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| 14 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Jordan's best friend Emily really loves watching movies! As they walked home from the cinema, Emily wondered aloud, "Which country's population, as a whole, has the easiest access to movie theaters?" After an intense weekend of internet research, she created the following table.
Number of Theaters | |
---|---|
China | 54 164 |
U.S. | 40 246 |
India | 11 000 |
France | 5 741 |
Canada | 3 114 |
South Africa | 800 |
Well, China has the greatest amount of theaters, so people in China must have the easiest access — so it seems. But wait! Countries have different populations. What if the number of theaters is largest simply because the population is largest? Emily decides it is better to calculate the number of movie theaters in the respective country per 10 000 people.
Number of Theaters | Population | Number of Theaters per 10 000 People | |
---|---|---|---|
China | 54 164 | 1 433 783 686 | 0.37 |
U.S. | 40 246 | 329 064 917 | 1.22 |
India | 11 000 |
1 366 417 754 | 0.08 |
France | 5 741 | 65 129 728 | 0.88 |
Canada | 3 114 | 37 411 047 | 0.83 |
South Africa | 800 | 58 558 270 | 0.14 |
A relation in which two values, such as the number of theaters and the number of people, are compared is called a 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. |
When two ratios are equal in value, an equals sign can be written between them, which results in creating a proportion.
A proportion is an equation showing the equivalence of two ratios, or fractions, with different numerators and denominators. a/b = c/d or a:b=c:d The first and last numbers in the proportion are called the extremes, while the other two numbers are called the means. ↓ a0.75em extremes means : ↑ b= ↑ c:↓ d
As an example of proportionality, consider slices of pizza. Depending on the number of times it has been sliced, the same amount of pizza could be cut into 1, 2, or 4 pieces.Consider the given ratio. Then, analyze the values of the ratios in each answer and choose which one forms a proportion with the given ratio.
Up to this point, each proportion has dealt with fractions. However, every proportion can be rewritten as an equation without any fractions. This is a property that is especially useful when solving proportions for an unknown variable.
In a proportion, the product of the extremes is equal to the product of the means.
a/b=c/d ⇒ ad=bc
This property is also known as cross-multiplication or Means-Extremes Property of Proportion.
LHS * b=RHS* b
a/c* b = a* b/c
LHS * d=RHS* d
Commutative Property of Multiplication
Jordan and Emily, exhausted from watching so many movies, sat down to finally do their Math homework. In one exercise, they were asked to determine whether the two given ratios 1518 and 3.754.5 form a proportion. Both girls used the Cross Products Property, but their solutions were different. Jordan gasped in bewilderment.
In the first step of Jordan's solution, she writes the given ratios as a potential proportion. To highlight the fact that it is not yet known whether the ratios form a proportion, she put a question mark above the equals sign.
Then, it is shown that Jordan applied the Cross Products Property to obtain the following result.
In order to determine whether that property was applied properly, recall what it states.
a/b=c/d ⇒ ad= bc
According to the property, in any proportion, the product of the extremes is equal to the product of the means. Identify the extremes and means in the exercise girls were solving.
To properly apply the property, the means and the extremes should be multiplied and set equal. However, Jordan mistakenly multiplied the numerator and denominator of each fraction. 15* 18 &= 3.75* 4.5 * 15* 4.5 &= 18* 3.75 ✓ Hence, Jordan's solution is not correct.
Next, Emily's solution can be analyzed. After writing the ratios as a potential proportion, she also chose to apply the Cross Products Property.
As can be seen, instead of multiplying the extremes and means of the fractions, Emily multiplied the numerators and denominators of the fractions and then set them equal. 15* 3.75 &= 18* 4.5 * 15* 4.5 &= 18* 3.75 ✓ Therefore, Emily's solution is also incorrect.
Solve the given proportion for the unknown variable x.
In real-life situations, two quantities that are being compared often have different units. Such ratios have a specific name.
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 |
After finishing their homework, Jordan and Emily decided to bake some cookies together. The recipe they found calls for 2 cups of water for every 5 cups of flour. However, the girls wanted to cook more cookies to be able to share with their classmates. They used 12 cups of flour and now they need to figure out how many cups of water they need.
Help Jordan and Emily solve this problem so that they can share the delicious cookies!
Cross multiply
Multiply
.LHS /5.=.RHS /5.
Calculate quotient
Rearrange equation
? cups of flour/1 cup of water It is given that the recipe calls for 2 cups of water for every 5 cups of flour. This is equivalent to saying that for every 5 cups of flour 2 cups of water are required. 5cups of flour/2cups of water Now, to find how many cups of flour needed for just 1 cup of water, both the numerator and denominator of the fraction should be divided by 2. 5 /2cups of flour/2 /2cups of water ⇕ 2.5cups of flour/1cups of water Therefore, the unit rate of flour to water is 2.5 cups of flour to 1 cup of water.
