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 3.1
Diego can paint one-third of a wall in an hour. Mark can complete one-sixth of the wall in an hour. How long will it take to paint the wall if they work together?
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We are told that Diego can paint one-third of a wall in an hour. Mark paints one-sixth of the wall during this time These relationships can be written as two unit rates.
| Diego | Mark |
|---|---|
| 1/3of a wall:1hour | 1/6of a wall:1hour |
We can find the unit rate for the portion of the wall these two guys can do in an hour together by adding each one's portion. 1/3+1/6of a wall:1hour Let's simplify this rate by adding the fractions in the left-hand side.
Diego and Mark can complete one-half of a wall in an hour working together. Since we want the time it will take them to paint a whole wall, we need to calculate the unit rate in hours per portion of the wall painted. Let's change the order of the obtained rate to get the hours on the left side of the rate. Hours to Portion of Wall Painted 1hour:1/2of a wall We can now divide both units of this rate by 12 to get the unit rate for the wall painted.
This means that it will take Diego and Mark 2 hours painting a wall together.
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Dylan, Kevin, and Heichi run in a 120-meter race.
When Dylan finishes, Kevin is 20 meters behind him. When Kevin finishes, Heichi is 24 meters behind him. The three of them run at a constant speed through the race. How far behind Dylan was Heichi when Dylan finished?
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Dylan ran 120 meters when he finished the race. This means that Kevin has run 120−20= 100 meters of the total race. We can write this relationship as the ratio of the distance Dylan runs to the distance Kevin runs. Ratio of Dylan to Kevin 120m: 100m Now, when Kevin finishes the race, Heichi has run 120-24= 96meters. Let's write the ratio of the distance Kevin runs to the distance Heichi runs. Ratio of Kevin to Heichi 120m: 96m We need now to create equivalent ratios to this last ratio. In doing so, we try to get on its left-hand side of this equivalent ratio 100. This way, we can know how far Heichi was when Dylan finished the race.
This means that when Dylan finished the race, Heichi had run 80 meters. Therefore, Heichi was 120-80= 40 meters behind Dylan when Dylan finished the race.
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Ratio and Rate
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