Ratios and Proportions

On a social network, an influencer's video has got $450$ "likes" after being viewed by $3000$ people. Calculate the ratio of the number of "likes" to the number of views, simplify the result, and interpret what the obtained value means.

In a math exam with $20$ questions, Lucy has answered $15$ correctly and got $5$ wrong. Write the ratio of:

1. a) The number of correct answers to the number of wrong answers.
2. b) The number of correct answers to the total number of questions.

Indicate in each case which is the antecedent and which is the consequent.

Given the ratio $\frac{4}{5}$, invent a real-life context for teenagers (for example, related to movies, clothes, or sports) where these quantities are used and explain their meaning. State which quantities you are comparing.

A video game store offers a pack of $3$ controller covers for $\$12,60$. Express the ratio of the price to the number of covers. What would we usually call this result in everyday language?

An electric scooter travels $12$ miles in $30$ minutes. Calculate the ratio of the distance traveled to the time taken (expressed in hours). What physical quantity does this ratio represent?

On a streaming platform, a moderator has counted that in the last $45$ minutes, $1350$ messages have appeared in the broadcast chat.

1. a) Calculate the ratio of the number of messages to the time elapsed in minutes.
2. b) Explain what the obtained result means in the context of the problem.
3. c) If the ratio remains constant, how many messages will be received in a full hour?

To prepare an orange paint mixture, an artist uses $3$ cans of red paint for every $2$ cans of yellow paint.

1. a) Express the ratio of red paint to yellow paint.
2. b) If he wants to keep the same shade and uses $6$ cans of yellow paint, how many cans of red paint will he need?

In a basketball training session, Mark has scored $8$ shots out of $12$ attempts, while Paula has scored $6$ out of $9$ attempts.

1. a) Write the success ratio for each of them.
2. b) Check if the two ratios form a proportion. Who has better aim?

In a middle school, the ratio of the number of students to the number of teachers is $18$.

1. a) Explain in a sentence what this ratio value means.
2. b) If there are currently $32$ teachers in the school, how many students are enrolled?
3. c) If the number of students is expected to increase to $630$ next year, how many teachers should there be to maintain the same ratio?

For an end-of-year party, the students are comparing two offers of bags of chips to see which is cheaper. The brand "Crunch" offers a $250$ oz bag for $\$1,75$, while the brand "Snack" sells a $400$ oz bag for $\$2,60$.

1. a) Calculate the ratio of price to weight (in dollars per ounce) for each brand.
2. b) Using the ratios obtained, indicate which of the two brands is cheaper.

In an archery competition, an archer's score is analyzed using the ratio of "arrows in the center" to "total arrows shot".

• Archer A has achieved a ratio of $7:12$ in $48$ shots.

• Archer B has achieved a ratio of $5:8$ in $40$ shots.

1. a) Which of the two archers has been more effective? Justify this by comparing the ratios using a common denominator.

1. b) If Archer A wants to equal Archer B's effectiveness ratio in his next round of $24$ arrows, how many of those new arrows must he necessarily hit in the center?

In an art painting workshop, two shades of emerald green are being created by mixing blue and yellow paint. Mixture A contains $5$ parts of blue paint for every $3$ parts of yellow paint. Mixture B uses $8$ parts of blue paint for every $5$ parts of yellow paint.

1. a) Determine the ratio of blue to yellow in both mixtures, expressing it as a decimal number.

1. b) If we want the mural to be as "bluish" as possible, which mixture should we choose? Justify your answer by comparing the ratios.

1. c) If in Mixture A we decide to add $2$ more parts of each color, will the initial ratio change? Justify this mathematically.

Calculate the ratio of the base to the height of the screen of the following tablet and explain if the result would be the same if the tablet were twice as big while keeping the same shape.

A scale model of a new municipal sports center has been built. The real building has a total length of $75$ feet, while on the model that same distance measures $30$ inches.

1. a) Calculate the ratio of the model's measurement to the real measurement (remember to express both in the same unit).
2. b) What is the scale of the plan used for the model?

A scale model of a skyscraper is to be built. The scale factor between the model and the real building is $1:250$.

1. a) If the surface area of the base of the real building is $1\thinspace200$ $m^2$, what is the ratio of the surface area of the model's base to that of the real skyscraper?

1. b) Calculate the area of the base of the model in square inches. (Careful: remember that the ratio of the surface areas is the square of the ratio of the lengths).

To achieve an olive green paint shade, a painter mixes cans of white, yellow, and black paint. The ratio of the amount of yellow paint to white paint is $3/4$, and the ratio of black paint to yellow paint is $1/5$.

If he has used $2$ gallons of white paint for a room:

1. a) How many gallons of yellow paint has he needed?
2. b) What is the ratio of black paint to white paint in this mixture?

A courier company uses drones to deliver parcels. It is known that the ratio of the weight of the parcel ($P$, in lb) to the battery consumption ($B$, in % per mile) follows a constant relationship of $1,5 : 0,4$.

1. a) If a parcel weighs $4,5$ lb, what will be the battery consumption for each mile traveled?

1. b) If a drone only has $12\%$ of battery remaining and must make a delivery $5$ miles away, what is the maximum weight (weight/distance ratio) it could carry to just reach the destination?

Population density is the ratio of the number of inhabitants of a territory to its surface area expressed in $km^2$. Observe the data for these two fictional counties:

• County A: $450\,000$ inhabitants and $15\,000$ $km^2$.

• County B: $280\,000$ inhabitants and $7\,000$ $km^2$.

1. a) Calculate the population density of each county and interpret the result.
2. b) Which of the two counties is more "populated" in relation to its size?

The "body mass index" (BMI) is the ratio of a person's mass ($m$, in lb) to the square of their height ($h$, in feet), that is, $IMC = \frac{m}{h^2}$.

1. a) If an athlete who is $1,80$ ft tall has a BMI of $22,5$, what is their mass?

1. b) If another athlete weighs $85$ lb and is $1,92$ ft tall, calculate their BMI. What is the ratio of the second athlete's BMI to the first's? Express the result to two decimal places.

In a study on bioclimatic architecture, the ratio of the surface area of the windows to the total surface area of the walls in a room is analyzed to optimize natural light. In a room with $25$ $m^2$ of walls, there are two windows: one measuring $120$ in. $\times$ $150$ in. and a circular one with a radius of $80$ in.

1. a) Calculate the ratio of the total glazed surface area to the surface area of the walls. (Use $\pi \approx 3,14$).

1. b) If the ideal recommendation is a ratio of $1:5$, how many $m^2$ of glass are needed or in excess to meet it?