What Is the Unit Rate Formula?
A unit rate is a rate that compares the first quantity to one unit of the second quantity, expressed as a ratio. A unit rate tells us how many units of the first quantity relate to just one unit of the second quantity.
Put simply, a unit rate is when the first quantity is described for one unit of the second quantity. Typically, it is expressed as a fraction or ratio.
How much something costs per unit in relation to another quantity is calculated using the unit rate formula.
When calculating a rate, the denominator represents the independent variable, while the numerator represents the dependent variable. Unit rate is a comparison of quantities with different units, often linking a dependent variable (numerator) to an independent variable (denominator), helping understand their relationship.
It’s a powerful tool for analyzing real-world scenarios involving different measurements. Time usually acts as the independent variable, while money often serves as the dependent variable in rate calculations. For example, miles per hour, dollars per gallon, etc.
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Unit Rate: Definition
Unit rate can be defined as the ratio between two measurements with the second term fixed at 1 unit.
It helps us understand how one quantity changes concerning another quantity when the second quantity is one unit. The unit rate allows for direct comparisons and is expressed in the form of “quantity per one unit.”
Unit Rate Formula
The formula for finding a unit rate for any two quantities say a and b can be given as,:
Unit Rate = Quantity of Interest ÷ Number of Units $= \frac{a}{b}$
where
Quantity of Interest or a: The quantity or amount you want to find the rate for.
Number of Units or b: The number of units or items associated with the quantity of interest.
Example: Consider the scenario where you need to figure out the unit rate for the daily sales of apples. If 200 apples are sold in 5 days, the unit rate would be:
Unit Rate = 200 apples ÷ 5 days
Unit Rate = 40 apples per day
So, the unit rate in this case is 40 apples per day. It tells you that on average, 40 apples are sold each day.
Remember that the unit rate should have consistent units in both the numerator and the denominator. In the example above, it’s “apples per day.” This ensures that the rate is accurately represented and can be easily compared with other rates.
How to Calculate the Unit Rate Using the Unit Rate Formula
To calculate the unit rate, divide the measured quantity by the chosen unit. In a unit rate, the second quantity is consistently 1. Determine the unit rate by dividing the top number by the bottom number, making the bottom number equal to 1.
Finding unit rates using the formula is a straightforward process. Let’s understand the steps with an example.
Example 1: Mia purchased 10 pounds of apples for $20. What is the unit rate of the cost per pound of apples?
Step 1: Identify the quantities.
Quantity of the second item (dependent variable): Cost of apples $= \$20$
Quantity of the first item (independent variable): Weight of apples = 10 pounds
Step 2: Apply the formula and perform the calculation.
Unit rate $= \frac{Quantity\; of\; second\; item}{Quantity\; of \;first\; item}$
Unit rate (cost per pound) $= \frac{\$20}{10}$ pounds
Unit rate (cost per pound) $= \$2$ per pound
Step 3: Interpret the result.
The unit rate of the cost per pound of apples is $\$2$. This means that each pound of apples costs $2.
Example 2: How to solve unit rates with fractions
Lets walk through an example to understand this.
A car travels $\frac{2}{3}$ of a mile in $\frac{1}{4}$ of an hour. Calculate the unit rate of miles per hour.
Given: Distance traveled $= \frac{2}{3}$ mile, Time taken $= \frac{1}{4}$hour
To find: Miles per hour
Unit rate $= \frac{Distance\; traveled}{Time \;taken} = \frac{\frac{2}{3} mile}{\frac{1}{4} hour}$
To divide fractions, we invert the divisor $(\frac{1}{4})$ and multiply.
Unit rate $= (\frac{2}{3} mile) \times (\frac{4}{1} hour) = \frac{8}{3}$ mile per hour
Therefore, the unit rate is $\frac{8}{3}$ miles per hour.
Facts about Unit Rate Formula
- Unit rate is frequently used in everyday circumstances where you need to compare different numbers, such as when calculating speed (in miles per hour), cost per unit (in dollars per item), efficiency (in things produced per hour), and many more instances.
- Quantities can be scaled up or down using the unit rate. For instance, if you are aware of the unit price of 2 apples per $1, you can simply determine that 10 apples would cost $5.
- Depending on the situation and preferred format, unit rates can be expressed as either fractions or decimals. Decimals may be more useful and practical in certain circumstances.
- The slope of a line on a graph can be used to depict the unit rate when comparing two quantities graphically. Higher unit rates are associated with steeper lines, and vice versa.
- When two quantities are inversely proportional, their unit rates will vary inversely. For instance, if the number of workers on a project decreases, the unit rate of work completed per worker will increase.
Conclusion
In this article, we learned about the unit rate formula. Understanding the unit rate formula is essential for various mathematical applications and real-world scenarios, as it simplifies comparisons and enables meaningful interpretations of data involving different quantities. Now, let’s solve some examples and practice problems to understand the concept better.
