Compound proportion
Five taps fill a tank in 8 hours. But what if the size of the tank changes as well? Now there are three quantities involved: taps, hours and litres. Compound proportion combines several direct and inverse relationships at the same time.
Three quantities, two relationships
If more workers do the same job in fewer days (the workers–days relationship is inverse), but a bigger job needs more days with the same number of workers (the job–days relationship is direct), both relationships act at the same time.
Three packaging machines produce $500$ parts in $4$ hours of continuous operation. How many parts will $5$ identical machines produce if they operate for $6$ hours?
Three packaging machines produce $500$ parts in $4$ hours of continuous operation. How many parts will $5$ identical machines produce if they operate for $6$ hours?
The method: one relationship at a time
A team of $4$ painters manages to paint a wall of $100 \text{ m}^2$ in $5$ days. If $2$ more painters are hired to work at the same rate, how many days in total will the…
A team of $4$ painters manages to paint a wall of $100 \text{ m}^2$ in $5$ days. If $2$ more painters are hired to work at the same rate, how many days in total will the $6$ painters need to paint a wall of $150 \text{ m}^2$?
In an office, $6$ printers working simultaneously print $1200$ pages in $2$ hours. How many printers with the same specifications are needed to print $2400$ pages in $3$…
In an office, $6$ printers working simultaneously print $1200$ pages in $2$ hours. How many printers with the same specifications are needed to print $2400$ pages in $3$ hours?
When a relationship is inverse
To harvest a cereal field of $12 \text{ ha}$, $5$ combine harvesters working $8$ hours a day need $3$ full days. How many days will $4$ combine harvesters working $10$…
To harvest a cereal field of $12 \text{ ha}$, $5$ combine harvesters working $8$ hours a day need $3$ full days. How many days will $4$ combine harvesters working $10$ hours a day need to harvest a field of $15 \text{ ha}$?
A printing house uses $4$ rotary presses working $10$ hours a day and takes $15$ days to print the first edition of a novel. If the publisher demands that a second…
A printing house uses $4$ rotary presses working $10$ hours a day and takes $15$ days to print the first edition of a novel. If the publisher demands that a second identical edition be finished in $12$ days and the printing house has $5$ machines, how many hours a day must the machines work to meet the deadline?
Quantities squared or cubed
Sometimes one quantity does not depend linearly on another: the surface area of a sphere depends on the square of the radius, and its volume on the cube.
To paint a spherical tank of radius $R$, a team of $5$ painters needs $8$ days working $6$ hours a day. How many days will $4$ painters, working $8$ hours a day, need to…
To paint a spherical tank of radius $R$, a team of $5$ painters needs $8$ days working $6$ hours a day. How many days will $4$ painters, working $8$ hours a day, need to paint a spherical tank of radius $2R$? (Reminder: the surface area to paint on a sphere is proportional to the square of its radius).
Ten weavers, working $8$ hours a day, produce a piece of cloth $40 \text{ m}$ long and $2 \text{ m}$ wide in $5$ days. How many weavers, working $10$ hours a day, will…
Ten weavers, working $8$ hours a day, produce a piece of cloth $40 \text{ m}$ long and $2 \text{ m}$ wide in $5$ days. How many weavers, working $10$ hours a day, will be required to produce a piece of cloth $60 \text{ m}$ long and $3 \text{ m}$ wide in $6$ days? (Note that the amount of work done is directly proportional to the total surface area of the cloth).
Back to the start: if we wanted to fill a tank with three times as many litres using 6 taps instead of 5, it would take $8 \cdot \dfrac{5}{6} \cdot 3 = 20$ hours (the more taps, the less time; the more litres, the more time).