How To Find The Unit Rate

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How to find the unit rate

Here you will learn how to find the unit rate including what the unit rate is and how to solve direct and inverse proportion word problems that involve finding the unit rate.

Students will first learn about how to find the unit rate as part of ratios and proportional relationships in $6$ th grade.

What is the unit rate?

The unit rate is a ratio that compares a quantity to one unit of another quantity.
It describes how much of something occurs, is produced, or exists per one unit of another measure, such as “miles per hour” or “price per item.”

To find the unit rate, you need to determine the value for one unit and then determine the value for the quantity you need.

For example,

If $5$ pens cost $\$ 1,$ then $1$ pen costs $20¢.$

$\$ 1 \div 5$ pens $=20¢$ per pen.

This would be the unit rate or the cost per pen (unit cost).

You can find out how much $x$ pens cost by multiplying $20$ by the number of pens, $x.$

Finding the unit rate is useful for conversions between different units of measurement as you regularly represent one unit equivalent to another, such as $1 \, cm= 10 \, mm.$

As with most direct and inverse proportion topics, there are a lot that relate to daily life.

Applications of unit rate include calculating the speed of an object, fuel prices, grocery store prices, exchange rates, the amount of work required to complete a job, earnings from a day’s work, proportions of monthly expenditure on clothes, food, and bills, etc.

What is the unit rate?

[FREE] Ratio Check for Understanding Quiz (Grade 6 to 7)

Use this quiz to check your grade 6 to 7 students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

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[FREE] Ratio Check for Understanding Quiz (Grade 6 to 7)

Use this quiz to check your grade 6 to 7 students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

DOWNLOAD FREE

Common Core State Standards

How does this relate to $6$ th grade and $7$ th grade math?

Grade 6 – Ratios & Proportional Relationships (6.RP.A.3)
Use ratio and rate reasoning to solve real-world and mathematical problems, example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Grade 6 – Ratios & Proportional Relationships (6.RP.A.3.B)
Solve unit rate problems including those involving unit pricing and constant speed.

For example, if it took $7$ hours to mow $4$ lawns, then at that rate, how many lawns could be mowed in $35$ hours? At what rate were lawns being mowed?

Grade 7 – Ratios and Proportional Relationships (7.RP.A.1)
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

For example, if a person walks $\cfrac{1}{2}$ mile in each $\cfrac{1}{4}$ hour, compute the unit rate as the complex fraction ${\cfrac{1}{2}}/{\cfrac{1}{4}}$ miles per hour, equivalently $2$ miles per hour.

In order to find the unit rate to solve a word problem:

Find the unit rate.
Find the value of the required number of units.
Write the answer.

How to find the unit rate examples

Example 1: direct proportion

If $2$ tins of beans cost $96¢,$ find the cost of $7$ tins of beans.

Find the unit rate.

You have been given the cost of $2$ tins, so you divide by $2$ to find the cost of $1$ tin.

96\div{2}=48

$1$ tin of beans costs $48¢.$

2Find the value of the required number of units.

You have found the cost of $1$ tin, so now you can multiply by $7$ to find the cost of $7$ tins.

48 \times 7=336

Remember $336¢=\$ 3.36.$

3Write the answer.

$7$ tins of beans cost $\$ 3.36.$

Example 2: direct proportion

$12$ small pies require $2$ cups of flour for the pastry. How much flour is needed to make $8$ pies?

Find the unit rate.

You have been given the amount of flour for $12$ pies, so you divide the cups of flour by $12$ to find the amount of flour for $1$ pie.

$2 \div 12=\cfrac{1}{6}$ cup

$\cfrac{1}{6}$ cup of flour per pie.

Find the value of the required number of units.

For $8$ pies, you need to multiply the amount of flour for $1$ pie by $8.$

$\cfrac{1}{6} \times 8=\cfrac{8}{6}=1 \cfrac{2}{6}$

Write the answer.

We can simplify $1 \cfrac{2}{6}$ to $1 \cfrac{1}{3}.$

So $8$ pies require $1 \cfrac{1}{3}$ cups of flour.

Note that fewer pies would require less flour, so the answer should be less than $2$ cups.

Example 3: direct proportion

$5$ miles is approximately equal to $8$ kilometers. How many kilometers is $23$ miles approximately equal to?

