The robotics club runs a T-shirt fundraiser. Their 3 heat presses print 90 shirts in 3 hours. A big order comes in, so 5 presses will run for 4 hours. How many shirts will they print? There are three quantities here (presses, hours, shirts), so a single proportion won't do it. You need a compound proportion.
Three Quantities at Once
In a "job" problem, the amount of work done depends on two things at the same time: how many workers (or machines) you have, and how long they work. If each press prints at the same steady speed, you can find the speed of one press for one hour. That is a unit rate.
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In a clothing customization shop, $3$ heat press machines manage to print $120$ T-shirts working for $4$ hours. If, due to an urgent order from a school, $5$ identical…
In a clothing customization shop, $3$ heat press machines manage to print $120$ T-shirts working for $4$ hours. If, due to an urgent order from a school, $5$ identical machines are started up, how many T-shirts will they be able to print in $6$ hours?
Five decorative water fountains, with the same flow rate, consume $300$ gallons of water if they are turned on for $8$ hours. Calculate how many gallons $10$ identical…
Five decorative water fountains, with the same flow rate, consume $300$ gallons of water if they are turned on for $8$ hours. Calculate how many gallons $10$ identical fountains will consume if they are kept on for only $2$ hours.
Unit rates work well, but there is a faster way when you only need one answer. Start from the known value of the unknown and scale it once for each quantity that changes.
Check it against the unit-rate answer from before: $90 \cdot \tfrac{5}{3} \cdot \tfrac{4}{3} = 200$ shirts. Same answer.
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An animal shelter needs $480$ pounds of feed to feed $40$ dogs for $30$ days. If the shelter receives a donation and increases its capacity to $60$ dogs, how many pounds…
An animal shelter needs $480$ pounds of feed to feed $40$ dogs for $30$ days. If the shelter receives a donation and increases its capacity to $60$ dogs, how many pounds of feed will they need to feed them for $20$ days?
To light an art exhibition hall for $12$ days, $15$ energy-saving LED spotlights have generated a cost of $\$54$. What will the electricity cost be if $20…
To light an art exhibition hall for $12$ days, $15$ energy-saving LED spotlights have generated a cost of $\$54$. What will the electricity cost be if $20$ identical spotlights are used during an event lasting $10$ days?
To feed the $15$ cows of a farm school for $12$ days, $540$ pounds of feed have been needed. Due to a new grant, the farm has acquired $5$ more cows. How many pounds of…
To feed the $15$ cows of a farm school for $12$ days, $540$ pounds of feed have been needed. Due to a new grant, the farm has acquired $5$ more cows. How many pounds of feed will be needed to feed all the animals for $20$ days?
Now the unknown is time. More workers means less time, but more work means more time. Ask the one-at-a-time question for each quantity.
Unknown: hours needed
If this goes up...
Type
number of workers
hours go down
inverse
amount of work to do
hours go up
direct
Cost problems work the same way. If cost is directly proportional to both the number of servers and the hours they run, then for a fixed budget, more servers means fewer hours.
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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 on 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 on the same server using $5$ identical connections, how many minutes will they take?
A courier company uses $8$ drones to deliver $160$ packages in $4$ hours of flight. If the company decides to increase its fleet to $10$ drones and wants to deliver…
A courier company uses $8$ drones to deliver $160$ packages in $4$ hours of flight. If the company decides to increase its fleet to $10$ drones and wants to deliver $250$ packages, how many total flight hours will the drones need to complete the task?
A video game server farm uses $8$ cooling units to maintain the optimal temperature of $120$ servers running for $24$ hours a day, with an electricity cost of…
A video game server farm uses $8$ cooling units to maintain the optimal temperature of $120$ servers running for $24$ hours a day, with an electricity cost of $480\text{ \$}$ a week. If the company decides to expand its capacity to $200$ servers and the price of electricity rises, so that the maximum permitted cost is $600\text{ \$}$, for how many hours a day can the servers be turned on if the same $8$ cooling units are kept? (Consider that the cost is directly proportional to the number of servers and the hours of use).
