Inverse proportion
If 4 painters take 6 days to paint a façade, 8 painters (twice as many) do not take twice as long: they take half the time. The more people work, the less time is needed. This relationship, where one quantity goes up while the other goes down in the same proportion, is inverse proportion.
Inversely proportional quantities
When $x$ is doubled, $y$ is divided by 2; when $x$ is divided by 3, $y$ is multiplied by 3. The curve gets closer to the axes but never touches them: neither $x$ nor $y$ can be $0$.
State which of the following pairs of quantities are inversely proportional: The speed at which a train travels and the time taken to cover a fixed distance. The number…
State which of the following pairs of quantities are inversely proportional:
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The speed at which a train travels and the time taken to cover a fixed distance.
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The number of kilograms of apples you buy and the total price you pay.
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The number of workers on a building site and the days taken to finish it.
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A person's age and their height.
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The flow rate of water from a tap and the time needed to fill a swimming pool.
Complete the following table knowing that quantities $A$ and $B$ are inversely proportional, and calculate the constant of proportionality $k$. | Quantity $A$ | $2$ |…
Complete the following table knowing that quantities $A$ and $B$ are inversely proportional, and calculate the constant of proportionality $k$.
| Quantity $A$ | $2$ | $4$ | $x$ | $10$ |
|---|---|---|---|---|
| Quantity $B$ | $15$ | $y$ | $6$ | $z$ |
The inverse rule of three
A team of $4$ painters takes $6$ days to paint the facade of a building. If the company decides to send only $3$ painters, how many days will it take them to do exactly…
A team of $4$ painters takes $6$ days to paint the facade of a building. If the company decides to send only $3$ painters, how many days will it take them to do exactly the same job?
A car travels at a constant speed of $90\text{ km/h}$ and takes $4$ hours to reach its destination. If on the return journey it wants to cover the same route in $3$…
A car travels at a constant speed of $90\text{ km/h}$ and takes $4$ hours to reach its destination. If on the return journey it wants to cover the same route in $3$ hours, at what constant speed must it travel?
A gear with $20$ teeth is connected to a motor and makes $30$ revolutions per minute. If it is coupled to another smaller gear that has $15$ teeth, how many revolutions…
A gear with $20$ teeth is connected to a motor and makes $30$ revolutions per minute. If it is coupled to another smaller gear that has $15$ teeth, how many revolutions per minute will this second gear make?
Inversely proportional sharing
Divide the amount of $300$ € into parts inversely proportional to the numbers $2$ and $3$.
Divide the amount of $300$ € into parts inversely proportional to the numbers $2$ and $3$.
A grandfather decides to divide $4500$ € among his three grandchildren in inverse proportion to their ages, which are $10$, $12$ and $15$ years old. Calculate the exact…
A grandfather decides to divide $4500$ € among his three grandchildren in inverse proportion to their ages, which are $10$, $12$ and $15$ years old. Calculate the exact amount of money each grandchild receives.
Combined work: several working at once
When two workers (taps, workers, machines) do the same job at the same time, you do not add their times: you add their "rates" (the fraction of the job done per unit of time).
A tap $A$ can fill a tank in $6$ hours, while tap $B$ would fill it in $4$ hours. If both taps are opened simultaneously starting with an empty tank, how long will it…
A tap $A$ can fill a tank in $6$ hours, while tap $B$ would fill it in $4$ hours. If both taps are opened simultaneously starting with an empty tank, how long will it take to fill it? Express the result in hours and minutes.
An irrigation water tank has two inlet taps and a drain. The first tap would fill it from empty in $4$ hours, and the second tap would fill it in $6$ hours. Meanwhile,…
An irrigation water tank has two inlet taps and a drain. The first tap would fill it from empty in $4$ hours, and the second tap would fill it in $6$ hours. Meanwhile, the drain is capable of emptying the tank completely in $8$ hours. If the tank is empty and all three pipes are opened simultaneously, how long will it take to fill? Express the result in hours, minutes and seconds.
Algebraic approach: interrupted jobs
A construction company calculates that a workforce of $20$ workers will finish a building in $40$ days. After $10$ days of uninterrupted work, a materials transport…
A construction company calculates that a workforce of $20$ workers will finish a building in $40$ days. After $10$ days of uninterrupted work, a materials transport strike forces work to stop for $5$ days. To avoid breaching the contract, the company decides to hire more staff. How many additional workers will need to be hired to finish the building within the total timeframe originally planned?
An industrial mechanism consists of three gears $A$, $B$ and $C$ connected in line ($A$ drives $B$, and $B$ drives $C$). Gear $A$ has $24$ teeth, $B$ has $36$ teeth and…
An industrial mechanism consists of three gears $A$, $B$ and $C$ connected in line ($A$ drives $B$, and $B$ drives $C$). Gear $A$ has $24$ teeth, $B$ has $36$ teeth and $C$ has $16$ teeth. If the motor turns gear $A$ at a speed of $120$ revolutions per minute (rpm), calculate the rotational speed of gear $C$. Does the number of teeth on the intermediate gear $B$ affect the final speed of gear $C$? Justify your answer.
Three business partners must split the repair costs for their premises in inverse proportion to the age (in years) of their company vehicles, which are $2$, $4$ and $x$…
Three business partners must split the repair costs for their premises in inverse proportion to the age (in years) of their company vehicles, which are $2$, $4$ and $x$ years old. If the total repair cost comes to $1400$ € and it is known that the first partner (the one with the $2$-year-old vehicle) had to pay $800$ €, what is the age of the third partner's car and how much will the second partner pay?