Exercises: Ratio

In a Year 7 class, there are $12$ pupils who play basketball and $18$ pupils who play football. Write the ratio of the number of basketball players to football players. Explain in your own words what the result means after simplifying the fraction.

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

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

• Write the ratio of white paint to red paint.

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

A radio-controlled car travels a distance of $45$ metres in $15$ seconds. Calculate the ratio of distance to 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 of the number of potions to the number of chests. On average, how many health potions are there in each chest?

Un paquete de $500$ gramos de cereales cuesta $3,50$ €. Expresa la razón entre el precio y el peso en gramos. Después, calcula el valor decimal de esa razón para saber cuánto cuesta $1$ solo gramo de cereales.

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?

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

• Calculate the ratio of pages to hours for Marta.

• Calculate the ratio of pages to hours for Javier.

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

Lucas is building a scale model of an EMT Madrid city bus. The real bus is $12\text{ m}$ long and his model is $15\text{ cm}$ long. To indicate the scale on the project's data sheet, Lucas writes the following:

'The scale is $\dfrac{15}{12} = \dfrac{5}{4} = 1,25$, which means that the model is $1,25$ times bigger than the bus'.

1. a) Explain the two errors that Lucas has made (one related to the units of measurement and the other to the interpretation of the resulting ratio).
2. b) Correct the approach and calculate the correct numerical scale of the model, expressing it in the usual format $1:n$.

In a secondary school's vegetable garden, the simplified ratio between the number of tomato plants and the number of lettuce plants planted is $\dfrac{3}{5}$. Furthermore, the Biology teacher tells you that the total number of plants (tomatoes plus lettuces) is greater than $30$ and less than $50$.

1. a) Exactly how many tomato plants and how many lettuce plants can there be? Write all possible pairs of solutions that meet these conditions.
2. b) Justify why it would be impossible for this garden to have a total of $35$ plants while maintaining that same ratio.

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 of stardust to 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.

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

1. a) Calculate the ratio of the number of likes to the number of views.
2. b) If in the next photo the ratio is $0,1$, which of the two photos was proportionally more successful among her followers?

In a basketball training session, Lucía scored $18$ three-pointers out of $30$ shots taken, while her teammate Marcos 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.

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

1. a) Write the ratio of the amount of red paint to yellow paint. Express it as an irreducible fraction.
2. b) If he wants to keep the same shade but only has $0,5\text{ litres}$ of yellow paint, what amount of red paint should he use? (Use the ratio obtained in the previous part).

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

1. a) Calculate the ratio between the hummingbird's wingbeats and the vulture's. What does that number mean?

1. 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 that ratio have?

On a treasure island map from a film, the distance between "Pirate Harbour" and "Skull Mountain" is $15\text{ cm}$. In reality, that distance is $4,5\text{ km}$.

1. a) Calculate the ratio of the map measurement to the real-life measurement (remember to convert both to the same unit, for example, centimetres).
2. b) What is the name of this special ratio used on maps?

In your Year 7 class there are $32$ pupils in total. It is known that for every $3$ pupils who prefer PE classes, there are $5$ pupils who prefer Art.

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

A sweet shop offers two types of bags of gummy sweets:

• 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-to-weight ratio for each bag.
2. b) Which of the two bags is cheaper value for money? Justify your answer using the values obtained.

At the end of a round of the Liga Endesa basketball league, the coaching staff compares the performance of two rival teams:

• Team A has won $17$ of the $28$ games played.

• Team B has won $13$ of the $24$ games played.

1. a) Before carrying out exact divisions, estimate whether both teams have won more or less than half of their matches (benchmark $0,5$). Reason whether that first estimation is enough for you to state with certainty which of the two has a better performance.
2. b) Check your estimate by calculating the ratio of wins per game played for each team as a decimal rounded to two decimal places. Which one is more effective?
3. c) If a third team has a win ratio of $0,75$ having played $20$ games, calculate how many games it has won and explain what that decimal means with respect to the total.

Using data from the municipal register (INE), we analyse the demographic structure of two municipalities with very different characteristics:

• In Aldeavieja (a rural municipality), there are $60$ residents under 18 years old and $180$ people over 65 years old.

