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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. Rate = 30minutes/2pizzas Now divide the numerator and denominator of this fraction by 2 to find the unit rate. 30÷ 2/2÷ 2=15/1 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. 15/1* 20= 300 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÷ 2/2 ÷ 2=80/1 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. 80/1* 3.5= 280 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.

Level 1

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Level 3

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?

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.

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?

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.

Ratio and Rate

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