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Consider some fun scenarios. Is there enough salsa for all these chips? What if the gas tank of the school bus has less gas than what it needs to make it to school? Amounts are compared to each other to find the right proportions. This lesson will explore similar 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.**

Explore

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

Consider similar situations where two things are mixed to create a third one. How can situations like this be described?

Discussion

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 $10:42.$ |

$Tearrik’s Ratio1527 Zain’s Ratio2545 $

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 | $1527 $ | $GCF(27,15)=3$ | $5⋅3 9⋅3 $ | $59 $ |

Zain | $2545 $ | $GCF(45,25)=5$ | $5⋅5 9⋅5 $ | $59 $ |

Pop Quiz

The applet shows different ratios using colon notation. Write the simplest form of the indicated ratio. Consider that some ratios might already be in their simplest form.

Example

It’s a beautiful Friday. Zain and their mother are celebrating their other parent's return home tonight after a week-long work trip. Zain's mother asks them for help to make homemade pizzas for dinner.

Their recipe is strict and claims that for every six olives on the pizza, two mushrooms must be added. If Zain plans to put $30$ olives on the pizza, how many mushrooms must the pizza have?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"mushrooms","answer":{"text":["10"]}}

Write the ratio of the number of olives to the number of mushrooms. Write an equivalent ratio to the original ratio where the numerator of the new ratio is $30.$ What number multiplied by $6$ gives $30?$ Multiply $2$ by the number found previously to find the number of mushrooms needed.

The recipe claims that for every six olives on the pizza, two mushrooms must be added. At the moment, the number of olives to the number of mushrooms on the pizza is the following ratio.

$6:2⇔26 $

Now, Zain and their mother have $30$ olives to put on the pizza. They want to know how many mushrooms they need to add to the pizza following the recipe exactly. This number can be found by writing an equivalent ratio to $6:2.$ The new ratio will have a numerator of $30.$
$26 =? 30 $

Note that $6⋅5=30.$ This means that $2$ must be multiplied by $5$ to find the missing number of the equivalent fraction.
$2⋅56⋅5 =1030 $

In this equivalent ratio, $30$ represents the olives that Zain and their mother will put on the pizza. Additionally, $10$ represents the number of mushrooms corresponding to $30$ olives. What a delicious smell! The pizza is almost ready.
Example

The two pizzas Zain and their mother made are still in the oven, smelling amazing. They decide to pass time by planning another weekend project — paint their house 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's mother says they will need a total of $18$ liters of this shade of orange. Then, they will need to calculate how many liters of yellow and red paint are necessary to create the mixture. Which option describes the right amount?{"type":"choice","form":{"alts":["<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">5<\/span><\/span><\/span><\/span> liters of red paint and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">4<\/span><\/span><\/span><\/span> liters of yellow paint","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span> liters of red paint and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span> liters of yellow paint","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span> liters of red paint and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span> liters of yellow paint","<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span> liters of red paint and <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span> liters of yellow paint"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Find the total amount described by the given ratio. Use this total amount to write a part-to-whole ratio for each color. Write an equivalent ratio for one of the part-to-whole ratios. The denominator of this equivalent ratio is $18.$ What number multiplied by $9$ gives $18?$ Find the numerator of the equivalent ratio

Consider the ratio of red to yellow paint that creates the specific shade of orange the family chose.

The amount of red and yellow paint needed for the $18$ liters of paint will be found using equivalent ratios. A ratio equivalent to the red paint ratio will be written. This equivalent ratio will have a denominator of $18.$

$Ratio of the Mixture5:4⇔45 $

This is a part-to-part ratio because it describes how much red paint there is in the mixture compared to the yellow paint. Note that the orange paint will consist of $9$ parts in total.
$5+4=9 $

The amount of red paint compared to the total amount of paint in the mixture can be determined using this information. This means it is a part-to-whole ratio. This also allows for the calculation of the ratio for the yellow paint. Part-To-Whole Ratios | |
---|---|

Red Paint | Yellow Paint |

$95 $ | $94 $ |

$95 =18? $

Both denominators are present. Check them and what is noticeable? Well, $9⋅2=18.$ This means that the numerator of the equivalent ratio is given by calculating the product of $5$ and $2.$ $9⋅25⋅2 =1810 $

This ratio can be expressed in words. Of the $18$ liters of paint, $10$ must be red to create the specific shade of orange Zain wants. The number of liters of yellow paint can be found by subtracting $10$ from $18.$
$Liters of Yellow Paint18−10=8 $

Zain's family needs $10$ liters of red paint and $8$ liters of yellow paint to create $18$ liters of the desired shade of orange. Their house is going to look so stylish!
Discussion

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

The two pizzas Zain and their mother prepared earlier were a such a great success. Now they are thinking to prepare more pizzas for a local charity.

External credits: @macrovector

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Write the rate for this situation. Divide the numerator and denominator of the rate by its denominator to find the unit rate. Multiply the unit rate by the number of pizzas that are needed. Divide the result by $60$ to get the number of hours of prep time.

This situation compares the time it takes to prepare the pizzas. Recall that it took Zain and their mother $30$ minutes to prepare $2$ pizzas. This rate can be written as a fraction.

$Rate:2pizzas30minutes $

Now, the numerator and denominator of this fraction will be divided by $2$ to find the unit rate. $Rate:2pizzas30minutes Unit Rate:2÷2pizzas30÷2minutes =1pizza15minutes $

This means that a pizza takes $15$ minutes to be ready. The time it will take to prepare $20$ pizzas can now be calculated by multiplying the unit rate by $20.$ $1pizza15minutes ⋅20Pizzas=300minutes $

Zain's family will spend about $300$ minutes to preparing the $20$ pizzas for the local charity. Notice that the answer is asked in hours. Divide $300$ by $60$ to get how many hours this time represents. $60minutes300minutes =5hours $

After $5$ hours, the $20$ pizzas are ready. Zain's family is taking them to the local charity. What a good lesson for Zain. Example

Zain realizes that their parent is so late arriving home. Zain calls. "Where are you?" The parent picks up and says, "Hey, Zain! I am $160$ miles away. I will be home in $2$ hours."

External credits: Hari Panicker

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Begin by writing the rate of the distance traveled to the time it takes as a fraction.

$Rate:2hours160miles $

Now, divide the numerator and denominator of this rate by $2$ to find the unit rate of this situation.
$Rate:2hours160miles Unit Rate:2÷2hours160÷2miles =1hour80miles $

The distance that Zain's parent traveled in $one$ hour is $80$ miles. Now, it is given that the total trip will take $3.5$ hours at this rate. Multiply the unit rate by $3.5$ to determine how far Zain's parent's work trip was.
$1hour80miles ⋅3.5hours=280miles $

Wow! Zain's parent's work trip was $280$ miles away. That is far, far away. Zain feels relieved to have figured out how soon their parent will be home.
Closure

Think about going shopping at the market. 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 decide whether buying more smaller packages or one large package offers a 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? Write the rates as fractions. That will help to find the unit rate for each bar later.

Standard-Size | Giant-Size | |
---|---|---|

Rate | $$1.191.55oz $ | $$8.597oz $ |

Divide the numerator and denominator of the standard size rate by $1.19$ to get its unit rate. Similarly, divide the numerator and denominator of the giant size ratio by $8.59.$

Standard-Size | Giant-Size | |
---|---|---|

Rate | $$1.191.55oz $ | $$8.597oz $ |

Unit Rate | $$11.30oz $ | $$10.81oz $ |

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