medida de unidades

Unit Rates

A unit rate is a special type of ratio (also called a single-unit rate). It will compare $1$ unit of some quantity to a different number of units of a different quantity. Unit rates are often used in real life situations, like when you need to convert between measurement systems. Any ratio that is not a unit rate can be turned into one. We'll discuss this a bit more later.

Examples

Let's look at some more examples:

$$25 \text{ students} : 1 \text{ teacher}$$ $$7 \text{ homework assignments} : 1 \text{ student}$$ $$3 \text{ final exams} : 1 \text{ day}$$

Here are some special examples. They are all unit conversions for units of length. These include feet, inches, and meters.

$$12 \text{ inches} : 1 \text{ foot}$$

$$3.2084 \text{ feet} : 1 \text{ meter}$$ $$1.60934 \text{ kilometers} : 1 \text{ mile}$$ $$100 \text{ centimeters}: 1 \text{ meter}$$

The unit with the $1$ will always appear on the right-hand side. This is a matter of convention. Here's one final example you probably see all the time riding around in a car:

$$35 \text{ miles}: 1 \text{ hour}$$

This rate, miles per hour, is sometimes abbreviated $\text{mph}$. It literally means the number of miles driven in a single hour.

Converting Ratios to Unit Rates

Sometimes, non-unit-rate ratios can be used to calculate unit rates! All you have to do is divide both sides of the ratio by the number on the right side. Consider the ratio: $$20 \text{ dollars} : 10 \text{ comics}$$

Before we begin any conversions, let's talk for a moment about what this means. This ratio says that the total price of $10$ comics is $20$ dollars.

We convert by dividing both sides of the ratio by $10$! We start by taking the price, divide it by $10$. We choose to divide by $10$ since that is the number to the right of the $:$ symbol.

$$\begin{align} 20 \text{ dollars} &: 10 \text{ comics}\\ \\ \dfrac{20}{10} \text{ dollars} &: \dfrac{10}{10} \text{ comics}\\ \\ 2 \text{ dollars} &: 1 \text{ comic}\\ \end{align}$$

There is always some special number we can multiply or divide by that'll give a unit rate. Of course, that number won't always be $10$, but it's out there. Also, the actual units, dollars and comics, don't change in the problem. We can do this procedure no matter what units we start with, though.

When the unit on the left is dollars, it's called a unit price.

Now, try a problem for yourself.

We divide both sides by $3$! $$\begin{align} 45 \text{ pepperonis} &: 3 \text{ pizzas}\\ \\ \dfrac{45}{3} \text{ pepperonis} &: \dfrac{3}{3} \text{ pizzas}\\ \\ 15 \text{ pepperonis} &: 1 \text{ pizza}\\ \end{align}$$

The Reverse Question

In the last problem, we were given a rate without a $1$ in it and asked to turn it into a unit rate. This time we'll do the reverse to see how long it will take to travel between two major cities.

Question: The distance driving from Minneapolis, Minnesota to Lincoln, Nebraska is $424$ miles. If you are driving $50$ miles per hour the entire trip and don't stop, how long will it take to make the entire trip?

Explanation: We are given the rate of $50$ miles per hour. This means you drive $50$ miles in a single hour. We want to know how long the trip will take if you continue at this rate (or speed). This is actually a proportion word problem in disguise, and so we'll have to solve an equation. Our variable is the total time for the whole trip. We'll call this variable $t$. Then we want to turn our given rate into one where the distance is $424$ miles. The proportion is

$$\frac{50 \text{ miles}}{1 \text{ hour}} = \frac{424 \text{ miles}}{t \text{ hours}}$$

In case you aren't familiar, the technique here is to cross-multiply. To do this it's easier to temporarily hide the units and just worry about numbers and variables.

$$\frac{50}{1} = \frac{424}{t}$$

That means we multiply both sides of the equation by something to get rid of the denominators. Let's multiply both sides by $t:$

$$50t = 424$$

This equation is much more familiar. We can solve in one step by dividing both sides by $50$:

$$t = \frac{424}{50} = 8.48$$

Remember that our time unit was hours. Thus it takes $8.48 \text{ hours}$ of driving at $50 \text{ miles per hour}$ to make the entire trip.

(Video) Unit Rates

by Mathademics!

This video by Mathademics! explains unit rates.

