Exercises: Compound proportion

In a custom clothing shop, $3$ thermal printing machines manage to print $120$ T-shirts working for $4$ hours. If due to an urgent order from a secondary school $5$ identical machines are started up, how many T-shirts will they be able to print in $6$ hours?

An animal shelter needs $480$ kg of feed to feed $40$ dogs for $30$ days. If the shelter receives a donation and increases its capacity to $60$ dogs, how many kilograms of feed will they need to feed them for $20$ days?

To upload the files of a new video game to a server, $4$ identical fibre optic connections take $15$ minutes to transfer $120$ GB. If we want to transfer $200$ GB to the same server using $5$ identical connections, how many minutes will it take?

A group of $6$ friends decides to distribute promotional material for their local e-sports team. In $3$ hours they managed to deliver $450$ flyers to letterboxes. If $2$ more friends join the group next Saturday and work for $4$ hours, how many flyers will they distribute in total?

In a water recycling plant, $8$ identical treatment units filter $2400$ cubic metres of water over a period of $5$ days. How many cubic metres of water will $12$ treatment units filter if they run for $7$ days?

Five decorative water fountains, with the same flow rate, consume $300$ litres of water if they are turned on for $8$ hours. Calculate how many litres $10$ identical fountains will consume if they are kept on for only $2$ hours.

A courier company uses $4$ vans to deliver $640$ parcels in $5$ days. If the company decides to hire $2$ more vans to boost the service, how many parcels will they be able to deliver in a period of $3$ days?

To illuminate an art exhibition hall for $12$ days, $15$ low-energy LED spotlights generated an expense of $54$ euros. What will the electricity cost be if $20$ identical spotlights are used during an event lasting $10$ days?

To paint the fences of a municipal sports centre, a crew of $3$ people took $8$ days working $6$ hours a day. The town council commissions an identical job to another crew made up of $6$ people who will work $4$ hours a day. Lucas solved the problem by writing the following in his notebook:

"Since there are twice as many people and fewer hours per day, I set up the proportion: $x = 8 \times \frac{6}{3} \times \frac{6}{4} = 24$ days".

1. a) Without performing numerical calculations, explain why a result of $24$ days is inconsistent with common sense in this case.
2. b) Identify the conceptual mistake Lucas made when analysing the relationship of each magnitude with the number of days.
3. c) Correct the approach and calculate the exact number of days the new crew will need to finish the job.

In a municipal plant nursery, $4$ identical sprinklers watered a meadow of $600\text{ m}^2$ for $3$ consecutive hours. The workers now need to water another meadow of $1\,200\text{ m}^2$ using $6$ sprinklers identical to the previous ones.

1. a) Before calculating, make a qualitative estimate: taking into account that the surface area to be watered is exactly double, but there are also $50\,\%$ more sprinklers, do you expect the required time to be greater than, less than, or equal to the initial $3$ hours? Justify your reasoning.
2. b) Classify each magnitude as directly or inversely proportional to time, set up the compound rule of three, and calculate precisely how many hours watering the second meadow will take.
3. c) Compare the value obtained with the estimate you made at the beginning and indicate whether the result confirms your reasoned deduction.

A group of $4$ travel content creators managed to edit $12$ videos for their channel working $6$ hours a day for a week. If $2$ more collaborators join the team next month and decide to work $8$ hours a day, how many videos will they be able to edit in the same period of time?

To feed the $15$ cows at a city farm for $12$ days, $540$ kg of feed was required. Thanks to a new grant, the farm has acquired $5$ more cows. How many kilograms of feed will be needed to feed all the animals for $20$ days?

A courier company uses $8$ drones to deliver $160$ parcels in $4$ flight hours. If the company decides to increase its fleet to $10$ drones and wants to deliver $250$ parcels, how many total flight hours will the drones need to complete the task?

A team of $6$ municipal workers cleans a beach area of $1500$ $m^2$ in $5$ hours. For the next cleaning day, the council has $8$ workers, but the area to be cleaned is $2800$ $m^2$. Calculate how long it will take them to finish the job.

In an electronics components factory, $5$ identical machines produce $2000$ processors running for $8$ hours a day. Due to an urgent order, the factory must deliver $3500$ processors in the same number of days. If $2$ more machines are started up, how many hours per day must all the machines run?

Three identical taps fill a tank of $10$ $m^3$ in $6$ hours. If we want to fill a pond of $25$ $m^3$ using $5$ taps with the same flow rate as before, how many hours will we need?

To print $1200$ advertising posters, $4$ industrial printers have been running for $15$ hours. If one of the printers breaks down and we need to print $1800$ posters for a new campaign, how many hours must the remaining printers work?

A team of $12$ archaeologists excavated a $60$-metre section of an ancient Roman road working for $9$ days at a rate of $7$ hours per day. If they want to excavate another $80$-metre section in $14$ days, how many hours per day must they work if the team keeps the same number of people?

A courier company uses electric vans to deliver urban parcels. During the morning shift, $4$ vans delivered $450$ parcels working for $6$ hours. For the afternoon, $900$ parcels have accumulated and must be delivered in a reduced shift of $4$ hours.

To find out how many vans must be deployed in the afternoon, two team members suggest two different strategies:

• Aoife proposes applying a compound rule of three directly, analysing the direct or inverse proportionality ratios of each magnitude with respect to the number of vans.

• Cian prefers to reduce to the unit first: calculating how many parcels a single van delivers in one hour, and from that figure deducing the required number of vans.

1. a) Solve the problem following Aoife's method, detailing the classification of each magnitude and the fraction by which you multiply.
2. b) Solve the problem applying Cian's method, indicating the real meaning and the unit of measurement of each intermediate result.
3. c) Analyse both approaches: what are the advantages and disadvantages of each method, and which one would you find more reliable if the problem involved four or more magnitudes at the same time?

