Exercises: Inverse proportion

State with reasons whether the following pairs of quantities have an inversely proportional relationship:

1. a) The number of friends who get together to buy a gift of $60 \text{ €}$ and the amount of money each one must pay.
2. b) The time you leave a mobile phone charging and the percentage of battery it recovers.
3. c) The speed at which a hiker walks and the time it takes to complete a route of $10 \text{ km}$.
4. d) The number of pages in a book and its total weight.

Complete the following table in your exercise book knowing that quantities A and B are inversely proportional. First calculate the constant of inverse proportionality $k$.

A group of $4$ Year 10 students has decided to clean a stretch of beach after a music festival. They calculate that it will take them $6$ hours to leave everything spotless. If $4$ more friends join them in the end (making $8$ in total), how long will it take them to finish the clean-up if everyone works at the same pace?

For the end-of-year trip, the students have hired a bus. If $40$ students go, each one must pay $15 \text{ €}$. At the last minute, $10$ students fall ill and cannot go. How much will each of the remaining students have to pay to cover the total cost of the bus?

A farmer has fodder stored to feed his $20$ cows for $30$ days. If he decides to sell $5$ of his cows to reduce expenses, for how many days will he be able to feed the remaining cows with the same amount of fodder?

Three identical taps take $10$ hours to fill the water tank of a sports center. If we want the tank to fill in only $6$ hours so that we can open the facilities earlier, how many taps identical to the previous ones will we need to have open in total?

A cyclist travelling at a constant speed of $20 \text{ km/h}$ takes $45$ minutes to cover the distance between two villages. If they increased their speed to $30 \text{ km/h}$, how many minutes would it take them to complete the same route?

In a small printing shop, $2$ machines working at full capacity take $40$ minutes to print the flyers for a local festival. If one of the machines breaks down and only one can work, how long (in minutes and hours) will it take to complete the same job? And what if they had $4$ identical machines running?

Marcos is planning a $360\text{ km}$ car trip between Madrid and Zaragoza. He knows that driving at a regulatory average speed of $120\text{ km/h}$ it takes him exactly $3$ hours. To consume less fuel, he decides to travel at an average speed of $90\text{ km/h}$ and writes down the following setup in his notebook to calculate the time $t$ it will take:

He concludes, convinced, that it will take him $2$ hours and $15$ minutes.

1. a) Without performing additional calculations, reason why the result obtained by Marcos contradicts common sense and explain what conceptual error he has made regarding the proportionality relationship between speed and time.
2. b) Correct the exercise: find the inverse proportionality constant $k$, state what physical quantity it represents in this context, and calculate how long Marcos will really take at $90\text{ km/h}$ (express the final result in whole hours).

A group of $4$ friends rents a country house in the Sierra de Cazorla to spend a weekend. The total rental price is fixed and they have agreed to divide it equally, corresponding to exactly $75$ € each. Just before making the final booking, $2$ more friends join the trip, so the final group becomes $6$ people.

1. a) Before calculating the exact price, make a reasoned estimate: will the price that each one pays be between $37,50$ € and $75$ €, or will it be less than $37,50$ €? Justify it by analyzing what would have happened if the initial number of friends had doubled.
2. b) Check your estimate by calculating the inverse proportionality constant $k$ (the total price of the accommodation) and the exact cost per person for the $6$ people. Does the result match your previous prediction?

A group of 9th-grade students (3.º de ESO) is organizing an end-of-year dinner. The restaurant has given them a fixed budget for the rental of the hall and the special menu, which will be split equally among all the attendees.

Copy the following table into your notebook and complete it to analyze how the price per person changes depending on how many students finally sign up:

Calculate the constant of inverse proportionality $k'$ and explain what it means in this context.

To complete a workout app challenge, Lucía needs to travel a distance of $12$ km on her electric scooter. If she maintains a constant speed of $15$ km/h, it takes her exactly $48$ minutes.

If the next day she decides to take it easier at a speed of $10$ km/h to save battery, how many minutes will she take to complete the same route? And if she wanted to take only half an hour, at what speed should she travel?

An animal shelter has received a donation of bags of dry food. With the $45$ dogs currently in the shelter, there is enough food for $20$ days.

