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Ratio How to write a ratio Multiplying and dividing decimals Multiplying and dividing fractionsHere 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.

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 20Β’.

\$ 1 \div 5 pens =20Β’ 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 dayβs work, proportions of monthly expenditure on clothes, food, and bills, etc.

Use this quiz to check your grade 6 to 7 studentsβ 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 FREEUse this quiz to check your grade 6 to 7 studentsβ 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 FREEHow 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.**

If 2 tins of beans cost 96Β’, 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}=481 tin of beans costs 48Β’.

2**Find 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=336Remember 336Β’=\$ 3.36.

3**Write the answer.**

7 tins of beans cost \$ 3.36.

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.

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.

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.

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.

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.

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).

- 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.

**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.

- Ratio problem solving
- Simplifying ratios
- Dividing ratios
- Unit rate math
- How to calculate exchange rates
- Ratio to fraction
- Ratio to percent
- Ratio scale
- Constant of proportionality

1. The total price of 5 pencils is 70Β’. What is the total price of 3 pencils?

67Β’

62Β’

47Β’

42Β’

70\div{5}=14

1 pencil costs 14Β’.

14\times{3}=42

3 pencils cost 42Β’.

2. A car costs \$ 270 to rent for 3 days. How much would it cost to rent the car for 14 days?

\$ 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.

3. 100 \, ml of milk contains 44 calories. Find how many calories are in 65 \, ml of milk.

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.

4. A recipe needs \cfrac{1}{4} cup of sugar to make caramel custard for 4 people. How much sugar is needed to make the caramel custard for 10 people?

\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.

5. It takes 7 people 10 days to build a wall. How long will it take 5 people to build the same wall?

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.

6. It costs 3 people \$ 25 each to hire a taxi. How much would it cost 5 people each to hire the same taxi?

\$ 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.

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.

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

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.

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 }}β.

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

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).

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[FREE] Common Core Practice Tests (3rd to 8th Grade)

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