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3. Ratio and Rate
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eCourses /
Pre-Algebra /
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
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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.

Explore

Mixing Colors to Create Shades of a Third Color

Play with the amount of each color to create shades of a third color.
Consider similar situations where two things are mixed to create a third one. How can situations like this be described?
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 to where is a non-zero number.

The ratio means that for every units of one quantity, there are 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 The number of sophomores to freshmen on the basketball team is The number of sophomores to all basketball team members is
Example The number of mangoes to jackfruits the vendor has is The number of mangoes to all fruits the vendor has is
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.
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
Zain
These ratios are equivalent because both simplify to Equivalent ratios can be created by multiplying or dividing the numerator and denominator of a ratio by the same number.
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.

An applet that generates random ratios. It asks for the simplest form of the given ratio.
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.

Pizza1.png

Their recipe is strict and claims that for every six olives on the pizza, two mushrooms must be added. If Zain plans to put olives on the pizza, how many mushrooms must the pizza have?

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 What number multiplied by gives Multiply 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.
Now, Zain and their mother have 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 The new ratio will have a numerator of
Note that This means that must be multiplied by to find the missing number of the equivalent fraction.
In this equivalent ratio, represents the olives that Zain and their mother will put on the pizza. Additionally, represents the number of mushrooms corresponding to olives. What a delicious smell! The pizza is almost ready.
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

Zain's mother says they will need a total of 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?

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 What number multiplied by gives 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.
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 parts in total.
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
The amount of red and yellow paint needed for the 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
Both denominators are present. Check them and what is noticeable? Well, This means that the numerator of the equivalent ratio is given by calculating the product of and
This ratio can be expressed in words. Of the liters of paint, must be red to create the specific shade of orange Zain wants. The number of liters of yellow paint can be found by subtracting from
Zain's family needs liters of red paint and liters of yellow paint to create liters of the desired shade of orange. Their house is going to look so stylish!
Discussion

Rate

A rate is a ratio that compares two quantities measured in different units. For example, a certain species of bamboo grows feet in height in years. Then, is its rate of growth. Here are some other possible examples of rate.
Some example rates and the corresponding scenarios
Rates might be most useful when finding how much of something is per unit of something else. Such a comparison is called a unit rate. If the given rates are not already unit rates, they can be determined by some calculations. Dive deeper into two of the previous examples.
Scenario Rate Unit Rate
Kriz finds Pokémon every days. Pokémon per days,
Pokémon per days 		Pokémon per day,
Pokémon per year
At a party, candies were eaten by kids. candies per kids,
candies per kids 		candies per kid
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.

Two pizzas placed together.
External credits: @macrovector
Recall that it took minutes to prepare two pizzas. How much time do they need to prepare pizzas if this rate is kept?

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 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 minutes to prepare pizzas. This rate can be written as a fraction.
Now, the numerator and denominator of this fraction will be divided by to find the unit rate.
This means that a pizza takes minutes to be ready. The time it will take to prepare pizzas can now be calculated by multiplying the unit rate by
Zain's family will spend about minutes to preparing the pizzas for the local charity. Notice that the answer is asked in hours. Divide by to get how many hours this time represents.
After hours, the pizzas are ready. Zain's family is taking them to the local charity. What a good lesson for Zain.
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 miles away. I will be home in hours."

External credits: Hari Panicker
Zain's parent challenges them to find out how far this work trip was if it takes hours in total and the same rate is kept. How far was it?

Hint

Write the rate of the distance traveled to the time it takes as a fraction. Find the unit rate. Multiply the unit rate by hours.

Solution
Begin by writing the rate of the distance traveled to the time it takes as a fraction.
Now, divide the numerator and denominator of this rate by to find the unit rate of this situation.
The distance that Zain's parent traveled in hour is miles. Now, it is given that the total trip will take hours at this rate. Multiply the unit rate by to determine how far Zain's parent's work trip was.
Wow! Zain's parent's work trip was miles away. That is far, far away. Zain feels relieved to have figured out how soon their parent will be home.
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.

A really confused person.

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.

The prize and weight of the standard size chocolate bar and the giant size bar.

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

Divide the numerator and denominator of the standard size rate by to get its unit rate. Similarly, divide the numerator and denominator of the giant size ratio by

Standard-Size Giant-Size
Rate
Unit Rate
Review the standard-size rate. The amount of chocolate when paying one dollar is ounces. On the other hand, the giant-size rate is ounces of chocolate per one dollar. This means buying smaller bars is better value for the buyer. The content of this lesson applies not only to chocolate, but to many real-life situations.



Level 1
Level 2
Level 3
Ratio and Rate
Ignacio and Izabella have a collection of cards. Initially, for every cards Ignacio had, Izabella had Ignacio gives half of his cards to Izabella. Izabella now has more cards than Ignacio. Initially, how many more cards did Ignacio have than Izabella?

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.

The ratio of boys to girls in an art class is If seven boys join the class today, how many girls need to join the class to keep the ratio of boys to girls?

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.

Maya's family is traveling by car to her relatives' house. Maya's father drives miles in hours.

Diagram of Maya's house and her relative's house
Their destination is miles away. Which statement best describes Maya's family location after hours of driving at the same rate?

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.

In practices Zosia can run kilometers in minutes. Zain can run kilometers in minutes.

Who runs faster?
Assume they keep their same practice rates during the race. How much faster, in minutes, will the faster runner finish ahead of the other?

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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