should always rely on medical experts and trusted sources like the National Institutes of Health or Mayo Clinic, the underlying math often looks like this.

Suppose a medication guideline says “3 milligrams per pound of body weight,” and someone weighs \(w\) pounds. The total dose is

[
D(w) = 3w.
]

If we want the dose per pound but there’s also a fixed base amount (say 50 mg to get started), then

[
R(w) = \frac{50 + 3w}{w}.
]

This is a rational expression:

[
R(w) = \frac{50}{w} + 3.
]

Here are two more health-related examples of rational expressions:

• Concentration of a drug in the bloodstream over time, simplified in a basic model, might look like \(C(t) = \frac{100}{t+1}\), where \(t\) is time in hours.
• Body mass index (BMI), used by organizations like the CDC, is given by
  [
  \text{BMI} = \frac{703w}{h^2},
  ]
  where \(w\) is weight in pounds and \(h\) is height in inches. That’s a rational expression in the variable \(h\).

These real examples include the same patterns you see in algebra class: a quantity divided by a polynomial in some variable.

Breaking down our 3 main practical examples of rational expressions

Let’s organize what we’ve seen so far into examples of rational expressions: 3 practical examples that you can keep in your mental toolkit.

1. Travel time as a function of speed

• Expression: \(T(v) = \frac{180}{v}\)
• Story: Time needed for a 180-mile trip depending on your speed.
• Domain note: \(v \neq 0\). Also, negative speeds don’t make sense here, so we usually take \(v > 0\).

2. Cost per person in a shared subscription

• Expression: \(C(n) = \frac{20 + 3n}{n} = \frac{20}{n} + 3\)
• Story: Monthly cost per user when a \(20 base fee and \)3 per profile are split among \(n\) people.
• Domain note: \(n\) must be a positive integer, and \(n \neq 0\).

3. Cost per gigabyte in a data plan

• Expression: \(P(g) = \frac{15 + 0.10g}{g} = \frac{15}{g} + 0.10\)
• Story: Average cost per gigabyte when you pay a fixed access fee plus a per-gigabyte rate.
• Domain note: \(g > 0\); you can’t divide by zero gigabytes.

These are the best examples of rational expressions for beginners because:

• They’re easy to visualize.
• The variables have clear meanings.
• You can plug in real numbers and interpret the results immediately.

More real examples: rational expressions in 2024–2025 life

To deepen your understanding, let’s add several more real examples of rational expressions that feel very 2024–2025.

Internet speed and download time

With remote work and streaming still going strong, internet speed matters. Suppose you’re downloading a 5-gigabyte file. If your internet speed is \(s\) megabits per second, a simplified model for download time might look like

[
T(s) = \frac{40000}{s}.
]

Here, 5 GB has been converted into about 40,000 megabits (using 1 byte = 8 bits and some rounding). This is another textbook-style example of a rational expression with a real-world twist.

Average emissions per mile

Environmental data often uses rational expressions. If a car emits a fixed startup amount of pollution (say \(E_0\) grams) plus \(k\) grams per mile, then total emissions for \(m\) miles are

[
E(m) = E_0 + km.
]

The average emissions per mile are

[
A(m) = \frac{E_0 + km}{m} = \frac{E_0}{m} + k.
]

Again, a rational expression with a clear interpretation: as \(m\) gets larger, \(\frac{E_0}{m}\) gets smaller, so the startup cost gets “spread out.”

Average grade over assignments

Students see rational expressions when they average scores. Suppose you have \(n\) assignments, each worth the same amount, and your total points are \(S\). Your average score per assignment is

[
G(n) = \frac{S}{n}.
]

If you model total points as something like \(S = 90n - 10\) (maybe you lost 10 points somewhere), then

[
G(n) = \frac{90n - 10}{n} = 90 - \frac{10}{n}.
]

This shows your average grade approaching 90 as you complete more assignments.

These extra examples include the same structure as our earlier examples of rational expressions: 3 practical examples, but now you can see them across technology, environment, and education.

How to recognize and simplify examples of rational expressions

At this point, you’ve seen a lot of situations. Let’s extract a simple checklist you can use when you’re handed a new expression.

Recognizing a rational expression

An expression is rational if:

• It’s written as a fraction.
• The numerator and denominator are polynomials in the variable(s).

So these examples include rational expressions:

• \(\frac{2x+5}{x-1}\)
• \(\frac{180}{v}\)
• \(\frac{20+3n}{n}\)
• \(\frac{703w}{h^2}\)

But these are not rational expressions:

• \(\frac{\sqrt{x}}{x+1}\) (square root of x in the numerator)
• \(\frac{\sin x}{x^2}\) (sine is not a polynomial)

Simplifying a rational expression: one of our examples

Take the subscription example again:

[
C(n) = \frac{20 + 3n}{n}.
]

We can split it into two fractions:

[
C(n) = \frac{20}{n} + \frac{3n}{n} = \frac{20}{n} + 3.
]

That’s it. The simplified form tells a better story: \(3 per user, plus a shared \)20 fee.

If you had something like

[
R(x) = \frac{x^2 - 9}{x^2 - x - 6},
]

you could factor numerator and denominator:

[
R(x) = \frac{(x-3)(x+3)}{(x-3)(x+2)} = \frac{x+3}{x+2}, \quad x \neq 3, x \neq -2.
]

This is the kind of algebra that supports all the real examples we’ve talked about.

FAQ: common questions about examples of rational expressions

What is a simple example of a rational expression?

A very simple example of a rational expression is \(\frac{2x+1}{x-4}\). Both the numerator and denominator are polynomials in \(x\), and the denominator is not zero (so we say \(x \neq 4\)).

Which real-life examples of rational expressions are most useful for students?

Some of the best examples for students are:

• Travel time: \(T(v) = \frac{180}{v}\)
• Cost per person: \(C(n) = \frac{20 + 3n}{n}\)
• Cost per gigabyte: \(P(g) = \frac{15 + 0.10g}{g}\)
• BMI: \(\text{BMI} = \frac{703w}{h^2}\)

These examples of rational expressions connect directly to daily life, health, and money.

How do I know if something is an example of a rational expression or not?

Check the top and bottom. If both are polynomials in the variable and the bottom is not zero, it’s an example of a rational expression. If you see square roots of the variable, trig functions, or variables in exponents (like \(2^x\)), then it’s not a rational expression.

Why do teachers use so many word problems as examples of rational expressions?

Because the structure “something divided by something else” shows up constantly in real life: speed, density, average cost, average grade, concentration, and more. Word problems give a chance to turn those situations into algebra. It’s not just busywork; it’s training you to recognize when a fraction with variables represents a real situation.

Where can I learn more about rational expressions and related algebra topics?

For deeper study, you can check out:

• Algebra resources from Khan Academy (nonprofit, free lessons)
• Open course materials from universities like MIT OpenCourseWare or Harvard
• General math background from sites like NCTM (National Council of Teachers of Mathematics)

As you work through more problems, keep coming back to these examples of rational expressions: 3 practical examples—travel time, shared costs, and data pricing. They’re simple enough to remember and rich enough to model a lot of real-world situations.