Exercises: Ratio

On a social network, an influencer's video has received $450$ "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 value obtained means.

A video game shop 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?

Calculate the ratio of the width to the height of the screen of the following tablet, and explain whether the result would be the same if the tablet were twice as large while maintaining the same shape.

In a Maths exam of $20$ questions, Nerea answered $15$ correctly and got $5$ wrong. Write down the ratio of:

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

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

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

1. a) Express the ratio of red paint to yellow paint.
2. b) If they want to keep the same shade and use $6$ tins of yellow paint, how many tins of red paint will they need?

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

In a basketball practice, Seán has scored $8$ shots out of $12$ attempts, while Aoife has scored $6$ out of $9$ attempts.

1. a) Write the ratio of successful shots for each of them.

1. b) Check whether the two ratios form a proportion. Who has better aim?

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

In a 2nd Year class, there are $12$ girls and $18$ boys. Cian says out loud: "The ratio of girls to boys is $\dfrac{12}{30}$, which simplifies to $\dfrac{2}{5}$, meaning that for every $2$ girls there are $5$ boys in our class."

1. a) Identify and explain the conceptual error that Cian made when constructing the terms of the ratio. What quantity did he actually place in the denominator?
2. b) Write the correct ratio of girls to boys, simplify it to an irreducible fraction, and interpret in your own words what the result means in this context.
3. c) If in the neighbouring secondary school exactly the same ratio of girls to boys is maintained and we know there are $24$ boys, how many girls are there in that school? Write the corresponding proportion and find the unknown value.

In an artisan pastry workshop in Kilkenny, traditional marzipan is made using only ground almonds and sugar. The master pastry chef states that the ratio of the mass of almonds to the mass of sugar must be exactly $3$ to $2$ (that is, $\dfrac{3}{2}$).

1. a) Propose two different combinations of quantities of almonds and sugar (in grams) that strictly respect that ratio, such that the total mass of the mixture does not exceed $600\text{ g}$ in either case.
2. b) If you have an unopened $450\text{ g}$ packet of ground almonds and want to use it completely without having anything left over, what exact amount of sugar must you add to maintain the proportion? Set up the formal proportion before solving for the unknown.
3. c) An apprentice claims that mixing $350\text{ g}$ of almonds with $250\text{ g}$ of sugar maintains the proportion because "he added $50\text{ g}$ to each ingredient compared to the mixture of $300\text{ g}$ and $200\text{ g}$". Justify mathematically using ratios why his reasoning is flawed.

On a streaming platform, a moderator counted $1350$ messages in the stream's chat over the last $45$ minutes.

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

For an end-of-term party, students are comparing two offers on bags of crisps to see which is more economical. The "Crunch" brand offers a $250$ g bag for $1.75$ €, while the "Snack" brand sells a $400$ g bag for $2.60$ €.

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

A graphic designer is working with two image formats. The first format has dimensions of $12$ cm wide by $18$ cm high. The second format is $20$ cm wide by $30$ cm high.

1. a) Calculate the ratio of width to height for each of the formats.
2. b) Are these dimensions proportional? Justify your answer by comparing the ratios.

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

If they used $2$ litres of white paint for a room:

1. a) How many litres of yellow paint did they need?
2. b) What is the ratio of black paint to white paint in this mixture?

In a basketball training session, Aoife scored $18$ of her $30$ shots, while Cian scored $14$ of his $20$ shots.

1. a) Calculate each player's success ratio (shots scored to total shots) as a decimal.
2. b) Which of the two players had a better performance in the training session?

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

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

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

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

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

1. a) Calculate the population density of each province and interpret the result.
2. b) Which of the two provinces is more "populated" relative to its size?

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

1. a) Explain in one 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 for next year the number of students is expected to increase to $630$, how many teachers should there be to maintain the same ratio?

In a medium-mountain stage of the Rás Tailteann cycling race, two mountain passes are analysed:

• Windy Gap has a horizontal length of $6\text{ km}$ and overcomes a cumulative elevation gain of $420\text{ m}$.

• Healy Pass has a horizontal length of $9\text{ km}$ and overcomes a cumulative elevation gain of $540\text{ m}$.

(The ratio between the elevation gain in metres and the distance in kilometres determines the average difficulty of the pass in $\text{m/km}$).

1. a) Before calculating any exact quotient, mentally round the data and make a reasoned estimate of which of the two passes you think is steeper (higher elevation gain/distance ratio). Explain what you focused on to form your estimate.
2. b) Precisely calculate the ratio between the elevation gain and the distance for each pass, expressing the result as a decimal number. Check whether your initial estimate was correct.
3. c) The organisers want to design a new $5\text{ km}$ connecting section with the same elevation gain/distance ratio as Windy Gap. Set up the formal proportion and calculate how many metres of elevation gain this section must have. Assess whether the result obtained is consistent with the gradients of the stage.

According to the municipal census data published by the Central Statistics Office (CSO), County Leitrim has about $90\,000$ inhabitants in an area of approximately $10\,300\text{ km}^2$, while County Dublin has about $945\,000$ inhabitants in an area of about $4500\text{ km}^2$.