During a drought alert period managed by the Hydrographic Confederation, the management of a campsite records the water consumption of its facilities. According to their measurements, a group of $120$ people staying for $6$ days generated a consumption of $32\,400$ litres of drinking water. For the month of August, the occupancy forecast is $180$ people for $15$ days.

1. a) Identify the magnitudes in the problem and analyse the type of proportional relationship each one has with the volume of water consumed.
2. b) If people maintained exactly the same consumption habit as in the previous record, calculate using a compound rule of three how many litres of water would be used in August.
3. c) The municipal emergency plan requires that during those $15$ days the campsite must not exceed a strict limit of $67\,500$ litres in total. Determine how many litres per person per day daily consumption must be limited to in order to comply with the regulation, and calculate what percentage reduction relative to the initial daily consumption this restriction represents.

In an animal rescue centre, it has been calculated that $15$ bags of dry food of $20\text{ kg}$ each are sufficient to feed $40$ dogs for $12$ days. Due to a rescue campaign, the number of dogs increases by $20$ more animals. If the centre buys bags of $30\text{ kg}$ of the same food, how many bags will they need to feed all the dogs for $25$ days?

A municipal cleaning team, consisting of $12$ workers, managed to clean an area of $3,600\text{ m}^2$ of woodland in $6$ days, working $8$ hours per day. For next week, the council needs to clean an area of $5,000\text{ m}^2$. If $3$ workers are off sick and cannot attend, how many hours per day must the remaining workers work to finish the task in only $5$ days?

A video game server farm uses $8$ cooling units to maintain the optimal temperature for $120$ servers running for $24$ hours a day, with an electricity cost of $€480$ a week. If the company decides to expand its capacity to $200$ servers and the price of electricity rises, so that the maximum allowed cost is $€600$, for how many hours a day can the servers be switched on if the same $8$ cooling units are kept? (Assume that the cost is directly proportional to the number of servers and the hours of use).

To print the flyers for a music festival, $3$ industrial printers working simultaneously take $15$ hours to print $4,500$ copies. One of the printers breaks down and must be removed. How long will it take the remaining printers to print $6,000$ copies if, to compensate for the missing third machine, the printing speed of the other two is increased by $25\text{ \%}$? (Hint: think about how the increase in speed affects the time required).

A group of $5$ identical taps takes $8$ hours to fill $3$ tanks of $600$ litres each. If we want to fill $4$ tanks of $900$ litres each in a maximum time of $10$ hours, how many taps with the same characteristics will we need to turn on? If the result is not an integer, explain the minimum number of taps we should install.

Six tanker trucks of $12,000\text{ L}$ capacity each made $4$ trips to empty an industrial waste container in $2$ days. If $2$ more trucks of $15,000\text{ L}$ capacity each are hired to do a similar job emptying a container that is $50\text{ \%}$ larger than the previous one, how many days will it take them to complete the emptying if each truck makes $5$ trips per day?

A team of $4$ programmers developed a mobile app with $180$ screens in $15$ days, dedicating $6$ hours per day each. For a new project of a more complex app with $300$ screens, the company has $20$ days available. If they decide to hire $2$ more programmers and increase the working day to $8$ hours per day, calculate whether they will be able to deliver the project on time and how many screens per day on average the full team will develop now.

A sports footwear factory produced $2,400$ pairs of trainers in $10$ days working with $12$ automatic machines for $16$ hours a day. Due to an urgent order from an international distributor, the factory must produce $3,600$ pairs in only $8$ days. If they bought $3$ additional machines with the same characteristics, how many extra hours per day, above the usual $16$, will all the machines have to run to meet the deadline?

In a secondary school's technology workshop, $4$ 3D printers operating $5$ hours a day took $3$ days to manufacture $30$ identical parts for a robotics project.

1. a) For a science outreach fair, a batch of $96$ identical parts needs to be manufactured. The department now has $6$ identical 3D printers and schedules $4$-hour working days. Determine how many whole days it will take them to have the complete batch ready.
2. b) Write the full wording of an inverse problem to the previous section, where the unknown is the number of hours per day that the $6$ printers must operate, setting as known data the days obtained in section (a) and the objective of manufacturing the $96$ parts.
3. c) Assume that the work on the new batch begins with the $6$ printers at $4$ hours a day, but at the end of the fourth day of work $2$ machines break down and become unusable. Set up an equation based on compound proportionality to calculate how many hours a day the remaining machines will have to work during the remaining days so as not to delay the final delivery.

In a photovoltaic solar plant, a maintenance contractor employs $36$ workers organised in crews to clear and inspect an area of $45\text{ hectares}$ of panels, taking $10$ days with $8$-hour daily shifts.

1. a) For an expansion of the plant that will cover $90\text{ hectares}$, the company has a deadline of $16$ days, but the working day must be limited to $6$ hours per day due to heatwave work restrictions. Use the compound rule of three to deduce how many workers must be hired in total to meet the planned deadline.
2. b) To boost productivity, management distributes a financial bonus of $\text{€}\ 21\,600$ among three crews ($A$, $B$, and $C$) according to two joint criteria: the number of people in the crew and the days worked in the month. Crew $A$ has $10$ people and worked $18$ days; crew $B$ has $12$ people and worked $15$ days; and crew $C$ has $15$ people and worked $12$ days. Calculate the exact amount each crew receives and justify why the distribution is fair despite having different headcounts and schedules.
3. c) A supervisor states: "There is no need to work with four magnitudes. Since days and daily hours multiply to give total hours per worker, we can reduce the problem to just three magnitudes: workers, total hours per worker, and hectares". Demonstrate whether his proposal is mathematically valid by analysing the proportional relationship of this new combined magnitude with respect to the number of workers.