# Unit Rate Calculator

By Maciej Kowalski, PhD candidate
Last updated: Sep 11, 2020

Welcome to Omni's unit rate calculator, where we'll explain how to find the unit rate for a given fraction. "But what is a unit rate?" Don't you worry, we'll get to the unit rate definition soon enough. For now, let's just say that it's connected to the ratio of two numbers, although ratios and rates are something slightly different. Still, we'll explain in detail how to find the unit rate and give a few numerical unit rate examples.

## Ratios and rates. Unit rate definition

You know how, when you get a good grade, your parents say, "But your friend Jack got a better one. Why didn't you?" And when you get an average grade, but it's one of the best ones in class, you hear, "You shouldn't look at other kids!" Funny how people can choose to compare two values only when it suits their purposes.

Mathematically speaking, this comparison is called the ratio. We often present it as a fraction, although not always. For instance, say that you got `75%` on the test, and the class average was `62%`. Then the ratio `75 / 62` describes your score in terms of the mean. If it's larger than `1` (i.e., the first number is larger than the second), then you're above the average. If not, then perhaps it's best to spend some more time studying?

The rate, on the other hand, tells you how much of the first number corresponds to how much of the second. This vague definition is better explained with an example.

Say that you have a dog and you give him treats every now and then. If, in a week, it eats twenty-one treats, then the corresponding rate is `21` treats per `7` days. Note how colloquially, we'd react to that with something like, "At this rate, I'll have to buy more in a couple of days." The word "rate" is no coincidence here.

## What is a unit rate?

So what is a unit rate? It is the same thing but with the second number equal to `1`. Or, if you prefer a more scientific unit rate definition, it is a way to translate the rate as we have it into the equivalent fraction with the denominator `1`. After all, it might be useful to know that the `21` dog treats per week translate to three treats a day.

As we've seen, ratios and rates are connected, but it's quite impossible to say that one is better than the other. They're just different. To convince yourself of that, recall how a map scale is always given in form `1 : 30,000,000`, which is a ratio, while we measure the density of objects in ounces (or grams) per cubic inch (or centimeter), which is a rate. The latter is a unit rate example, and, in fact, most of physics is.

Now that we have the unit rate definition out in the open, it's time to see how to find this unit rate. The doggy example above should have got the idea through to you, but, nevertheless, why don't we commit a whole section to describe it in detail?

## How to find the unit rate?

Suppose that you have two numbers, `a` and `b`. Then their rate is `a / b`, but what is the unit rate?

From the unit rate definition, we know that it is the equivalent fraction with denominator `1`. In other words, we want to find the `c` which satisfies the equation

`a / b = c / 1`,

which is simply

`a / b = c`.

This already suggests what we must do: we divide `a` by `b`, and the result is our answer. Simple as that. A piece of cake, wouldn't you say?

A piece of cake it might be, but let's still see how to find the unit rate when we actually have numbers instead of symbols. The theory is fine and all, but if it's unit rate examples that you're looking for, then the next section is the one for you!

## Unit rate examples: using the unit rate calculator

You're finally able to make your dream come true - you're going on a road trip! The plan is to get there by plane, and, once you've landed, find a car to rent and visit a few places in one week.

All in all, you want to visit four cities, call them `A`, `B`, `C`, and `D`, so you have three trips to make between them: `A -> B`, `B -> C`, and `C -> D`. From what you found on the net, the distances are `80 mi`, `140 mi`, and `110 mi` respectively. Also, your GPS tells you that the drives should respectively take `1.5 hr`, `3 hrs`, and `2.5 hrs`. Well, it's quite a lot of driving, but even the gas costs can't spoil your enthusiasm!

But what speed will you travel at? It's one thing to know how far and how long it will take, but it might also be useful to know what kind of road you can expect.

For the first trip, the rate at which you'll be driving is simply the fraction `80 mi / 1.5 hr`. However, to get the actual velocity, we should divide the two numbers and find (surprise, surprise) the unit rate.

Take a look at the unit rate calculator and the formula at the top. Accordingly, in order to find the unit rate, we need to input the values of `a` and `b`. In our case, this means that for the `A -> B` trip, we have to input

`a = 80`, `b = 1.5`.

Similarly, for the other two trips, we input

`a = 140`, `b = 3`,

and

`a = 110`, `b = 2.5`.

From the previous section, we know quite well how to find the unit rate: we divide the two numbers. So why don't we grab a piece of paper, and check if we agree with the unit rate calculator?

Denote the consecutive velocities by `v₁`, `v₂`, and `v₃`. Then, we have

`v₁ = 80 mi / 1.5 hr ≈ 53.33 mph`,

`v₂ = 140 mi / 3 hr ≈ 46.67 mph`,

`v₃ = 110 mi / 2.5 hr ≈ 44 mph`.

Well, it looks like the road is going to get tougher and tougher. Perhaps the terrain becomes more hilly?

Still, it's going to be worth it! You have your beach body ready, so, without further ado... sandy beaches, here we come!

Maciej Kowalski, PhD candidate a
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Result
Unit rate
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