Find the unit rate.

As $5$ miles is approximately $8$ kilometers, you divide the number of kilometers by $5$ to find the number of kilometers per mile.

$8\div{5}=1.6$

$1$ mile is approximately $1.6 \, km.$

Find the value of the required number of units.

You need to calculate the number of kilometers for $23$ miles, so you need to multiply $1.6$ by $23.$

$1.6\times{23}=36.8$

Write the answer.

$23$ miles is approximately $36.8 \, km.$

Example 4: direct proportion – ratio

The ratio of lemonade to calories for a drink is $100 \, ml\text{:}60 \, kcal$ (kilocalories). Find out how many calories there are in a $330 \, ml$ can of lemonade.

Find the unit rate.

As you need to find the number of calories per milliliter, you need the number of milliliters to equal $1$ and so you must divide both sides of the ratio by $100$ to get the ratio $1\text{:}0.6.$

Find the value of the required number of units.

As $1$ milliliter of lemonade contains $0.6$ calories, and you need the number of calories for $330 \, ml,$ you need to multiply $0.6$ by $330.$ This works out to be,

$0.6 \times 330=198$

Write the answer.

$330 \, ml$ of lemonade contains $198$ calories.

Example 5: inverse proportion

If $3$ people take $12$ days to complete a job, find how long it would take $9$ people to do the same job. Assume that each person works at the same rate.

Find the unit rate.

If the number of people on the job increases, the time it takes to complete the job decreases and so you need to determine how long it would take $1$ person to complete the job on their own first.

If you divide the number of people by $3,$ the time taken must be multiplied by $3$ as the two values are inversely proportional.

$12 \times 3=36$

So $1$ person would take $36$ days to complete the job.

Find the value of the required number of units.

As you need the time taken for $9$ people to complete the job, you multiply the number of people by $9,$ and divide the number of hours by $9.$

$36 \div 9=4$

Write the answer.

It would take $9$ people $4$ days to complete the job.

Example 6: inverse proportion

It takes $5$ machines $9$ hours to complete an order. How long would it take $3$ machines to complete the same order? Assume that each machine works at the same rate.

Find the unit rate.

As the number of machines decreases, the length of time to complete the order would increase.

If you divide the number of machines by $5,$ you need to multiply the number of hours by $5$ to get the work rate of $1$ machine.

$9 \times 5=45$

So it takes $1$ machine $45$ hours to complete the order.

Find the value of the required number of units.

As the number of machines increases, the length of time to complete the order decreases and so if you have $3$ machines, the time taken for $1$ machine would be divided by $3$ to get

$45\div{3}=15.$

Write the answer.

It would take $3$ machines $15$ hours to complete the order.

Example 7: inverse proportion

The same volume of water is poured into two cylindrical tubes, tube $A$ and tube $B.$ The depth of water in each tube is inversely proportional to the cross-sectional area of the tube. The cross-sectional area of tube $A$ is $28 \, cm^{2}.$

The depth of water in tube $A$ is $12 \, cm.$ The depth of water in tube $B$ is $17 \, cm.$ Calculate the cross-sectional area for tube $B.$

Find the unit rate.

As tube A has a cross-sectional area of $28 \, cm^{2}$ and the depth of water is $12 \, cm,$ the cross-sectional area would be $12$ times greater if the depth of water is $1 \, cm.$ Multiplying $28 \, cm^{2}$ by $12,$ you get

$28\times{12}=336$

So a tube with a depth of $1 \, cm$ would have a cross-sectional area of $336 \, cm^{2}.$

Find the value of the required number of units.

As the depth of tube $B$ is $17 \, cm,$ you need to multiply the depth by $17,$ and so you have to divide the cross-sectional area by $17$ to get

$336\div{17}=19.76\text{ (nearest hundredth)}$

Write the answer.

The cross-sectional area of tube $B$ is $19.76 \, cm^{2}$ (nearest hundredth).

Teaching tips for how to find the unit rate

Visuals like tables, graphs, and diagrams can help students see the relationship between quantities. For example, a bar graph showing different quantities and their total cost can help students understand how to find the unit rate.

