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Rate

A rate is a ratio in which the two terms being compared have different units.

Instead of using the "to" form for ratios (for example, $3$ red marbles to $2$ blue marbles), rates often use the word per.

A rate that you see every day is on a speed limit sign. In the US, the speed limit might be $65$ miles per hour. This is actually a ratio of $65\text{ miles}$ to $1\text{ hour}$. Because the speed and time have different units (miles vs. hours), this is an example of a rate.

Here are some examples of common rates: Speed limit, interest rate, crime rate, profit rate, birth rate, death rate, etc. Maybe there's a unit of time or if you're looking at heart rate (beats per minute). Rates are everywhere in our lives, which is why they are so important to understand.

Here, we're looking at violent crime rate per 100,000 people. Say we zoom into Pennsylvania: we can see that the violent crime rate is (according to the key on the right-hand side), 243.28 to 364.92 violent crimes per 100,000 people. Because we used rates, we can compare different states using this handy map!

In the graph above, we can see Russian mortality rates at deaths per 10,000 people. Again, using rates allows us to compare the values over the years.

You probably see this sign everyday! The speed limit is 65 mph (miles per hour). This is also called unit rates. You can also learn about the relationship between distance-rate-time in this lesson.

If Marcus' pay were to change, we would call that a change per unit. Maybe Marcus gained or lost pay - either way, his hourly rate, currently $$14$, would change!

(Video) What is Rate and Unit Rate

by BroandSisMathClub

Here's a video by BroandSisMathClub defining rate and unit rates and using them to solve problems.

$\large{\text{Summary}}$

A rate is a ratio of two quantities measured in different units. A common example is miles per hour. In the video, she uses: $$\frac{40\text{ pushups}}{5\text{ minutes}}$$ Pushups and minutes are two different measurements so this is a rate. It's telling us that someone can do $40\text{ pushups}$ per every $5\text{ minutes}$.

Rates, like ratios, can also be represented using a colon: $40\,{:}\,5$.

A unit rate is a rate where the second measurement is a quantity of $1$. In the example above, the unit rate would be the number of pushups done in $1\text{ minute}$. This can be found dividing both the top and bottom by the denominator (or using the multiplication property. $$\frac{40\text{ pushups}\div5}{5\text{ minutes}\div5}=\frac{8\text{ pushups}}{1\text{ minute}}$$

Another way to find the unit rate is by using cross products which you'll learn about later.

Rates

A rate is a relationship between two different types of units. A rate is often represented using the word ‘per’.

Some examples of rates are miles per hour, price per item, dollars per hour, meters per second. For example, if an apple cost $\$4$ for $2$ pounds, then the price of apples is $\$2$ per pound. Let’s try an example.

Sarah can travel $500$ meters in $5$ minutes. What is the rate of meters per minute?

Sarah can travel $500$ meters in $5$ minutes, and we are trying to calculate how many meters she can travel in a minute. $$\begin{align} 500 \text{ meters}: 5 \text{ minutes} \\ 500 \text{ meters} \cdot \dfrac{1}{5}: 5 \text{ minutes} \cdot \dfrac{1}{5} \\ 100 \text{ meters} : 1 \text{ minute} \\ \end{align}$$

Now, try an example.

$$\begin{align} 25 \text{ questions}: 100 \text{ points} \\ 25 \text{ questions} \cdot \dfrac{1}{25}:100 \text{ points} \cdot \dfrac{1}{25} \\ 1 \text{ question} : 4 \text{ points} \\ \end{align}$$

Watch out when you're solving word problems - you may need to convert between units of measurement!.