Compound rule of three
A tortilla shop has 3 machines that make 120 kg of tortillas in 4 hours. On Saturday, 5 machines will work for 6 hours. How many kilograms will they make? Here there are three quantities at once: machines, hours, and kilograms. A simple rule of three is not enough; we need the compound rule of three.
Three quantities at once
In a custom clothing store, $3$ thermal printing machines manage to print $120$ T-shirts working for $4$ hours. If due to an urgent order from a middle school $5$…
In a custom clothing store, $3$ thermal printing machines manage to print $120$ T-shirts working for $4$ hours. If due to an urgent order from a middle 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…
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?
Both quantities are direct
In a water recycling plant, $8$ identical treatment units filter $2400$ cubic meters of water over a period of $5$ days. How many cubic meters of water will $12$…
In a water recycling plant, $8$ identical treatment units filter $2400$ cubic meters of water over a period of $5$ days. How many cubic meters of water will $12$ treatment units filter if they run for $7$ days?
Five decorative water fountains, with the same flow rate, consume $300$ liters of water if they are turned on for $8$ hours. Calculate how many liters $10$ identical…
Five decorative water fountains, with the same flow rate, consume $300$ liters of water if they are turned on for $8$ hours. Calculate how many liters $10$ identical fountains will consume if they are kept on for only $2$ hours.
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…
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?
One direct and one inverse
Many times the unknown is a time. Then the number of people or machines is usually inverse (more people, less time) and the amount of work to do is direct (more work, more time).
To upload the files of a new video game to a server, $4$ identical fiber optic connections take $15$ minutes to transfer $120$ GB. If we want to transfer $200$ GB to the…
To upload the files of a new video game to a server, $4$ identical fiber 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?
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…
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?
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 $250$…
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?
More than three quantities
The method does not change even if there are four or five quantities: one ratio for each one. With workers, days, and hours per day, you have to decide carefully about each one.
A team of $12$ archaeologists excavated a $60$-meter section of an ancient Roman road working for $9$ days at a rate of $7$ hours per day. If they want to excavate…
A team of $12$ archaeologists excavated a $60$-meter section of an ancient Roman road working for $9$ days at a rate of $7$ hours per day. If they want to excavate another $80$-meter section in $14$ days, how many hours per day must they work if the team keeps the same number of people?
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,…
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 municipality needs to clean an area of $5,000\text{ m}^2$. If $3$ workers are out sick and cannot attend, how many hours per day must the remaining workers work to finish the task in only $5$ days?
A sports footwear factory produced $2,400$ pairs of sneakers in $10$ days working with $12$ automatic machines for $16$ hours a day. Due to an urgent order from an…
A sports footwear factory produced $2,400$ pairs of sneakers 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?
Data you need to prepare first
In the hard problems, you have to fix up the data before setting up the ratios.
-
"2 more join" or "3 drop out": calculate the real total. With 12 operators and 3 dropping out, 9 remain.
-
"A machine 25% faster": its output is multiplied by $1.25$.
-
"20 kg sacks," "600 L tanks": the total amount is number × size. Calculate the total and use it as a single quantity.
-
A result that is not a whole number (faucets, machines, people): since you cannot use half a faucet, you round up so that there is enough.
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…
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?
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…
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).
In an animal rescue center, 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…
In an animal rescue center, 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 center buys bags of $30\text{ kg}$ of the same food, how many bags will they need to feed all the dogs for $25$ days?