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In this lesson, using real-world examples, the comparison of two quantities will be practiced using a particular concept. Additionally, it will be shown how to use the concept to form and solve equations. ### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Challenge

Jordan trained all year to be able to run in the esteemed Summer Ultra-Marathon, which took place in her hometown Tallahassee, Florida. She comfortably ran the first $6$ miles of the ultra-marathon in $54$ minutes.

If she is able to maintain the same pace, how long will it take her to finish the remaining $29$ miles? Give the exact answer in hours.

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Explore

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 | $54164$ |

U.S. | $40246$ |

India | $11000$ |

France | $5741$ |

Canada | $3114$ |

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 $10000$ people.

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 | $800$ | $58558270$ | $0.14$ |

Discussion

A relation in which two values, such as the number of theaters and the number of people, are compared is called a ratio.

Concept

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.

$Fractionba Colon Notationa: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 $10:42.$ |

Explore

Consider different pairs of ratios. Compare their values and determine whether they are equal or not.

If the ratios are equal, how can this fact be written algebraically using only mathematics symbols?

Discussion

When two ratios are equal in value, an equals sign can be written between them, which results in creating a proportion.

Concept

A proportion is an equation showing the equivalence of two ratios, or fractions, with different numerators and denominators.

$ba =dc ora:b=c:d $

The first and last numbers in the proportion are called the extremes, while the other two numbers are called the means.
$ad↓means:↑b =↑c : extremes 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.
In this case, one-third of a pizza is the same amount of pizza as two-sixths or four-twelfths. If the simplified forms of two fractions are equal, then they are said to be proportional. For example, one-third is proportional to two-sixths and four-twelfths.

Pop Quiz

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.

Discussion

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.

Rule

In a proportion, the product of the extremes is equal to the product of the means.

$ba =dc ⇒ad=bc $

This property is also known as **cross-multiplication** or **Means-Extremes Property of Proportion**.

The Cross Products Property can be proven by using the Properties of Equality.

$ba =dc $

MultEqn

$LHS⋅b=RHS⋅b$

$a=dc ⋅b$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$a=dcb $

MultEqn

$LHS⋅d=RHS⋅d$

$ad=cb$

CommutativePropMult

Commutative Property of Multiplication

$ad=bc$

Example

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 $1815 $ and $4.53.75 $ form a proportion. Both girls used the Cross Products Property, but their solutions were different. Jordan gasped in bewilderment.

a Is either of the girls correct? Find and correct any mistakes made.

b Using the same method, determine whether $72 $ and $216 $ form a proportion.

a No, both girls made mistakes in their solutions.

b Yes, the ratios form a proportion.

a Recall what the Cross Products Property states. Identify the extremes and means in the proportion set by the girls. Are the second steps in either one's solutions correct?

b Write the given ratios as a potential proportion. Then, confirm if it is true by using the Cross Products Property.

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

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.

$ba =dc ⇒ad=bc$

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⋅1815⋅4.5 =3.75⋅4.5×=18⋅3.75✓ $

Hence, Jordan's solution is 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⋅3.7515⋅4.5 =18⋅4.5×=18⋅3.75✓ $

Therefore, Emily's solution is also b In order to determine whether $72 $ and $216 $ 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.

Pop Quiz

Solve the given proportion for the unknown variable $x.$

Discussion

In real-life situations, two quantities that are being compared often have different units. Such ratios have a specific name.

Concept

A rate is a ratio that compares two quantities measured in different units. For example, a certain species of bamboo grows $27$ feet in height in $2$ years. Then, $2years27ft $ is its rate of growth. Here are some other possible examples of rate.

Rates might be most useful when finding how much of something is per $1$ 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 $20$ 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

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!

a How many cups of water do the girls need to follow the recipe precisely?

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a What is the rate of water to flour given in the recipe? Set a proportion and solve it by using the Means-Extremes Property of Proportion.

b Think of how many cups of flour are needed per $1$ cup of water.

a In the recipe, it is given that for every $2$ cups of water $5$ cups of flour are needed. Using this information, the following ratio representing the rate of water to flour can be formed.

$5cups of flour2cups of water $

It is also known that the girls used $12$ cups of flour. Let $x$ represent the number of cups of water they need for that amount of flour. Using these two values, a new ratio can be formed. $12cups of flourxcups of water $

Since Jordan and Emily want to follow the recipe precisely, this ratio should be equivalent to the rate of water to flour given in the recipe. Therefore, the ratios can be set equal to form a proportion.
$5cups of flour2cups of water =12cups of flourxcups of water $

By solving this proportion for $x,$ the amount of needed cups of water can be calculated. The equation will be solved by applying the Means-Extremes Property of Proportion.
$52 =12x $

CrossMult

Cross multiply

$2⋅12=5x$

Multiply

Multiply

$24=5x$

DivEqn

$LHS/5=RHS/5$

$524 =x$

CalcQuot

Calculate quotient

$4.8=x$

RearrangeEqn

Rearrange equation

$x=4.8$

b Start by recalling that a unit rate is rate that tells the amount of one item in comparison to $1$ of another item. Therefore, finding the unit rate of flour to water means finding the number of cups of flour needed per $1$ cup of water.

