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Pre-Algebra View details

3. Ratio and Rate

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Pre-Algebra /

Chapter 1

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.

Student Learning Objectives:
Write rates and ratios using fractions and colons Simplify ratios Convert between rates

	10 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

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.

Greatest Common Factor GCF

Explore

Mixing Colors to Create Shades of a Third Color

Play with the amount of each color to create shades of a third color.

Similar situations occur when two quantities are mixed to create a new mixture. How can these situations be described mathematically?

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

These ratios are equivalent because both simplify to 95. 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 in colon notation. Write the simplest form of the indicated ratio. 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

How Many Mushrooms Are Needed for a Homemade Pizza?

Zain's mother asks them for help to make homemade pizzas for dinner.

For every 6 olives on the pizza, Zain wants to add 2 mushrooms. If they plan to put 30 olives on the pizza, how many mushrooms must the pizza have?

Hint

Write the part-to-part ratio of the number of olives to the number of mushrooms. Then, write an equivalent ratio with a numerator of 30.

Solution

For every 6 olives on the pizza, Zain wants to add 2 mushrooms. This gives us the following part-to-part ratio of olives to mushrooms. 6:2 ⇔ 6/2 Zain wants to put 30 olives on the pizza. To find the number of mushrooms they need, let's write a ratio equivalent to 6:2 with a numerator of 30. 6/2=30/? Since 6* 5= 30, we should multiply 2 by 5 to find the missing denominator of the equivalent ratio. 6* 5/2* 5=30/10 The pizza should have 10 mushrooms.

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 they need to create the mixture?

Hint

Find the total number of parts of paint needed for the mixture and write a part-to-whole ratio for each color. Then, write an equivalent ratio with a denominator of 18 for one of the part-to-whole ratios.

Solution

Let's recall the ratio of red to yellow paint that Zain needs for their shade of orange. 5: 4 ⇔ 5/4 They need 5 parts of red paint and 4 parts of yellow paint. So, the orange paint will consist of 9 parts in total. 5+ 4=9 Let's use this fact to write two part-to-whole ratios, one for the red paint and the other for the yellow paint.

Part-To-Whole Ratios
Red Paint	Yellow Paint
5/9	4/9

We can use equivalent ratios to find the amount of red paint needed for the 18 ounces of orange paint. This equivalent ratio will have a denominator of 18. 5/9=?/18 Since 9* 2=18, we should multiply the numerator by 2 to get the equivalent ratio. 5* 2/9* 2=10/18 So, 10 of the 18 ounces of paint should be red. We can find the amount of yellow paint by subtracting 10 from 18. 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.

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.

Rates are useful when finding how much of something there is per 1 unit of something else. These comparisons are called unit rates. If the given rates are not already unit rates, they can be determined by simplifying the rate until one unit is 1.

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

Example

What if a Greater Amount of Pizzas Is Needed?

The Zain is thinking of making pizzas for a local charity.

It took Zain 30 minutes to prepare two pizzas. How much time do they need to prepare 20 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.

Solution

Zain took 30 minutes to prepare 2 pizzas. We can write this rate as a fraction. rc Rate: & 30minutes/2pizzas Now divide the numerator and denominator of this fraction by 2 to find the unit rate. 30÷ 2minutes/2÷ 2pizzas=15minutes/1pizza It takes 15 minutes to make one pizza. Let's multiply the unit rate by 20 to find the time it will take to make 20 pizzas. 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. 300/60= 5 Zain will need 5 hours to make the 20 pizzas.

Example

Finding Missing Distances Using Rates

Zain's family is taking a small road trip to visit family.

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?

Hint

Write the rate of the distance traveled to the time it takes as a fraction. Find the unit rate and multiply it by 3.5 hours.

Solution

Let's start by writing the rate of the distance the family travels to the time it takes as a fraction. Rate = 160 miles/2 hours To find the unit rate of speed, we will divide the numerator and denominator of the rate by 2. 160÷ 2 miles/2 ÷ 2 hours=80 miles/1 hour Zain's family travels 80 miles in 1 hour. Next, let's multiply the unit rate by 3.5 hours to find how far will Zain's family drive in that time. 80miles/1hour* 3.5hours= 280 miles Zain's family will drive 280 miles in 3.5 hours.

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

Consider the standard-size rate. This bar costs $0.77 per ounce, but the giant-size bar costs $1.22 per ounce. This means buying smaller bars is better value for the buyer.

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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?

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 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?

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 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?

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 2 kilometers in 28 minutes. Zain can run 3 kilometers in 36 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.

Ratio and Rate

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