However, after a rescue campaign, the number of dogs increases to $60$. How many days will the same amount of dry food last now? If they wanted the food to last $25$ days, what is the maximum number of dogs they could keep in the shelter?

Three identical taps take $10$ hours to fill a water tank intended for watering a school vegetable garden. The goal is to reduce the filling time so that the water is ready before classes start.

How many additional taps of the same type would need to be added to fill the tank in just $6$ hours? Express the answer as the number of additional taps.

In a video game design competition, the panel awards a prize of $2800$ € inversely proportional to the number of errors detected in the testing phase.

Team "Pixel" had $2$ errors, team "Voxel" had $4$ errors and team "Sprite" had $5$ errors. How much money will each team receive?

A team of $8$ workers from an events company takes $12$ hours to set up the entire stage and lighting for a music festival, working at a constant pace.

If the festival opens earlier and they only have $6$ hours for setup, how many workers should work in total? If they end up having $16$ workers, how long would it take them to finish the job?

To bottle a batch of organic olive oil, $400$ bottles of $0,75$ litres each were used. For marketing reasons, the cooperative decides to change the bottle format to one-third-of-a-litre containers ($1/3$ of a litre).

How many bottles of this new size will be needed to bottle the same amount of oil? If we have $1200$ half-litre bottles ($0,5$ litres), will we be able to bottle the whole batch?

Five state-of-the-art 3D printers take $4$ hours to print the parts for $20$ school robot models.

Calculate how many printers with the same specifications would be needed to manufacture those same $20$ models in just $100$ minutes. (Be careful with the units of time before doing the calculations).

In an oil mill in Jaén during the olive harvesting season, a crew of $8$ day laborers equipped with mechanical shakers takes $15$ hours of continuous work to harvest and fill a special large-capacity trailer.

1. a) Model the situation: define the two variables involved, write the formula for the inverse proportionality function $y = \frac{k}{x}$ that relates them, and interpret what specific physical meaning the constant $k$ has in this agricultural work context.
2. b) To prevent the olives from losing organoleptic properties, the oil mill requires the trailer to be filled within a maximum of $6$ hours. Assuming that all workers perform at the same pace, what is the minimum number of day laborers that must work simultaneously to achieve this?
3. c) From a mathematical and physical point of view, explain why the graph of this function (a hyperbola) can never intersect the y-axis ($x = 0$) or the x-axis ($y = 0$).

An inverse proportionality function has the algebraic expression $y = \frac{k}{x}$ (with $x > 0$). Its graph passes through an unknown point $(a, b)$. It is observed that if $6$ units are added to the independent variable (that is, it becomes $a + 6$), the value of the dependent variable is reduced to one-fourth of its original value, becoming $\frac{b}{4}$. In addition, it is known that for $x = 8$, the value of the function is $y = 4,5$.

1. a) Set up an equation equating the cross-products of the two situations involving the constant $k$ to deduce the exact value of $a$.
2. b) From the value of $a$ and the additional data, calculate the constant $k$, write the final formula of the function, and give the exact coordinates of the point $(a, b)$.
3. c) Invent a plausible everyday situation (indicating quantities and units of measurement) that is accurately described by the formula you obtained.

A logistics company uses drones to deliver packages in a residential area. It has been shown that $12$ drones, working $8$ hours a day, take $6$ days to deliver all the orders for a week. Due to a software update, the efficiency of each drone increases by $20\%$. If the company decides to reduce the drones' working day to just $5$ hours daily and the volume of orders has increased so that there are now $25\%$ more packages, how many drones will be needed to complete the delivery in just $4$ days?

The drive system of a racing bicycle consists of a chainring (connected to the pedals) and a sprocket (connected to the rear wheel). When the cyclist uses a chainring with $52$ teeth and a sprocket with $13$ teeth, for each full pedal stroke, the rear wheel makes $4$ turns. If the cyclist switches to a sprocket with $26$ teeth to climb a hill, how many turns will the rear wheel make for each pedal stroke? If at the top they switch the chainring to one with $34$ teeth while keeping the sprocket with $26$, calculate the new gear ratio (turns of the wheel per pedal stroke). Explain why the relationship between the number of teeth and the turns is inversely proportional.

