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ESO Y2 · Mathematics
Arithmetic · Proportionality

Compound rule of three

3 stamping machines make 120 T-shirts in 4 hours. If 5 machines now work for 6 hours, how many T-shirts will they stamp? Here there are three quantities at once: machines, hours and T-shirts. To solve this we need something more than an ordinary rule of three: the compound rule of three.

Before you start, make sure you know: Direct proportion (1st ESO)
Before you start, make sure you know: Inverse proportion (1st ESO)

Three quantities related at once

Concept diagram with three boxes showing how machines and hours jointly influence the number of T-shirts made. machines3 → 5 hours4 → 6 T-shirts120 → x the first two act on the third at the same time
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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 $5$…

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?

DIRECT MACHINES 3 5 · DIRECT HOURS 4 6 = UNKNOWN T-SHIRTS 120 x
Solve
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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?

Solve

Both quantities are direct

Diagram showing the solution of the T-shirt problem by multiplying the initial value by the direct ratios of machines and hours. 120 T-shirts × 5/3 = 200 × 6/4 x = 300 T-shirts 120 · 5/3 · 6/4 = 300
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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 $12$…

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?

Solve
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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?

Constant: 1 week Creators Hours/day Videos 4 6 12 6 4+2 8 x Direct (D) Direct (D) Setup 4 6 6 8 = 12 x
Solve

One direct and one inverse

Flow diagram showing the change from 120 servers, 24 hours and 480 euros to 200 servers, 18 hours and 600 euros. 120 serv. 24 h · 480 € ÷ serv., × cost 200 serv. 18 h · 600 €
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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?

Solve
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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…

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?

Dogs Days Food (kg) Direct Direct 40 12 15 bags · 20 kg 300 40 + 20 60 25 x bags · 30 kg 30·x
Solve
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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…

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 $480\text{ €}$ 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 $600\text{ €}$, 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).

Inverse Direct Servers Hours/day Cost (€) 120 24 480 200 x 600 Cooling units = 8 (Constant, does not affect calculation)
Solve

Shares with more than one criterion

Sometimes a sum is shared out according to two things at once: for example, the money invested and the length of time it was invested. You multiply the two numbers for each person and share out in proportion to those products.

Diagram illustrating compound proportional sharing by combining two criteria through the product of their quantities. 2 000 € · 6 → 1 976,47 € 3 000 € · 4 → 1 976,47 € 5 000 € · 2 → 1 647,06 €
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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?

Solve
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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,…

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.

Time (h) Taps Volume (L) INVERSE DIRECT 8 5 1800 3 tanks × 600 L 10 x 3600 4 tanks × 900 L
Solve

Summary

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