Some problems give workers, days, and hours per day. These are separate quantities. You scale once for each one. The method is the same; you just have more fractions.
You can double-check with a unit rate: $900 \div (6 \times 3 \times 4) = 12.5$ ft² per volunteer-hour. The new job has $9 \times 2 \times 8 = 144$ volunteer-hours, and $144 \times 12.5 = 1{,}800$ ft². ✓
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In an electronic components factory, $5$ identical machines produce $2000$ processors working $8$ hours a day. Due to an urgent order, the factory must deliver $3500$…
In an electronic components factory, $5$ identical machines produce $2000$ processors working $8$ 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 a day must all the machines work?
A team of $12$ archaeologists has excavated a $60$-foot section of an ancient Roman road, working $9$ days at a rate of $7$ hours a day. If they want to excavate another…
A team of $12$ archaeologists has excavated a $60$-foot section of an ancient Roman road, working $9$ days at a rate of $7$ hours a day. If they want to excavate another $80$-foot section in $14$ days, how many hours a day must they work if the team keeps the same number of people?
A municipal cleaning brigade, made up of $12$ workers, has managed to clean an area of $3.600\text{ m}^2$ of forest in $6$ days, working $8$ hours a day. For next week,…
A municipal cleaning brigade, made up of $12$ workers, has managed to clean an area of $3.600\text{ m}^2$ of forest in $6$ days, working $8$ hours a day. For next week, the city council needs to clean an area of $5.000\text{ m}^2$. If $3$ workers have taken sick leave and will not be able to attend, how many hours a day will the remaining workers have to work to finish the task in just $5$ days?
Sometimes the "amount of work" is hidden inside a count and a size, like "2 tanks of 600 gallons each". Build the total first, then treat it as one quantity.
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Six tanker trucks of $12.000\text{ L}$ capacity each have made $4$ trips to empty an industrial waste container in $2$ days. If $2$ more trucks of $15.000\text{ L}$…
Six tanker trucks of $12.000\text{ L}$ capacity each have 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 they take to complete the emptying if each truck makes $5$ trips a day?
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 the remaining printers take to print $6.000$ copies if, to compensate for the lack of the 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 needed).
A calculator gives you a number. A real situation asks for something you can actually do. When you count whole things (bags, faucets, trucks) and the answer must be enough, round up, even if the decimal is small. Round only at the very end.
Checking whether a team can meet a deadline works the same way. Compute the time needed, then compare it with the time available.
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A group of $5$ identical faucets takes $8$ hours to fill $3$ tanks of $600$ gallons each. If we want to fill $4$ tanks of $900$ gallons each in a maximum time of $10$…
A group of $5$ identical faucets takes $8$ hours to fill $3$ tanks of $600$ gallons each. If we want to fill $4$ tanks of $900$ gallons each in a maximum time of $10$ hours, how many faucets of the same characteristics will we need to open? If the result is not an integer, reason how many faucets we should install as a minimum.
In an animal shelter, it has been calculated that $15$ bags of feed of $20\text{ kg}$ each are enough to feed $40$ dogs for $12$ days. Due to a rescue campaign, the…
In an animal shelter, it has been calculated that $15$ bags of feed of $20\text{ kg}$ each are enough to feed $40$ dogs for $12$ days. Due to a rescue campaign, the number of dogs increases by $20$ more animals. If the shelter buys $30\text{ kg}$ bags of the same feed, how many bags will they need to feed all the dogs for $25$ days?
A team of $4$ programmers has developed a mobile app of $180$ screens in $15$ days, dedicating $6$ hours a day each. For a new project of a more complex app of $300$…
A team of $4$ programmers has developed a mobile app of $180$ screens in $15$ days, dedicating $6$ hours a day each. For a new project of a more complex app of $300$ screens, the company has $20$ days. If they decide to hire $2$ more programmers and increase the working day to $8$ hours a day, calculate if they will be able to deliver the project on time and how many screens a day on average the whole team will now develop.