There are online tools (such as a unit rate calculator) that can help visualize and practice unit rates. Incorporating these can make learning more engaging.

Include real-life scenarios on worksheets, such as calculating miles per gallon of gas, price per unit, or speed. This helps students see the relevance of unit rates in everyday life.

Easy mistakes to make

Ignoring units of measurement
Sometimes students omit units or fail to include them when calculating the unit rate, which can lead to confusion. Always include units (for example, dollars per apple, miles per hour) to clarify what the rate represents.

Not simplifying the ratio
Students may forget to simplify the ratio to a unit rate. For example, if $10$ apples cost $\$ 5,$ the unit rate should be $\$ 0.50$ per apple, not $\$ 5$ per $10$ apples.

Not assuming all units are equal in a word problem
In unit rate problems, you need to assume that all units are equal. If the problem involves the cost of items, you need to assume that there are no special offers to consider. If the problem involves the weight of apples, you assume that all apples weigh the same.

Practice how to find the unit rate questions

67¢

62¢

47¢

42¢

70\div{5}=14

$1$ pencil costs $14¢.$

14\times{3}=42

$3$ pencils cost $42¢.$

\$ 1,320

\$ 1,200

\$ 1,260

\$ 1,140

270\div{3}=90

It costs $\$ 90$ to rent the car for $1$ day.

90\times{14}=1,260

It costs $\$ 1,260$ to rent the car for $14$ days.

$26.8$ calories

$27.6$ calories

$29.7$ calories

$28.6$ calories

44\div{100}=0.44

There are $0.44$ calories in $1 \, ml$ of milk.

0.44\times{65}=28.6

There are $28.6$ calories in $65 \, ml$ of milk.

$\cfrac{3}{4}$ cup

$\cfrac{5}{8}$ cup

$\cfrac{1}{2}$ cup

$\cfrac{7}{8}$ cup

\cfrac{1}{4} \div 4=\cfrac{1}{16}

$\cfrac{1}{16}$ cup of sugar is needed for $1$ portion (enough for $1$ person).

\cfrac{1}{16} \times 10=\cfrac{10}{16}=\cfrac{5}{8}

$\cfrac{5}{8}$ cup of sugar is needed for $10$ people.

$7$ days

$12$ days

$14$ days

$18$ days

As the number of people decreases, the time taken will increase. This is an inverse proportion problem.

10\times{7}=70

It would take $1$ person $70$ days to build the wall.

70\div{5}=14

It would take $5$ people $14$ days to build the wall.

\$ 15

\$ 20

\$ 17

\$ 12

As the number of people increases the cost each person pays will decrease. This is an inverse proportion problem.

25\times{3}=75

The taxi costs $\$ 75$ for $1$ person.

75\div{5}=15

The taxi costs $\$ 15$ each for $5$ people.

How to find the unit rate FAQs

How do you find the unit rate step by step?

To find the unit rate, divide the total quantity by the number of units, and simplify if needed.

For example, if $12$ apples cost $\$ 6,$ the unit rate is $\$ 6 \div 12$ apples $=\$ 0.50$ per apple.

You can also set up the ratio as a fraction, then divide the numerator by the denominator and simplify.

What are some common examples of unit rates?

Common examples of unit rates include the number of miles per hour, price per item, cost per pound, and words per minute.

How can graphs help us find unit rates?

Graphs can help us find unit rates by showing the relationship between two quantities.

When one quantity is plotted on the $x$ -axis and the other on the $y$ -axis, the unit rate can be found by identifying the slope of the line, which represents how much the $y$ -value changes for each increase of one unit on the $x$ -axis.

How can you write a unit rate as a fraction?

You can write a unit rate as a fraction by placing the first quantity as the numerator (top number) and the second quantity as the denominator (bottom number).

For example, if the unit rate is “ $50$ miles per hour,” it can be represented as the fraction $\cfrac{50 \text { miles }}{1 \text { hour }}.$

The fraction shows how much of the first quantity corresponds to one unit of the second quantity.

What is the difference between unit rate and rate of change?

The unit rate compares two quantities with one of them being one unit (example, miles per hour), while the rate of change measures how one quantity changes relative to another (example, how the temperature increases per hour).

The next lessons are

Proportion
Converting fractions decimals and percentages
Percent

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