1. a) Mathematically model this situation by setting up the ratio that defines population density (number of inhabitants per square kilometre) for each of the two provinces. Calculate its decimal value rounded to the nearest tenth.
2. b) Interpret the real meaning of both decimal values: what does that number represent in daily life and in the spatial distribution of the inhabitants of each province?
3. c) If a district in the interior of County Dublin has an area of $350\text{ km}^2$ and maintains exactly the same provincial average density, how many inhabitants reside in it? Set up the formal proportion $\dfrac{a}{b} = \dfrac{c}{d}$ and calculate the unknown using the cross-multiplication property.

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

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

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 colour, will the initial ratio change? Justify your answer mathematically.

A video game designer is comparing two monitors for their new workstation. The first monitor has a "standard widescreen" format with a ratio of its width to its height of $16:9$. The second is an "ultrawide" monitor with a ratio of $21:9$.

1. a) If both monitors have exactly the same height of $36$ cm, what is the difference between their widths?

1. b) Calculate the ratio of the area of the ultrawide monitor to the area of the standard monitor. Express the result as an irreducible fraction.

On a high-precision map of the Wicklow Mountains, the scale is given as a ratio of $1:50\thinspace000$. A hiker measures the distance between two mountain huts on the map and gets $14.2$ cm. However, when checking a digital app, they notice that the actual distance is $7.5$ km.

1. a) Calculate the actual ratio (scale) between the measurement on the map and the measurement from the digital app.

1. b) What is the error in centimetres that the map shows compared to the distance on the app?

A courier company uses drones to deliver parcels. It is known that the ratio of parcel weight ($P$, in kg) to battery consumption ($B$, in % per kilometre) follows a constant relation of $1.5 : 0.4$.

1. a) If a parcel weighs $4.5$ kg, what will the battery consumption be for each kilometre travelled?

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

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

1. a) Calculate the ratio of the total glazed area to the wall area. (Use $\pi \approx 3.14$).

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

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

1. a) If an athlete of $1.80$ m has a BMI of $22.5$, what is their mass?

1. b) If another athlete weighs $85$ kg and is $1.92$ m tall, calculate their BMI. What is the ratio of the second athlete's BMI to the first athlete's BMI? Express the result to two decimal places.

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

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

1. b) Calculate the base area of the model in square centimetres. (Be careful: remember that the ratio of the areas is the square of the ratio of the lengths).

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

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

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

1. a) Which of the two archers was more effective? Justify your answer by comparing the ratios using a common denominator.

1. b) If Archer A wants to match Archer B's success ratio in their next round of $24$ arrows, how many of those new arrows must hit the centre?

To check whether two ratios form a proportion (that is, if $\dfrac{a}{b} = \dfrac{c}{d}$), three different methods can be used:

• Method 1 (Decimal value): Calculate the decimal quotient of each ratio and check whether they match.

• Method 2 (Irreducible fraction): Simplify both ratios by dividing by their greatest common divisor and compare the irreducible fractions.

• Method 3 (Cross products): Check whether the equality $a \times d = b \times c$ holds.

Two pairs of ratios obtained by comparing the fuel consumption of different vehicles are presented:

• Pair A: $\dfrac{7}{13}$ and $\dfrac{21}{39}$

• Pair B: $\dfrac{1200}{1800}$ and $\dfrac{1400}{2100}$

1. a) For Pair A, apply Method 1 and Method 3. Explain why Method 1 can create uncertainty when rounding repeating infinite decimal digits, whereas Method 3 guarantees an exact answer.
2. b) For Pair B, compare the calculation effort between Method 2 and Method 3 if you had to solve it by hand without a calculator. Justify which of the two is clearly faster and safer in that case.
3. c) Draw a reasoned conclusion: under what conditions is it advisable to choose the cross-products method, and in which ones is it preferable to simplify to the irreducible fraction?

In the emergency department of a regional hospital, the medical management sets that the optimal ratio of the number of waiting patients to the number of on-duty doctors must be at most $8$ to $1$ (that is, a ratio of $\dfrac{8}{1}$).

On a night with a high influx of patients, there are $78$ people waiting in the room and only $6$ doctors attending.

1. a) Model the situation by calculating the current ratio of patients per doctor as a decimal number. Interpret the result obtained and determine whether the service meets the recommendation or if the staff is overloaded.
2. b) If throughout that night it is not possible to bring in any additional doctor, what is the maximum number of patients that should be waiting for the ratio to be exactly the recommended $8:1$? How many patients would have to be diverted to other primary care centres to achieve this?
3. c) Suppose that during the early morning new patients arrive and the total number of waiting people rises to $96$. To address the situation with higher quality of care, management sets a new, more demanding ratio of $6$ patients per doctor. Set up an equation from the proportion $\dfrac{96}{6 + x} = \dfrac{6}{1}$ to calculate how many reinforcement doctors ($x$) must be called in immediately, and explain each step of the resolution.