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.
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Lesson Settings & Tools
| | 10 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Ratio and Rate
Slide of 10
Consider some fun scenarios. Is there enough salsa for all these chips? What if the gas tank of the school bus has less gas than what it needs to make it to school? Amounts are compared to each other to find the right proportions. This lesson will explore similar 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 Fraction && Colon Notation 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 using colon notation. Write the simplest form of the indicated ratio. Consider that some ratios might already be in their simplest form.
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Example
What Amount of Mushroom Is Needed for a Homemade Pizza?
It’s a beautiful Friday. Zain and their mother are celebrating their other parent's return home tonight after a week-long work trip. Zain's mother asks them for help to make homemade pizzas for dinner.
Their recipe is strict and claims that for every six olives on the pizza, two mushrooms must be added. If Zain plans 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
The recipe claims that for every six olives on the pizza, two mushrooms must be added. At the moment, the number of olives to the number of mushrooms on the pizza is the following ratio.
6:2 ⇔ 6/2
Now, Zain and their mother have 30 olives to put on the pizza. They want to know how many mushrooms they need to add to the pizza following the recipe exactly. This number can be found by writing an equivalent ratio to 6:2. The new ratio will have a numerator of 30.
6/2=30/?
Note that 6* 5= 30. This means that 2 must be multiplied by 5 to find the missing number of the equivalent fraction.
6* 5/2* 5=30/10
In this equivalent ratio, 30 represents the olives that Zain and their mother will put on the pizza. Additionally, 10 represents the number of mushrooms corresponding to 30 olives. What a delicious smell! The pizza is almost ready.
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Example
Using Ratios to Mix Colors to Create a Desired Shade
The two pizzas Zain and their mother made are still in the oven, smelling amazing. They decide to pass time by planning another weekend project — paint their house 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's mother says they will need a total of 18 liters of this shade of orange. Then, they will need to calculate how many liters of yellow and red paint are necessary to create the mixture. Which option describes the right amount?
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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 the family chose.
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. Note that 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 determined using this information. This means it is a part-to-whole ratio. This also allows for the calculation of the ratio for the yellow paint.
| Part-To-Whole Ratios | |
|---|---|
| Red Paint | Yellow Paint |
| 5/9 | 4/9 |
The amount of red and yellow paint needed for the 18 liters of paint will be found using equivalent ratios. A ratio equivalent to the red paint ratio will be written. This equivalent ratio will have a denominator of 18. 5/9=?/18 Both denominators are present. Check them and what is noticeable? Well, 9* 2= 18. This means that the numerator of the equivalent ratio is given by calculating the product of 5 and 2. 5* 2/9* 2=10/18 This ratio can be expressed in words. Of the 18 liters of paint, 10 must be red to create the specific shade of orange Zain wants. The number of liters of yellow paint can be found by subtracting 10 from 18. Liters of Yellow Paint 18- 10= 8 Zain's family needs 10 liters of red paint and 8 liters of yellow paint to create 18 liters of the desired shade of orange. Their house is going to look so stylish!
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Discussion
Rate
A rate is a ratio that compares two quantities measured in different units. For example, a certain species of bamboo grows 27 feet in height in 2 years. Then, 27ft2years is its rate of growth. Here are some other possible examples of rate.
| 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 two pizzas Zain and their mother prepared earlier were a such a great success. Now they are thinking to prepare more pizzas for a local charity.
External credits: @macrovector
Recall that it took 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. Multiply the unit rate by the number of pizzas that are needed. Divide the result by 60 to get the number of hours of prep time.
Solution
This situation compares the time it takes to prepare the pizzas. Recall that it took Zain and their mother 30 minutes to prepare 2 pizzas. This rate can be written as a fraction.
Rate: 30minutes/2pizzas
Now, the numerator and denominator of this fraction will be divided by 2 to find the unit rate. ccc
Rate: [0.2em]
30minutes/2pizzas [1.5em]
Unit Rate: [0.2em]
30÷ 2minutes/2÷ 2pizzas=15minutes/1pizza
This means that a pizza takes 15 minutes to be ready. The time it will take to prepare 20 pizzas can now be calculated by multiplying the unit rate by 20. 15minutes/1pizza* 20Pizzas= 300 minutes
Zain's family will spend about 300 minutes to preparing the 20 pizzas for the local charity. Notice that the answer is asked in hours. Divide 300 by 60 to get how many hours this time represents. 300minutes/60minutes= 5 hours
After 5 hours, the 20 pizzas are ready. Zain's family is taking them to the local charity. What a good lesson for Zain.
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Example
Finding Missing Distances Using Rates
Zain realizes that their parent is so late arriving home. Zain calls. "Where are you?" The parent picks up and says, "Hey, Zain! I am 160 miles away. I will be home in 2 hours."
External credits: Hari Panicker
Zain's parent challenges them to find out how far this work trip was if it takes 3.5 hours in total and the same rate is kept. How far was it?
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Solution
Begin by writing the rate of the distance traveled to the time it takes as a fraction.
Rate: 160 miles/2 hours
Now, divide the numerator and denominator of this rate by 2 to find the unit rate of this situation.
Rate: 160 miles/2 hours [1em] Unit Rate: 160÷ 2 miles/2 ÷ 2 hours=80 miles/1 hour
The distance that Zain's parent traveled in one hour is 80 miles. Now, it is given that the total trip will take 3.5 hours at this rate. Multiply the unit rate by 3.5 to determine how far Zain's parent's work trip was.
