Ratios and Rates

In a 6th grade class, there are $12$ students who play basketball and $18$ students who play soccer. Write the ratio between the number of basketball players and soccer players. Explain in your own words what the result you have obtained after simplifying the fraction means.

Look at the following drawing representing the pieces of fruit in a fruit bowl. Calculate the ratio between the number of apples (orange) and the number of pears (soft white).

In a strategy video game, to craft a "Super Speed" potion, the alchemist uses $150\text{ mg}$ of stardust for every $250\text{ ml}$ of magic water.

1. a) Express the ratio between the stardust and the magic water. Simplify the result and interpret what this value means.
2. b) Calculate the inverse ratio (magic water to stardust) and explain what information it gives us.

To prepare a pink paint mixture, an artist uses $3$ pots of white paint for every $2$ pots of red paint.

• Write the ratio between the white and red paint.

• If the artist decided to use $6$ pots of red paint, how many pots of white paint should they use to keep the same shade of pink?

Look at these two mixtures of water and strawberry syrup. Mixture $A$ has $2$ glasses of syrup for every $5$ glasses of water. Mixture $B$ has $4$ glasses of syrup for every $10$ glasses of water.

• Write the ratio for each mixture.

• Are these mixtures proportional? That is, will they taste the same?

To prepare a very special orange color, a graffiti artist mixes $1,2\text{ liters}$ of red paint with $0,8\text{ liters}$ of yellow paint.

1. a) Write the ratio between the amount of red and yellow paint. Express it as a fraction in simplest form.
2. b) If he wants to keep the same shade but only has $0,5\text{ liters}$ of yellow paint, how much red paint should he use? (Use the ratio obtained in the previous section).

A remote-controlled car travels a distance of $45$ feet in $15$ seconds. Calculate the ratio between distance and time. What physical quantity does this result represent?

In an adventure video game, your character has obtained $24$ health potions after opening $6$ magic chests. Calculate the ratio between the number of potions and the number of chests. How many health potions are there, on average, in each chest?

A $500$ ounce box of cereal costs $\$3,50$. Express the ratio between the price and the weight in ounces. Then, calculate the decimal value of that ratio to find out how much a $1$ single ounce of cereal costs.

Emma has read $60$ pages of a novel in $2$ hours, while her brother Ethan has read $45$ pages in $1,5$ hours.

• Calculate the ratio between pages and hours for Emma.

• Calculate the ratio between pages and hours for Ethan.

• Do they both read at the same speed? Justify your answer.

In a basketball training session, Sophia has scored $18$ three-pointers out of $30$ shots taken, while her teammate Marcus has scored $14$ three-pointers out of $20$ shots.

1. a) Express the success ratio of each player as a decimal number.
2. b) Which of the two has a better success ratio? Justify your answer by comparing both values.

A candy store offers two types of bags of gummy candies:

• Bag A: contains $250\text{ g}$ and costs $3,50\text{ \$}$.

• Bag B: contains $400\text{ g}$ and costs $5,20\text{ \$}$.

1. a) Calculate the price/weight ratio for each bag.
2. b) Which of the two bags is more economical? Justify your answer using the values obtained.

In your 6th grade class there is a total of $32$ students. It is known that for every $3$ students who prefer Physical Education classes, there are $5$ students who prefer Art.

1. a) Write the ratio between those who prefer Physical Education and those who prefer Art.
2. b) If these are the only two options and everyone has chosen one, how many students prefer each subject? (Hint: note that the sum of the terms of the simplified ratio is $3 + 5 = 8$).

An urban artist is creating a new color "Cyber Green" by mixing blue paint and yellow paint. In her first pot, she has mixed $0,75$ gallons of blue with $1,2$ gallons of yellow. Running out of paint, she decides to prepare a second mixture in a larger pot using $2,5$ gallons of blue. How many gallons of yellow must she add so that the ratio between blue and yellow is exactly the same as in the first mixture? Express the initial ratio as a fraction in simplest form.

In a basketball training session, Sophia has scored $18$ shots out of $25$ attempts, while Marcus has scored $24$ out of $32$ attempts.

1. a) Write the ratio of successful shots for each player.
2. b) Calculate the decimal value of each ratio.
3. c) If both maintain their efficiency and shoot $100$ times, how many baskets will each one score? Are these ratios equivalent? Justify your answer using cross-multiplication.

The ratio between the base length and the height of old movie theater screens was $4:3$ (almost square format), while in modern movie theaters the ratio is usually $21:9$ (widescreen format).

If a modern movie theater screen has a height of $6,3$ feet, what is its base length? What would be the base of an old screen with that same height of $6,3$ feet? Compare both results and explain which of the two screens is more "elongated".

On a treasure island map from a movie, the distance between "Pirate Port" and "Skull Mountain" is $15\text{ in}$. In reality, that distance is $4,5\text{ mi}$.

1. a) Calculate the ratio between the measurement on the map and the real measurement (remember to convert both to the same unit, for example, inches).
2. b) What name is given to this special ratio used on maps?

A hummingbird is able to flap its wings about $4.800$ times in a minute. On the other hand, a griffon vulture flaps them about $120$ times every minute.

1. a) Calculate the ratio between the hummingbird's wing flaps and the vulture's. What does this number mean?
2. b) If we compare the hummingbird's weight ($3\text{ g}$) with its length ($10\text{ cm}$), calculate the weight/length ratio. What units would this ratio have?

A hybrid car consumes $4,2$ gallons of gas to travel $100$ miles, while a motorcycle consumes $1,5$ gallons to travel $40$ miles.

1. a) Calculate the consumption ratio (gallons per mile) of each vehicle.
2. b) Which of the two vehicles consumes less proportionally?
3. c) If the ratio between consumption and distance of a third vehicle is $0,052$, how many gallons will it use on a $250$ mile journey?

Population density is the ratio between the number of inhabitants and the area they occupy (inhabitants/sq mi).

• City X has a population of $1.250.000$ people and an area of $450,5$ sq mi.

• City Y has $840.000$ inhabitants and occupies $120,2$ sq mi.

Calculate the population density of both cities, rounding to the nearest whole number. If city X tripled its population but maintained its area, what would its new density be? Would it still be less dense than city Y?

A travel influencer has reviewed her Instagram statistics. On her latest post, she got $4.500$ "likes" after receiving a total of $60.000$ views.

1. a) Calculate the ratio between the number of "likes" and the number of views.
2. b) If on the next photo the ratio is $0,1$, which of the two photos has been more successful proportionally among her followers?

The "engagement rate" of an influencer is calculated by comparing the number of interactions (likes and comments) with their total number of followers.

• User A has $12.500$ followers and has obtained $625$ interactions on their last photo.

• User B has $45.000$ followers and has obtained $2.160$ interactions.

Calculate the ratio of interactions per follower for each one. Which of the two has a proportionally more active audience? Express the result as a decimal and explain what that value means in this context.

In a nutrition study, the ratio between the amount of sugar and the total weight of two types of cereal is analyzed.

• "Energy" cereals have $12$ oz of sugar in a $40$ oz portion.

• "Health" cereals have $18$ oz of sugar in a $75$ oz box.

Calculate the ratio of sugar per ounce of product for both. What percentage of sugar does each one have? (Remember that to convert a ratio to a percentage you can multiply its decimal value by 100). Which of the two would you choose if you wanted to consume less sugar proportionally?