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 $1{,}200$ pesos and the amount of money each one must pay.
2. b) The time you leave a cell 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 $\$300$. 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 faucets take $10$ hours to fill the water tank of a sports complex. If we want the tank to fill in just $6$ hours to be able to open the facilities earlier, how many identical faucets 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?

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 $56\,000$ pesos 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 avocado oil, $400$ bottles of $0.75$ liters each were used. For marketing reasons, the cooperative decides to change the bottle format to one-third-of-a-liter containers ($1/3$ of a liter).

How many bottles of this new size will be needed to bottle the same amount of oil? If we have $1200$ half-liter bottles ($0.5$ liters), 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).

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 municipal government hires a team of $15$ workers to build a skatepark in $20$ days, working $6$ hours daily. After $8$ days of work, the municipality requests that the work be finished $4$ days earlier than planned to coincide with the town's patron saint festival. 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 $24$ liters. 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 $5$ liters, 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?