80miles/1hour* 3.5hours= 280 miles
Wow! Zain's parent's work trip was 280 miles away. That is far, far away. Zain feels relieved to have figured out how soon their parent will be home.
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Closure
Unit Rates Can Help to Identify the Best Deal
Think about going shopping at the market. 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 decide whether buying more smaller packages or one large package offers a 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? Write the rates as fractions. That will help to find the unit rate for each bar later.
| Standard-Size | Giant-Size | |
|---|---|---|
| Rate | 1.55oz/$1.19 | 7oz/$8.59 |
Divide the numerator and denominator of the standard size rate by 1.19 to get its unit rate. Similarly, divide the numerator and denominator of the giant size ratio by 8.59.
| Standard-Size | Giant-Size | |
|---|---|---|
| Rate | 1.55oz/$1.19 | 7oz/$8.59 |
| Unit Rate | 1.30oz/$1 | 0.81oz/$1 |
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Ratio and Rate
Exercise 2.1
Ignacio and Izabella have a collection of cards. Initially, for every 10 cards Ignacio had, Izabella had 4. Ignacio gives half of his cards to Izabella. Izabella now has 20 more cards than Ignacio. Initially, how many more cards did Ignacio have than Izabella?
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We want to know how many more cards Ignacio had initially than Izabella. We are told that Izabella had four cards for every ten cards Ignacio had. Let's write the ratio of Ignacio's cards to Izabella's cards. 10: 4 We can graph this ratio using squares arranged horizontally. Ten squares for Ingacio and four squares for Izabella.
Next, Ignacio gives half of his cards to Izabella. We can display this information in our diagram by moving five of the ten squares Ignacio has and adding them to the squares Izabella has.
Note that Izabella now has four extra squares than Ignacio. These four squares represent the 20 more cards Izabella has than Ignacio. Let's divide 20 by 4 to find how many cards each square represents. 20÷ 4 = 5cards In the first diagram representing the initial ratio, Ignacio had 6 squares more than Izabella. Each of these squares represents 5 cards. Let's multiply 6 by 5 to find how many more cards Ignacio had than Izabella initially. 6* 5=30cards This means initially Ignacio had 30 more cards than Izabella.
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The ratio of boys to girls in an art class is 7:5. If seven boys join the class today, how many girls need to join the class to keep the ratio of boys to girls?
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Consider the ratio of boys to girls in the art class. ccc Ratio &&Fraction Form 7: 5 &&7/5 We are told that today 7 boys joined the class. The number of girls that need to join the class to keep the ratio of 7: 5 is required. We can find this amount using an equivalent ratio to this ratio. The numerator of this equivalent ratio is given by adding the number of boys that joined the class today and the previous number of boys in the class. 7+ 7/ ⇔ 14/ The denominator of this new ratio is missing. We can determine this missing value by first finding what number times the numerator of the first ratio is 14. 7* 2= 14 This means that the missing value of the equivalent ratio is given by multiplying 5 by 2. 7* 2/5* 2 ⇔ 14/10 The number of girls in the class must be 10 to keep the ratio of 7: 5, given that 7 boys joined today. However, because there were previously 5 girls, we subtract this amount to get the number of girls that need to be joined. 10- 5=5girls This means that 5 girls need to join the class to keep the ratio of boys to girls.
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Maya's family is traveling by car to her relatives' house. Maya's father drives 120 miles in 2 hours.
Their destination is 500 miles away. Which statement best describes Maya's family location after 8 hours of driving at the same rate?
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We will identify which statement is true about Maya's family trip. Recall that Maya's father drives 120 miles in 2 hours. We can represent this situation as a rate. The rate of the distance traveled to the time it takes. Ratio of Distance to Time: 120Miles: 2 Hours We can now divide both units by 2. That will give us the unit rate because we will have 1 hour. Ratio of Distance to Time: 120/2Miles:2/2 Hours ⇕ 60Miles: 1 Hour This unit rate means that Maya's father drives 60 miles in 1 hour. Let's multiply this unit rate by 8 to see how far the family is after driving for 8 hours. 60* 8Miles: 1* 8 Hours ⇕ 480Miles:8 Hours Maya's relatives' house is 500 miles away. This means that if Maya's father keeps the same driving rate, the family has not arrived at their destination, yet. We can find how many miles are left to drive by subtracting 480 from 500. 500-480= 20Miles Maya's family still has 20 miles more to drive before reaching their destination.
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We need to determine the faster runner between Zosia and Zain. Let's first write the rate of time to distance traveled for each runner. In doing so, we can find the unit rates of each runner. Let's do it!
| Rate | |
|---|---|
| Zosia | 28min:2km |
| Zain | 36min:3km |
We can now divide both units of Zosia's rate by 2. This way, we can get Zosia's unit rate. On the other hand, we can divide Zain's rate by 3.
| Rate | Division | Unit Rate | |
|---|---|---|---|
| Zosia | 28min:2km | 28/2min:2/2km | 14min:1km |
| Zain | 36min:3km | 36/3min:3/3km | 12min:1km |
This means it takes Zosia 14 minutes to run 1 kilometer. Conversely, Zain is faster because they run 1 kilometer in 12 minutes.
We now multiply the unit rate of each runner by 5 to find how much it will take each to run the five-kilometer race. We can do this by using the fraction form of the rate. Let's first calculate the time for Zosia. 14 min/1 km* 5km=70min It will take Zosia 70 minutes to run the five-kilometer race. Now, let's calculate the time for Zain. 12 min/1 km* 5km=60min It will take Zain 10 minutes less than Zosia because he will run that distance in 60 minutes.
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Ratio and Rate
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