• In Valdenebro (a residential municipality), there are $2400$ residents under 18 years old and $1600$ people over 65 years old.

1. a) Model the 'ratio of youth to elderly' (number of minors divided by number of elderly people) for each municipality and express it as an irreducible fraction.
2. b) Interpret the meaning of each of these two irreducible fractions in your own words (for example: 'for every... elderly people there are... young people').
3. c) If $30$ new minors under 18 were registered in Aldeavieja (without changing the number of elderly people), would it reach Valdenebro's youth ratio? Justify your answer by comparing both ratios.

In competitive video gaming, server performance is measured by the ratio of connected players to bandwidth consumed (in Mbps). The "BattleArena" server has $120$ players and consumes $450$ Mbps. On the other hand, the "GalaxyQuest" server has $85$ players and consumes $320$ Mbps. Calculate the consumption ratio per player for both servers and determine which of the two is more efficient (the one consuming fewer Mbps per player). Express the ratios as decimals.

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

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

• User A has $12.500$ followers and received $625$ interactions on their latest photo.

• User B has $45.000$ followers and received $2.160$ 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 basketball practice, Lucía scored $18$ shots out of $25$ attempts, while Marcos scored $24$ out of $32$ attempts.

1. a) Write the success ratio 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 the cross-multiplication of extremes and means.

The ratio of the length of the base to the height of old cinema screens was $4:3$ (almost square aspect ratio), whereas in modern cinemas the ratio is usually $21:9$ (widescreen format).

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

A hybrid car consumes $4,2$ litres of petrol to travel $100$ km, whereas a motorcycle consumes $1,5$ litres to travel $40$ km.

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

In a nutrition study, the ratio of the amount of sugar to the total weight of two types of cereal is analysed.

• "Energy" cereal has $12$ g of sugar in a $40$ g serving.

• "Health" cereal has $18$ g of sugar in a $75$ g box.

Calculate the ratio of sugar per gram 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?

Population density is the ratio of the number of inhabitants to the area they occupy (inhabitants/km²).

• City X has a population of $1.250.000$ people and an area of $450,5$ km².

• City Y has $840.000$ inhabitants and occupies $120,2$ km².

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

You need to check whether the ratios $\dfrac{18}{24}$ and $\dfrac{27}{36}$ form a proportion. In class, three different ways are proposed:

• Method 1 (Candela): Check whether the product of the extremes equals the product of the means ($18 \cdot 36$ versus $24 \cdot 27$).

• Method 2 (Bruno): Simplify both fractions to their irreducible fraction and observe whether they match.

• Method 3 (Darío): Divide the numerator by the denominator in both ratios and compare the resulting decimal numbers.

1. a) Apply Method 1 and Method 2 to determine whether or not they form a proportion.
2. b) Analyse the advantages and disadvantages of each of the three methods. If the numbers had four digits (for example, $\dfrac{3216}{4824}$ and $\dfrac{2144}{3216}$) and you did not have a calculator, which of the three methods would you rule out first and which would you choose as the most efficient and reliable? Justify your choice.

In the Art workshop, you are preparing a custom shade of green paint by mixing pots of blue and yellow paint. To start with, the ratio of pots of blue to pots of yellow is $\dfrac{2}{3}$, having used $6$ pots of blue paint.

1. a) Determine how many pots of yellow paint were used in that initial mix.
2. b) To lighten the shade, the teacher states that the new ratio (blue to yellow) must be exactly $\dfrac{1}{2}$. If you only have extra pots of yellow paint available (the amount of blue does not change), how many yellow pots must you add to the original mix?
3. c) A classmate claims: 'To go from the ratio $\dfrac{2}{3}$ to the ratio $\dfrac{1}{2}$, the difference between the two is $\dfrac{2}{3} - \dfrac{1}{2} = \dfrac{1}{6}$, so it is enough to add one-sixth of the pots that were already there'. Explain in detail why this reasoning is false and what conceptual error he makes by treating a ratio as if it were an additive quantity.