$1cup of water?cups of flour $

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.
$2cups of water5cups of flour $

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 $2.$
$2/2cups of water5/2cups of flour ⇕1cups of water2.5cups of flour $

Therefore, the unit rate of flour to water is $2.5$ cups of flour to $1$ cup of water.
Example

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.

a How many invitations did Emily make in the first sitting?

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b How long does it take Emily to make $1$ invitation?

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c Who, Emily or her sister, makes invitations faster?

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a Write the expressions of Emily's rate of work in the first and seconds sittings. Then set and solve a proportion.

b Calculate the quotient of the time Emily worked on the first batch of invitations and the number of invitations she made.

c Note that Emily's sister finished all $5$ invitations in $7$ minutes. Compare Emily's rate of work and her sister's rate.

a Let $x$ be the number of invitations that Emily made in the first sitting. It is given that she worked on them for $21$ minutes, so the quotient of $x$ and $21$ represents the rate of Emily's work.

$21x $

Together with her sister, Emily made $2x$ invitations in $7$ minutes. By dividing $2x$ by $7,$ the rate of their work together can be found. However, in this case determining only Emily's rate would be more useful. To find it, subtract $5$ invitations that Emily's sister made and then divide by $7.$
$72x−5 $

Since Emily worked with the same rate the whole time, the following proportion can be formed.
$21x =72x−5 $

Finally, it can be solved for $x$ by using cross-multiplication.
$21x =72x−5 $

CrossMult

Cross multiply

$7x=21(2x−5)$

DivEqn

$LHS/7=RHS/7$

$x=3(2x−5)$

Distr

Distribute $3$

$x=6x−15$

SubEqn

$LHS−6x=RHS−6x$

$-5x=-15$

DivEqn

$LHS/(-5)=RHS/(-5)$

$x=3$

b In Part A, it was calculated that Emily made $3$ invitations in $21$ minutes. By dividing $21$ by $3,$ the amount of time she needs to make one invitation can be found.

$321 =7min $

c It is given that after working for $7$ minutes the girls made twice the amount of invitations Emily made by herself before combining forces. This means that Emily's sister finished $5$ invitations in $7$ minutes.

$Emily’s Sister’s Rate:7minutes5invitations $

At the same time, it was determined in Part B that Emily makes $1$ invitation in $7$ minutes.
$Emily’s Rate:7minutes1invitation $

By comparing those rates, it can be concluded that Emily's sister is working much faster.
$7minutes5invitations >7minutes1invitation $

It is a good thing she offered to help!
Closure

Using the information learned in this lesson, the challenge presented at the beginning can now be solved. Recall that Jordan participated in the Summer Marathon in her hometown Tallahassee, Florida. She comfortably ran the first $6$ miles of the marathon in $54$ minutes. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"hours","answer":{"text":["4.35"]}} ### Hint

### Solution

If she is able to maintain the same speed, how long will it take her to finish the remaining $29$ miles? Give the exact answer in hours.

Calculate Jordan's speed in miles per hour. Then, use it to set a proportion.

In order to determine the time Jordan needs to finish running the marathon, her speed should be calculated. It is given that Jordan ran $6$ miles in $54$ minutes. *per hour* would be more useful. To find it, expand the fraction by $60$ and then simplify.
Now, let $y$ represent the number of hours Jordan needs to run $29$ miles. Since it is said that Jordan maintains the same rate, the following proportion can be set.
If Jordan maintains the same rate, it will take her an impressive $4.35$ hours or $4$ hours and $21$ minutes to finish the remaining $29$ miles of the ultra-marathon.

$54min6mi $

By dividing both the numerator and denominator of the fraction by $54,$ Jordan's unit rate of miles per minute can be found.
$5454 min546 mi ⇔1min546 mi $

However, since the answer should be given in hours, the unit rate of miles $1min546 mi $

ExpandFrac

$ba =b⋅60a⋅60 $

$1⋅60min546 ⋅60mi $

▼

Simplify

OneMult

$1⋅a=a$

$60min546 ⋅60mi $

Rewrite

Rewrite $60min$ as $1h$

$1h546 ⋅60mi $

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$1h546⋅60 mi $

ReduceFrac

$ba =b/6a/6 $

$1h96⋅10 mi $

Multiply

Multiply

$1h960 mi $

ReduceFrac

$ba =b/3a/3 $

$1h320 mi $

$1h320 mi =yh29mi $

Use the Means-Extremes Property to solve it and calculate the value of $y.$
$1320 =y29 $

CrossMult

Cross multiply

$320 y=1⋅29$

OneMult

$1⋅a=a$

$320 y=29$

MultEqn

$LHS⋅3=RHS⋅3$

$20y=87$

DivEqn

$LHS/20=RHS/20$

$y=4.35$

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