Emily, eager to watch more movies, planned to host a movie night. To invite her friends, she started making colorful invitations. She made the first batch in just 21 minutes, but then her mother asked her to help in the kitchen. She had to take a break from making invitations.
When she returned to making invitations, her younger sister offered her help in creating 5 invitations. After 7 minutes, Emily realized that together they already have made twice the number of invitations she made in the first sitting.
Cross multiply
.LHS /7.=.RHS /7.
Distribute 3
LHS-6x=RHS-6x
.LHS /(- 5).=.RHS /(- 5).
21/3=7 min
Emily's Sister's Rate: 5invitations/7minutes At the same time, it was determined in Part B that Emily makes 1 invitation in 7 minutes. Emily's Rate: 1invitation/7minutes By comparing those rates, it can be concluded that Emily's sister is working much faster. 5invitations/7minutes > 1invitation/7minutes It is a good thing she offered to help!
Calculate Jordan's speed in miles per hour. Then, use it to set a proportion.
a/b=a * 60/b * 60
1* a=a
Rewrite 60min as 1h
a/c* b = a* b/c
a/b=.a /6./.b /6.
Multiply
a/b=.a /3./.b /3.
Determine whether each pair of ratios are equivalent ratios.
To determine if the given ratios are equivalent, we will set them equal to one another and use the Cross Products Property. If the resultant equation is true, then the ratios are equivalent.
Since this is a true statement, we can conclude that the ratios are equivalent.
In a similar fashion, we will set the given ratios equal and see if they form a true equation to see whether they are equivalent.
A false statement was obtained. Therefore, the ratios are not equivalent.
Again, to determine whether 1314 and 182196 are equivalent, we will first set them equal. Then, we will cross multiply to see if the formed equation is true.
Because a true statement was obtained, the ratios are equivalent.
Solve each proportion. If necessary, round to the nearest hundredth.
To solve the proportion, we will start by using the Cross Products Property. The goal is to find the value of m by isolating it on one side of the equation.
Similarly to Part A, we will solve the given proportion by using the Cross Products Property.
To solve the proportion, we can first use the Cross Products Property. Then, we will try to isolate the variable on one side of the equation.
The new advanced Shiny Glow Cleaner can wash 148 cars in 4 hours. At that rate, how many cars can it wash in 9 hours?
We know that the Shiny Glow Cleaner washes the cars at a constant rate. Because of that, we can find how many cars it can wash in 9 hours by setting a proportion. Since the cleaner can wash 148 cars in 4 hours, its rate is equal to the ratio of 148 and 4. Rate: 148cars/4hours Let c be the number of cars washed in 9 hours. Similarly, the rate of the car is equal to the ratio of c and 9. Rate: ccars/9hours By setting these two ratios equal, we can form a proportion. 148cars/4hours=ccars/9hours Let's solve it by using the Cross Products Property and then isolating c.
The Shiny Glow Cleaner can wash 333 cars in 9 hours. Not bad at all!
To find the height of the model, we will set a proportion. We are told that every 1 foot of the scale model represents 850.625 feet of the actual tower. Model/Tower → 1ft/850.625 ft Let's call h the height of Kriz's model. Using this variable and the height of the tower, 2722 feet, we can form another ratio. Model/Tower → hft/2722 ft Since these two ratios are equivalent, they can be set equal to one another, which creates a proportion. 1ft/850.625 ft = hft/2722 ft Finally, we can solve it by applying the Cross Products Property and then isolating h.
The model that Kriz will make is going to be 3.2 feet tall. Quite impressive!
We know that 2 out of every 7 students at Jefferson High School are members of some school club. This means that the other 5 students out of 7 are not in any club. We can use these values to form a ratio. 5not in clubs/7students To determine how many students in total are not in any club, we will equate this ratio to the ratio of the total number of students who are not in any club to all students at the school. 5not in clubs/7students=Students not in clubs/Total number of students The total number of students at Jefferson High School is 1813, so let's call the number of students not in any club n to form the final proportion. 5not in clubs/7students=n/1813 We can solve this proportion by using the Cross Products Property and isolating the variable.
Therefore, 1295 out of Jefferson High School's 1813 students are not members of any club.
In a survey, 15 % of the students said that their daily amount of screen time is less than 2 hours. There were 587 students whose screen time was less than 2 hours.
To find the total number of students who took part in the survey, we will write a proportion — an equation that shows the equality of two ratios.
We know that 15 % of the students who reported that their daily screen time is less than 2 hours. Let's rewrite this percentage as a ratio.
15 % = 15/100
This means that out of every 100 students, 15 reported less than 2 hours of screen time per day. To write the second ratio, recall that 587 students corresponds to the 15 % who had less than 2 hours of screen time per day. Let's call n the total number of students who took part in the survey.
587/n
This ratio can be read as out of n students, 587 reported less than 2 hours of daily screen time.
Finally, we can write our proportion by setting the two ratios equal to each other. 15/100= 587/n Let's solve the proportion by using the Cross Products Property.
The total number of students who participated in the survey was 3913.