A local council hires a team of $15$ workers to build a skatepark in $20$ days, working $6$ hours daily. After $8$ days of work, the council requests that the work be finished $4$ days earlier than planned to coincide with local festivities. At that exact moment, $3$ workers go on leave. How many hours a day will the remaining workers need to work to meet the new deadline?

A group of marine biologists is studying the relationship between the pressure and volume of a gas inside a sampling device (Boyle's Law). At sea level ($1$ atm pressure), the gas occupies a volume of $24$ litres. Upon diving to a certain depth, the pressure increases to $3,2$ atm. What will the new volume of the gas be? If the device can only withstand a maximum compression leaving the volume at $5$ litres, what is the maximum pressure it can withstand before breaking?

To render (process) an animated film of $90$ minutes, a studio uses a render farm of $40$ servers working at full capacity for $12$ days. The studio wants to release a new short film of only $15$ minutes, but needs the rendering to be ready in just $2$ days. However, the new servers they have purchased are $50\%$ more powerful than the old ones. How many of the new servers will be needed to render the short film on time?

An animal shelter has dry food reserves to feed its $80$ dogs for $45$ days, giving them $3$ daily portions of $200$ g each. After $10$ days, $20$ more dogs arrive, rescued from a flood. In addition, the vet recommends increasing the size of each portion to $250$ g, but reducing the number of daily portions to $2$ to improve digestion. How many days will the remaining dry food last for all the dogs under the new conditions?

In a physics laboratory, a magnetic motor is being tested. It is known that the speed of rotation ($V$) is inversely proportional to the weight of the load ($P$) and directly proportional to the applied voltage ($U$). With a voltage of $120$ V and a load of $10$ kg, the motor rotates at $2.400$ rpm. If we keep the motor rotating at $2.400$ rpm but increase the load to $15$ kg, what must the new voltage be? If we finally lower the voltage to $80$ V with that load of $15$ kg, at what speed will the motor rotate?

During the forest fire prevention campaign in Galicia, a brigade made up of $6$ forest firefighters consumes a water reserve of $18\,000\text{ liters}$ in $10$ days during their surveillance and brush-clearing tasks. The command center needs to calculate how many days a new tank of $25\,200\text{ liters}$ will last if the team is expanded to $7$ firefighters, assuming that the daily consumption per firefighter does not change.

Two emergency technicians propose different strategies:

• Method 1 (Unitary method): Calculate how many liters a single firefighter consumes in a single day, and from that rate calculate the days the new reserve will last for the $7$ firefighters.

• Method 2 (Compound proportionality): Identify the relationship (direct or inverse) of each quantity with the number of days and set up the corresponding equation of fractions.

1. a) Solve the problem applying both methods step by step and verify that exactly the same result is obtained.
2. b) Critically analyze the two procedures: which of the methods offers a greater understanding of the physical meaning of each intermediate step, and which would be computationally more agile if you had to program a spreadsheet to evaluate $20$ scenarios with different brigades and tanks? Justify your answer.

The Transport Consortium of a Spanish city manages a circular urban bus line whose total route measures $18\text{ km}$. During off-peak hours, the buses run at a constant average speed of $15\text{ km/h}$ (including stops and passenger exchanges). The frequency or waiting interval between consecutive buses at a stop ($y$, in minutes) depends exclusively on the number of vehicles assigned to the line ($x$).

1. a) Prove that the time it takes for a bus to complete a full round trip is $72$ minutes and deduce the expression of the inverse proportionality function $y = \frac{k}{x}$ that determines the frequency $y$ (in minutes) as a function of the number of buses $x$.
2. b) During rush hour, traffic congestion reduces the commercial speed of the line to $12\text{ km/h}$. The City Council contractually sets that the maximum headway between buses must not exceed $6$ minutes ($y \le 6$). If during off-peak hours the line operated with a frequency of $12$ minutes, how many additional buses must be added to the service during rush hour to meet the municipal requirement?
3. c) If the Consortium intended to reduce the waiting time between buses to exactly $0$ minutes ($y = 0$), what answer does the mathematical model provide? Explain why, on a real urban road, this inverse proportionality model collapses when the value of $x$ becomes excessively large.