Simple proportionality and the rule of three
A mechanic swaps a cog in a gear system and the machine starts to spin faster. A currency dealer gives you fewer pounds for your euros than yesterday. A more powerful pump empties a swimming pool sooner. In all three cases there are two related quantities, but they are not always related in the same way. Before setting up a rule of three, you need to know what the relationship is.
Direct and inverse: the review you can't get wrong
The graph gives it away: direct proportion is a straight line through the origin; inverse proportion is a curve (a hyperbola) that never touches the axes.
It is known that a quantity $Y$ is directly proportional to another quantity $X$. When $X = a$, the value of quantity $Y$ is $b + 3$. If the value of $X$ quadruples…
It is known that a quantity $Y$ is directly proportional to another quantity $X$. When $X = a$, the value of quantity $Y$ is $b + 3$. If the value of $X$ quadruples (that is, $X = 4a$), determine the algebraic expression that defines the new value of $Y$ in terms of $b$.
Two physical quantities $P$ and $Q$ are inversely proportional. If in a given process quantity $P$ undergoes an increase of $25\%$, by what exact percentage will…
Two physical quantities $P$ and $Q$ are inversely proportional. If in a given process quantity $P$ undergoes an increase of $25\%$, by what exact percentage will quantity $Q$ decrease? Prove your answer using the constant of proportionality.
The rule of three: two routes, the same result
At a greengrocer's, $5$ kg of apples cost $8,50$ €. How much will $3$ kg of the same apples cost?
At a greengrocer's, $5$ kg of apples cost $8,50$ €. How much will $3$ kg of the same apples cost?
To fill a water tank, $8$ identical taps pouring water at the same rate are used, taking $12$ hours to fill it completely. If we want the tank to be filled in $16$…
To fill a water tank, $8$ identical taps pouring water at the same rate are used, taking $12$ hours to fill it completely. If we want the tank to be filled in $16$ hours, how many taps identical to the previous ones must be left open?
A gear transmission system consists of a driving gear with $45$ teeth rotating at $120$ revolutions per minute (rpm). This gear is meshed with a second driven gear with…
A gear transmission system consists of a driving gear with $45$ teeth rotating at $120$ revolutions per minute (rpm). This gear is meshed with a second driven gear with $36$ teeth. At what speed (in rpm) will the second gear rotate?
Recognising the type before setting up the proportion
The costliest mistake is not a calculation error: it is setting up the proportion the wrong way round.
According to Boyle's Law, at constant temperature, the volume occupied by a gas is inversely proportional to the pressure to which it is subjected. A gas occupies a…
According to Boyle's Law, at constant temperature, the volume occupied by a gas is inversely proportional to the pressure to which it is subjected. A gas occupies a volume of $15,6$ litres when the pressure is $1,2$ atmospheres. What pressure in atmospheres will need to be applied for the gas to be compressed until it occupies exactly $10,4$ litres?
In a long-distance race, the average speed of a runner and the time taken to reach the finish line are inversely proportional quantities. Runner A finishes the course in…
In a long-distance race, the average speed of a runner and the time taken to reach the finish line are inversely proportional quantities. Runner A finishes the course in $t$ minutes. Runner B runs at an average speed that is $20\%$ higher than that of Runner A. Express the time taken by Runner B in terms of $t$.
Decimals, changes of unit and currencies
In technical problems the data are not usually round numbers: you have to convert minutes to seconds, litres to cubic metres, or work with exchange rates.
With a stable internet connection, downloading a $3,2$ GB file takes exactly $15$ minutes. If you need to download a $8,4$ GB update, how long will it take? Express the…
With a stable internet connection, downloading a $3,2$ GB file takes exactly $15$ minutes. If you need to download a $8,4$ GB update, how long will it take? Express the final result in minutes and seconds.
At a currency exchange bureau, a British tourist found that changing $125$ £ (pounds sterling) gave her $145$ €. The next day, at the same exchange rate, a Spanish…
At a currency exchange bureau, a British tourist found that changing $125$ £ (pounds sterling) gave her $145$ €. The next day, at the same exchange rate, a Spanish tourist wants to obtain $250$ £. How many euros will he have to give?
A water extraction pump operates at a flow rate of $2,5$ litres per second and takes $4$ hours and $30$ minutes to empty a swimming pool. If it is replaced by a more…
A water extraction pump operates at a flow rate of $2,5$ litres per second and takes $4$ hours and $30$ minutes to empty a swimming pool. If it is replaced by a more powerful pump with a flow rate of $3,0$ litres per second, how long will it take to empty the same pool?
Back to the beginning: The gear system and the pump are inverse proportions (more teeth or a greater flow rate mean fewer revolutions per minute or a shorter emptying time), whereas currency exchange is a direct proportion (the more euros you hand over, the more pounds you receive).