$\large{\text{Summary}}$

A rate is a comparison with different units of measure. $$\frac{18\text{ dollars}}{30\text{ minutes}}$$

A unit rate is a rate where the bottom number is $1$ (also called a single-unit rate). We can turn any rate into a unit rate. We do this by dividing both the top and bottom by the bottom number. Of course, the bottom number is often called the denominator.

$$\frac{18\text{ dollars}}{30\text{ minutes}}\rightarrow\frac{0.6\text{ dollars}}{1\text{ minutes}}$$

Suppose a $5$ ounce can of soup costs $\$1.25$. How much does $1$ ounce cost? $$\frac{1.25\hbox{ dollars}}{5\hbox{ ounces}}{{\div\,\,5}\atop{\div\,\,5}}=\frac{0.25\hbox{ dollar}}{1\hbox{ ounce}}$$ So each ounce costs $\$0.25$, or $25$ cents! This is called the unit price for soup.

Which Deal is the Best?

Now we do another problem that involves comparing two unit prices.

Example: You are at the grocery store and are looking to buy some boxes of cereal for your family.

There are two deals on boxes of cereal.

Deal 1: Buy any three boxes for $\$8.10$.

Deal 2: Buy any five boxes for $\$13.00$.

The question is, which deal is better? By this we mean, which purchase has a lower unit price?

To answer this question, we set up a ratio for each deal. Then we'll rewrite the ratios to be out of just a single box of cereal.

Deal 1: $8.10 \text{ dollars} : 3 \text{ boxes of cereal} = \frac{8.10 \text{ dollars}}{3 \text{ boxes of cereal}}$

Deal 2: $13.00 \text{ dollars} : 5 \text{ boxes of cereal} = \frac{13.00 \text{ dollars}}{5 \text{ boxes of cereal}}$

To turn the bottom [right] number in the ratios into $1$s, we divide by the current bottom number. In this case that means dividing a decimal by a whole number.

$$\begin{align} 8.10 \div 3 &= 2.70 \\ 13.00 \div 5 &= 2.60 \end{align}$$

This means we can simplify our ratios to resemble unit prices:

Deal 1: $\frac{8.10 \text{ dollars}}{3 \text{ boxes of cereal}} = \frac{2.70 \text{ dollars}}{1 \text{ box of cereal}}$

Deal 2: $\frac{13.00 \text{ dollars}}{5 \text{ boxes of cereal}} = \frac{2.60 \text{ dollars}}{1 \text{ box of cereal}}$

The unit price for Deal 2 is slightly lower (only by $10$ cents). As a result we say Deal 2 is better. Go enjoy your newly purchased cereal!

(Video) Solving Unit Rate Problems

by PlanetNutshell

This animation by PlanetNutshell explains unit rates.

$\large{\text{Summary}}$

Sometimes its hard to compare two rates. To effectively compare rates, we find the unit rates and compare those. The conversion process applies techniques from word problems about proportions.

A unit rate is a ratio between two units where the second rate is $1$. The first rate will be some other positive number.

$$\frac{30\text{ miles}}{1\text{ hour}}$$

Suppose Carl and Kate enter a pie eating contest. Carl can eat $4\text{ pies}$ in $8\text{ minutes}$ and Kate can eat $6\text{ pies}$ in $4\text{ minutes}$.

It sounds like Kate would win this contest. But how much pie can each person eat in $1\text{ minute}$? Let's start with Carl. He can eat: $$\frac{4\text{ pies}}{8\text{ minutes}}$$ Our goal is to make the denominator into a $1$. To do this, we divide both the numerator and denominator by $8$. This is a fancy way of multiplying by $1$, which does not change the value.

$$\frac{4\text{ pies}}{8\text{ minutes}}\frac{\div\,8}{\div\,8}=\frac{\frac12\text{ pies}}{1\text{ minute}}$$

So, Carl can eat $\frac12$ a pie every minute. If he keeps eating at this same rate, he'll have consumed $1$ pie in two minutes, $2$ pies in four minutes, and so on.

For Kate, we want to divide the top and bottom by $4$. $$\frac{6\text{ pies}}{4\text{ minutes}}\frac{\div\,4}{\div\,4}=\frac{\frac32\text{ pies}}{1\text{ minute}}$$ So, Kate can eat $\frac32$ pies every minute.

Put another way, Kate eats pie three times as fast as Carl does.

Try the following problem to check your understanding!