3. Ratio and Rate
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Chapter 1
3.
Ratio and Rate
This lesson delves into the mathematical concepts of ratio and rate, explaining how these are used in everyday life. For instance, it discusses how to calculate the unit rate to determine the best value when shopping for items like chocolate bars. It also explores how ratios can be used in cooking, such as determining the amount of ingredients needed for a homemade pizza. The concept of equivalent ratios is highlighted, which is useful for comparing different scenarios, like the cost per T-shirt or the speed of travel. Understanding these concepts is essential for making informed decisions in various aspects of life, including finance, cooking, and travel.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Ratio and Rate
Slide of 10
This lesson will explore situations where ratios and rates are used to create mixtures, calculate times, compare prices, and more.
Catch-Up and Review
Here is a recommended reading to go over before getting started with this lesson.
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Explore
Mixing Colors to Create Shades of a Third Color
Play with the amount of each color to create shades of a third color.
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Discussion
Ratio
A ratio is a comparison of two quantities that describes how much of one thing there is compared to another. Ratios are commonly represented using colon notation or as fractions. They are read as the ratio of a to b,
where b is a non-zero number.
ccc Colon Notation &&Fraction a:b && a/b
The ratio a:b means that for every a units of one quantity, there are b units of another quantity. Ratios can be part-to-part or part-to-whole.
| Part-To-Part | Part-To-Whole | |
|---|---|---|
| Explanation | Describes how two different groups are related | Describes the relationship between a specific group to a whole |
| Example 1 | The number of sophomores to freshmen on the basketball team is 7:15. | The number of sophomores to all basketball team members is 7:22. |
| Example 2 | The number of mangoes to jackfruits the vendor has is 10:20. | The number of mangoes to all fruits the vendor has is 10:42. |
Ratios that express the same relationship between quantities are called equivalent ratios. For instance, consider the ratios of pages read per minute by Tearrik and by Zain. Tearrik's Ratio& &Zain's Ratio 27/15& &45/25 These ratios can be simplified by finding the greatest common factor of their numerator and denominator. That factor can then be used to rewrite each ratio.
| Fraction Form | Greatest Common Factor | Rewrite | Simplify | |
|---|---|---|---|---|
| Tearrik | 27/15 | GCF(27,15)= 3 | 9* 3/5* 3 | 9/5 |
| Zain | 45/25 | GCF(45,25)= 5 | 9* 5/5* 5 | 9/5 |
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Pop Quiz
Simplifying Ratios
The applet shows different ratios in colon notation. Write the simplest form of the indicated ratio. Some ratios might already be in their simplest form.
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Example
How Many Mushrooms Are Needed for a Homemade Pizza?
Zain's mother asks them for help to make homemade pizzas for dinner.
For every six olives on the pizza, Zain wants to add two mushrooms. If they plan to put 30 olives on the pizza, how many mushrooms must the pizza have?
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Hint
Write the ratio of the number of olives to the number of mushrooms. Write an equivalent ratio to the original ratio where the numerator of the new ratio is 30. What number multiplied by 6 gives 30? Multiply 2 by the number found previously to find the number of mushrooms needed.
Solution
For every six olives on the pizza, two mushrooms must be added. At the moment, the ratio of olives to mushrooms on the pizza is as follows.
6:2 ⇔ 6/2
Zain wants to put 30 olives on the pizza. To find the number of mushrooms they need toput on the pizza, we can write a ratio equivalent to 6:2 with a numerator of 30.
6/2=30/?
Note that 6* 5= 30. This means we will multiply 2 by 5 to find the missing denominator of the equivalent fraction.
6* 5/2* 5=30/10
In this equivalent ratio, 30 represents the olives that Zain will put on the pizza. Then, 10 represents the number of mushrooms Zain needs for the 30 olives.
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Example
Using Ratios to Mix Colors to Create a Desired Shade
Zain wants to paint a birdhouse the color of an orange poppy flower. They want it to be a specific shade of orange. This shade is a result of a mixture of red and yellow in a ratio of 5:4.
Zain wants to create 18 ounces of this shade of paint. Which option describes the correct amounts of yellow and red paint are necessary to create the mixture?
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Hint
Find the total amount described by the given ratio. Use this total amount to write a part-to-whole ratio for each color. Write an equivalent ratio for one of the part-to-whole ratios. The denominator of this equivalent ratio is 18. What number multiplied by 9 gives 18? Find the numerator of the equivalent ratio
Solution
Consider the ratio of red to yellow paint that creates the specific shade of orange Zain wants.
Ratio of the Mixture 5: 4 ⇔ 5/4
This is a part-to-part ratio because it describes how much red paint there is in the mixture compared to the yellow paint. The orange paint will consist of 9 parts in total.
