Unlocking Proportions with Variables: Solving Proportional Equations and Ratios Explained

	Number of Theaters
China	$54164$
U.S.	$40246$
India	$11000$
France	$5741$
Canada	$3114$
South Africa	$800$

Number of Theaters

China

54164

U.S.

40246

India

11000

France

5741

Canada

3114

South Africa

800

	Number of Theaters	Population	Number of Theaters per $10000$ People
China	$54164$	$1433783686$	$0.37$
U.S.	$40246$	$329064917$	$1.22$
India	$11000$	$1366417754$	$0.08$
France	$5741$	$65129728$	$0.88$
Canada	$3114$	$37411047$	$0.83$
South Africa	$800$	$58558270$	$0.14$

Number of Theaters

Population

Number of Theaters per

10000

People

China

54164

1433783686

0.37

U.S.

40246

329064917

1.22

India

11000

1366417754

0.08

France

5741

65129728

0.88

Canada

3114

37411047

0.83

South Africa

800

58558270

0.14

	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 $10 : 42 .$

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 .

a First, each solution will be analyzed separately. Then, the exercise will be solved correctly.

Jordan's Solution

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.

$\frac{a}{b} = \frac{c}{d} \Rightarrow a d = b c$

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 \cdot 18 15 \cdot 4.5 = 3.75 \cdot 4.5 \times = 18 \cdot 3.75 ✓

Hence, Jordan's solution is not correct.

Emily's Solution

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 \cdot 3.75 15 \cdot 4.5 = 18 \cdot 4.5 \times = 18 \cdot 3.75 ✓

Therefore, Emily's solution is also incorrect.

Correct Solution

Now, it is time to solve the exercise properly.

\frac{1 5}{1 8} = ? \frac{3 . 7 5}{4 . 5}

CrossMult

Cross multiply

15 \cdot 4.5 = ? 18 \cdot 3.75

Multiply

67.5 = 67.5

Since the cross products are equal, the ratios can be concluded to form a proportion.

b In order to determine whether

\frac{2}{7}

and

\frac{6}{2 1}

form a proportion, set those ratios equal in the form of a potential proportion. Then, use the Cross Products Proportion to verify if the proportion is true.

\frac{2}{7} = ? \frac{6}{2 1}

CrossMult

Cross multiply

2 \cdot 21 = 7 \cdot 6

Multiply

42 = 42

A true equality was obtained. Therefore, the ratios form a proportion.

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

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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Catch-Up and Review

Predicting the Time to Run an Ultra-Marathon

Comparing the Number of Movie Theaters Across Countries

Defining a Ratio

Ratio

Comparing Two Ratios

The Concept of Proportion

Proportion

Finding Equal Ratios

Means-Extremes Property of Proportion

Cross Products Property

Proof

Learning to Apply Cross Products Property Correctly

Answer

Hint

Solution

Jordan's Solution

Emily's Solution

Correct Solution

Solving Proportions

Rate and Unit Rate

Rate

Using Rate and Unit Rate in Cooking

Hint

Solution

Calculating the Rates of Making Invitations

Hint

Solution

Calculating the Time to Finish an Ultra-Marathon

Hint

Solution

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