How to Find Unit Rates and Rates?

Rate is the ratio of two different quantities with different units, while the unit rate expresses the number of units of the first quantity for one unit of the second quantity. At the unit rate, the denominator is always one unit.

A step-by-step guide to unit rates and rates

In math, a rate is a ratio that compares two different quantities that have different units. For example, if we say Jim types \(60\) words in a minute, then his rate of typing is \(60\) words per minute. The word “per” indicates that we are dealing with a rate. The word “per” can be replaced with the “/” symbol in problems.

Unit rate

The unit rate is different from the rate at which a certain number of units of the first quantity are compared to one unit of the second quantity. In other words, we can say that the second quantity in the comparison is always \(1\). For example, there are \(60\) seconds in one minute. So as a unit rate, we can express it as \(60\) seconds per minute. Here, the word ‘per minute’ refers to one minute.

Ratio definition

A Ratio is used to compare two or more similar quantities or numbers with similar units. It is often written with a colon, and when used in words, we call the ratio of one quantity “to” the second quantity. For example, the ratio of girls and boys in a class is \(4\) to \(3\), or \(4:3\). Some problems may give you only two numbers, such as comparing two quantities, while some may have more than two quantities. For example, the ratio of different ingredients used in a recipe.

How is the rate calculated?

In general, we can write the formula for rate as the ratio between two quantities with various units. By putting this in the ratio format, we get,

\(\color{blue}{Rate= \frac {Quantity\ 1}{Quantity\ 2}}\)

The steps to calculate the rate are given below:

• Step 1: Write two quantities that are in different units.
• Step 2: Find the ratio of quantity \(1\) to quantity \(2\).
• Step 3: Express the ratio in the simplest way to find the rate.
• Step 4: The answer will proceed with \(\frac{unit\ 1}{unit\ 2}\) as the unit for the value of the rate found.

To better understand this issue, we give an example. John cycled for \(2\) hours and traveled \(24\) miles. To calculate the speed at which he rode, let us use the rate formula, which is \(Rate= \frac {Quantity 1}{ Quantity 2}\). Given, quantity \(1=24\) miles, quantity \(2=2\) hours. Substituting the values in the formula, we get, \(Rate= \frac{24\ miles}{2\ hours}\). Here, the speed is the rate. 

So, \(rate= \frac{12\ miles}{hour}\) or \(12\) miles per hour.

Unit rate is also a comparison between two quantities that have different units except that the quantity at the denominator is always one. Putting this in the ratio format, we get, \(Unit Rate= \frac{Quantity\ 1}{ One\ Unit\ of\ Quantity\ 2}\).

The rate “miles per minute” gives the distance traveled per unit of time. To calculate the unit rate, we divide the denominator with the numerator in such a way that the denominator becomes \(1\). For example, if \(250\) miles in \(2\) hours, the unit rate would be \(\frac {250\ miles}{ 2\ hours}\), which is equal to \(\frac {125\ miles}{hour}\). In other words, the denominator is always \(1\) at the unit rate.

Unit Rates and Rates – Example 1:

A printer prints \(80\) pages of an e-book in \(40\) seconds. Find the unit rate of the number of pages printed per second. 

Solution:

The number of copies created in \(40\) seconds \(=80\). To find the unit rate, divide the total number of pages printed by the total number of seconds. We get,

\(Unit\ rate = \frac {80}{40}\)

\(=2\)

Therefore, the printer prints \(2\) pages per second or \(2\)\(\frac{pages}{second}\).

Exercises for Unit Rates and Rates

1. Jane loves to cook and bakes wonderful cakes. He bakes \(45\) cakes in \(9\) hours. Can you figure out how many cakes he bakes per hour?
2. William and his family traveled \(1,500\) miles on vacation and consumed \(60\) gallons of gas. Can you help him calculate the average number of \(\frac{miles}{gallon}\)?

1. \(\color{blue}{5\ cakes\ per\ hour}\)
2. \(\color{blue}{25 \frac{miles}{gallon}}\)