5+ 4=9
The amount of red paint compared to the total amount of paint in the mixture can be used to create a part-to-whole ratio. This also us to calculate the ratio for the yellow paint.
| Part-To-Whole Ratios | |
|---|---|
| Red Paint | Yellow Paint |
| 5/9 | 4/9 |
We can find the amount of red and yellow paint needed for the 18 ounces of paint by using equivalent ratios, starting with the red paint ratio. This equivalent ratio will have a denominator of 18. 5/9=?/18 Since 9* 2=18, we can find the numerator of the equivalent ratio by calculating the product of 5 and 2. 5* 2/9* 2=10/18 We can express this ratio in words. Of the 18 ounces of paint, 10 must be red to create the specific shade of orange Zain wants. The amount of yellow paint can be found by subtracting 10 from 18. Ounces of Yellow Paint 18-10=8 Zain needs 10 ounces of red paint and 8 ounces of yellow paint to create 18 ounces of the desired shade of orange.
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Discussion
Rate
A rate is a ratio that compares two quantities measured in different units. For example, if a certain species of bamboo grows 27 feet in height in 2 years, then its rate of growth is 27ft2years. Here are some other example rates.
| Scenario | Rate | Unit Rate |
|---|---|---|
| Kriz finds 20 Pokémon every 10 days. | 20 Pokémon per 10 days, 10 Pokémon per 5 days |
2 Pokémon per 1 day, 730 Pokémon per 1 year |
| At a party, 42 candies were eaten by 6 kids. | 42 candies per 6 kids, 21 candies per 3 kids |
7 candies per 1 kid |
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Example
What if a Greater Amount of Pizzas Is Needed?
The Zain is thinking of making pizzas for a local charity.
External credits: @macrovector
It took Zain 30 minutes to prepare two pizzas. How much time do they need to prepare 20 pizzas if this rate is kept?
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Hint
Write the rate for this situation. Divide the numerator and denominator of the rate by its denominator to find the unit rate.
Solution
This situation compares the time it takes to prepare the pizzas. It took Zain 30 minutes to prepare 2 pizzas. This rate can be written as a fraction.
Rate: 30minutes/2pizzas
Now divide the numerator and denominator of this fraction by 2 to find the unit rate. ccc
Unit Rate: [0.2em]
30÷ 2minutes/2÷ 2pizzas=15minutes/1pizza
This means that it takes 15 minutes to make one pizza. The time it will take to make 20 pizzas can now be calculated by multiplying the unit rate by 20. 15minutes/1pizza* 20Pizzas= 300 minutes
Zain will spend about 300 minutes preparing the 20 pizzas for the local charity. We need to give the answer in hours, so let's divide 300 by 60 to get how many hours this time represents. 300minutes/60minutes= 5 hours
It will take Zain 5 hours to make the 20 pizzas.
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Example
Finding Missing Distances Using Rates
Zain's family is taking a small road trip to visit family.
External credits: Hari Panicker
If it takes them two hours to travel 160 miles, how far will they drive in 3.5 hours in total if they drive at a steady speed?
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Solution
Let's start by writing the rate of the distance the family travels to the time it takes to travel it as a fraction.
Rate: 160 miles/2 hours
Now we will divide the numerator and denominator of the rate by 2 to find the unit rate of speed.
Unit Rate: 160÷ 2 miles/2 ÷ 2 hours=80 miles/1 hour
Zain's family travels 80 miles in one hour. Now we want to find how long the family will travel after 3.5 hours. Let's multiply the unit rate by 3.5 to determine how far Zain's parent's work trip was.
80miles/1hour* 3.5hours= 280 miles
Zain's family will drive 280 miles in 3.5 hours.
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Closure
Identifying the Best Deal
Think about going to the grocery store. There are tons of different brands, and the same brand usually offers the same product packaged in different sizes. Deciding what to buy can be overwhelming.
People tend to think larger packages have a lower price per unit. Actually, that is true only sometimes. Comparing the unit rate will help us decide whether buying more smaller packages or one large package offers the better deal. The unit rate describes the cost per pound, quart, kilogram, or other corresponding unit of measure.
|
Compare unit prices to find the best value for money. |
Consider the following advertisement. Delicious standard-sized and giant-sized chocolate bars are on sale.
Is the giant-size bar a better option? Let's write the rates as fractions to help us find the unit rate for each bar later.
| Standard-Size | Giant-Size | |
|---|---|---|
| Rate | $1.19/1.55 oz | $8.59/7oz |
Divide the numerator and denominator of the standard size rate by 1.55 to get its unit rate of dollars per ounce. Similarly, we will divide the numerator and denominator of the giant size ratio by 7.
| Standard-Size | Giant-Size | |
|---|---|---|
| Rate | $1.19/1.55oz | $8.59/7oz |
| Unit Rate | $0.77/1oz | $1.23/1oz |
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Ratio and Rate
Exercise 1.1
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We want the ratio that compares the number of circles in the diagram to the number of triangles. In other words, we want a part-to-part ratio. Let's count the circles and triangles in the diagram.