Solved Examples on Unit Rate Formula
Example 1: A restaurant serves 180 meals in 3 hours. Calculate the unit rate to find the meals served per hour.
Solution:
Number of meals served = 180
Time taken = 3 hours
To find: Number of meals served per hour
Let a = number of meals served, b = time taken to serve
Unit rate formula: unit rate $= \frac{a}{b}$
Unit rate$ = \frac{Meals\; served}{Hours \;taken}$
Unit rate $= \frac{180}{3}$
Unit rate = 60 miles/hour
Therefore, the unit rate is 60 meals per hour.
Example 2: A car travels 360 miles on 12 gallons of fuel. Calculate the unit rate of miles per gallon (MPG).
Solution:
Distance traveled = 360 miles
Fuel used = 12 gallons
To find: Miles per gallon
Let a = distance traveled, b = fuel used
Unit rate formula: unit rate $= \frac{a}{b}$
Unit rate $= \frac{Distance\; traveled}{Fuel\; used}$
Unit rate $= \frac{360 \;miles}{12 \;gallons}$
Unit rate = 30 miles per gallon
Therefore, the unit rate is 30 miles per gallon.
Example 3: A factory produces 10 units of a product per hour. How many units will it produce in 6 hours?
Solution:
Units produced per hour = 10
Time taken = 6 hours
To find: Units produced in 6 hours
Let a = Units produced, b = hours taken
Unit rate $= \frac{a}{b} = \frac{Units \;produced}{Hours\; taken}$
Unit rate $= \frac{10\; units}{1\; hour} = 10$ units per hour
Number of units produced in 6 hours = 10 units per hour × 6 hours = 60 units
Therefore, the factory produces 60 units in 6 hours.
Example 4: A bakery bakes 60 cookies in 2 hours. How many cookies will be baked in 6 hours?
Solution:
Given: Cookies baked per hour = 60
Time taken = 6 hours
To find: Cookies baked in 6 hours
Let a = Cookies baked, b = Hours taken
Unit rate $= \frac{a}{b} = \frac{Cookies\; baked}{Hours\; taken}$
Unit rate$ = \frac{60\; cookies}{1\; hour} = 60$ cookies per hour
Number of cookies baked in 6 hours = 60 cookies per hour × 6 hours = 360 cookies
Therefore, the bakery bakes 360 cookies in 6 hours.
Example 5: A store offers a discount on buying 20 T-shirts for $200. Calculate the unit cost of a T-shirt.
Solution:
Price of 20 T-shirts $= \$200$
To find: Price per T-shirt
Let a = Price of T-shirts, b = Number of T-shirts
Unit rate $= \frac{a}{b} = \frac{Price\; of\; T-shirts}{Number \;of\; T-shirts}$
Unit rate $= \frac{\$20}{20\; T-shirts} = \$10$ per T-shirt
The unit cost of a T-shirt is 10 dollars per T-shirt
Practice Problems on Unit Rate Formula
How to Find Unit Rate Formula - Definition, Examples, FAQs
What is the definition of a unit rate?
A unit rate is a specific type of rate in which the second quantity is expressed in terms of one unit of the first quantity. It allows for easy comparison between different quantities.
In a unit rate, the denominator is always equal to
A unit rate is when the first quantity is described for one unit of the second quantity.
When comparing two ratios to find the unit rate, you can use
When comparing two ratios to find the unit rate, you can set up a proportion and cross-multiply to find the value of the unknown variable.
Which of the following correctly represents the unit rate of 3 kilograms of apples for $\$6$?
The unit rate is calculated by dividing the total cost ($6) by the quantity of apples (3 kilograms).
If a car travels at a speed of 60 miles per hour, how many miles will it travel in 2.5 hours?
To find the total distance traveled, multiply the unit rate (60 miles/hour) by the time (2.5 hours).
60 miles/hour x 2.5 hours = 150 miles.
Frequently Asked Questions about Unit Rate Formula
What do you mean by a unit rate and how does it differ from a regular rate?
A rate that compares two quantities and expresses the second quantity per one unit of the first quantity is known as a unit rate. Since it standardizes the comparison and makes it simpler to comprehend and compare various volumes, it varies from a regular rate.
Can the unit rate be greater than 1?
Yes, the unit rate can be greater than 1. It implies that there are several units of the second quantity for every unit of the first. For example, a unit rate of 3 means there are 3 units of the second quantity per one unit of the first quantity.
Are unit rates useful for converting between different units of measurement?
Yes, unit rates are helpful for converting between different units of measurement. By knowing the unit rate, you can easily convert quantities from one unit to another using simple multiplication or division.
Can unit rates be used for predicting future trends or outcomes?
Yes, unit rates can be used for predicting future trends or outcomes by analyzing the rate of change between two quantities. By studying past data and unit rates, projections can be made about future developments.
How do you find the unit rate when given a ratio?
To find the unit rate when given a ratio, simply divide the numerator of the ratio by the denominator. This will give you the quantity of the first item per one unit of the second item, expressed in the same units.