We can see that there are 18 circles and 30 triangles. The number of circles represents the numerator of the ratio. The number of triangles gives its denominator. Let's write it! Ratio of◯ to △: 18/30 We can simplify this ratio using the greatest common factor GCF of 18 and 30. Let's write the factors of these numbers to find their GCF. &Factors of 18:1, 2, 3, 6, 9, 18 &Factors of 30:1, 2, 3, 5, 6, 10, 15, 30 The GCF( 18, 30)= 6. We can now rewrite the numerator and denominator of the ratio to simplify it. Ratio of◯ to △: 18/30=3*6/5*6 = 3/5 This means that for every three circles in the diagram, there are five triangles.
Let's now determine the ratio of T-shirts to dresses in the diagram. This is also a part-to-part ratio. Let's count each of them to find how many of each item there are.
There are 12 T-shirts and 5 dresses. The number of T-shirts represents the numerator of the ratio. The number of dresses gives the denominator of the ratio. Let's write it! Ratio of T-shirts to Dresses: 12/5
This part asks for the ratio of oranges to all fruits in the diagram. This represents a part-to-whole ratio. Let's first determine the number of oranges and the number of fruits that are not oranges.
We can see that there are 7 oranges. We also have 11 fruits that are not oranges. We will add these two numbers to find the total number of fruits in the diagram. Total Number of Fruits: 7+ 11= 18 Now, the number of oranges represents the numerator of the ratio. The total number of fruits is its denominator. The Ratio of Oranges to All Fruits: 7/18
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Solution
The ratio of cups of flour to cups of milk for a pancake recipe is 3:2. Identify the options that are equivalent to the given ratio.
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Consider the ratio of cups of flour to cups of milk for the pancake recipe. 3:2 ⇔ 3/2 We will check each of the four ratios to see if any is equivalent to 3:2. Let's multiply or divide the numerator and the denominator of each of these ratios by the same number to see if it produces the ratio 3:2. Let's begin with the ratio 9:6. 9:6 ⇔ 9÷ 3/6÷ 3=3/2 We can see that dividing the numerator and the denominator of the ratio 9:6 by 3 results in the ratio 3:2. This means that this ratio is equivalent to the ratio 3:2. Now, let's check the ratio 8:7. 8:7 ⇔ 8/7 The denominator of this fraction is 7, which is a prime number. This number cannot be divided anymore. This means that this fraction cannot be simplified and cannot be equal to 3:2. Now, let's check the ratio 4:3. 4:3 ⇔ 4/3 This ratio has a similar situation to the previous one. Its denominator is a prime number. This ratio is also in its simplest form. This means that the ratio 4:3 is not equivalent to the ratio 3:2. Finally, let's look at the ratio 6:4. 6:4 ⇔ 6÷ 2/4÷ 2=3/2 The ratio 6:4 is equivalent to the ratio 3:2. Let's summarize our results in a table.
| Ratio | Equivalent to 3:2? |
|---|---|
| 9:6 | Yes |
| 8:7 | No |
| 4:3 | No |
| 6:4 | Yes |
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This situation compares dollars to T-shirts because it asks for the cost of a T-shirt. We can begin by writing this rate as a fraction. $16/4T-shirts We can now use this rate to find its unit rate. That will give us the cost of a T-shirt. Let's divide the numerator and the denominator of this rate by its denominator to get the unit rate.
| Rate | Unit Rate |
|---|---|
| $16/4T-shirts | 16÷ 4/4 ÷ 4=$4/1T-shirt |
This means that the cost per T-shirt is $4.
We want now to find how many miles Dylan travels. Let's first write the rate of distance traveled to the time it takes as a fraction. 48Miles/6 Hours We can now find the unit rate by following a similar process as in the previous part. Let's divide the numerator and the denominator of this rate by 6.
| Rate | Unit Rate |
|---|---|
| 48Miles/6 Hours | 48÷ 6/6 ÷ 6=8Miles/1Hour |
This means that Dylan travels 8 miles per hour.
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Solution
Read each situation carefully. Identify which is different from the others.
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Let's look at the given situations.
These situations represent rates because they compare two different measures, kilograms and meters. Let's write the rate for each to find its unit rate. This way, we can compare them and see which one is different. Let's look at the first one. 9kg : 3m Now, we can divide both units by 3 to get the unit rate. 9/3kg : 3/3m [0.3em] ⇕ [0.3em] 3kg: 1m We can continue the same way to find the rest of the ratios.
| Situation | Rate | Divide | Unit Rate |
|---|---|---|---|
| 9 kilograms for every 3 meters | 9kg : 3m | 9/3kg : 3/3m | 3kg:1m |
| 12 kilograms per 3 meters | 12kg : 3m | 12/3kg : 3/3m | 4kg:1m |
| 18 kilograms for every 6 meters | 18kg : 6m | 18/6kg : 6/6m | 3kg:1m |
| 15 kilograms per 5 meters | 15kg : 5m | 15/5kg : 5/5m | 3kg:1m |
Note that three out of four rates have the same unit rate of 3 kilograms per meter. This means that they are equivalent ratios. The only rate that does not belong with the other three is 12 kilograms per 3 meters
because it has a unit rate of 4 kilograms per meter.
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Ratio and Rate
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