Inverse Proportionality
For the end-of-year trip to Teotihuacan, your class rents a bus. If 40 students go, each one pays $\$300$. If 20 go, each one pays $\$600$. Half the students, double the price per person. When one quantity goes up and the other goes down in the same proportion, they are inversely proportional.
Inversely proportional quantities
| Students ($x$) | 10 | 20 | 30 | 40 | 60 |
|---|---|---|---|---|---|
| Pesos per student ($y$) | 1,200 | 600 | 400 | 300 | 200 |
| $x \times y$ | 12,000 | 12,000 | 12,000 | 12,000 | 12,000 |
The constant $k = 12{,}000$ pesos is the bus's total price, which doesn't change.
State with reasons whether the following pairs of quantities have an inversely proportional relationship: a) The number of friends who get together to buy a gift of…
State with reasons whether the following pairs of quantities have an inversely proportional relationship:
- a) The number of friends who get together to buy a gift of $1{,}200$ pesos and the amount of money each one must pay.
- b) The time you leave a cell phone charging and the percentage of battery it recovers.
- c) The speed at which a hiker walks and the time it takes to complete a route of $10 \text{ km}$.
- d) The number of pages in a book and its total weight.
Complete the following table in your exercise book knowing that quantities A and B are inversely proportional. First calculate the constant of inverse proportionality…
Complete the following table in your exercise book knowing that quantities A and B are inversely proportional. First calculate the constant of inverse proportionality $k$.
| Quantity A | $2$ | $4$ | $6$ | ... | $12$ |
|---|---|---|---|---|---|
| Quantity B | $12$ | ... | ... | $3$ | ... |
The graph: a hyperbola
The points of an inverse proportionality do not form a line: they form a curve called a hyperbola, which gets closer and closer to the axes but never touches them (neither $x$ nor $y$ is ever 0).
Notice as you move the slider: at any point, $x \times y$ always equals $k$.
A cyclist travelling at a constant speed of $20 \text{ km/h}$ takes $45$ minutes to cover the distance between two villages. If they increased their speed to…
A cyclist travelling at a constant speed of $20 \text{ km/h}$ takes $45$ minutes to cover the distance between two villages. If they increased their speed to $30 \text{ km/h}$, how many minutes would it take them to complete the same route?
The inverse rule of three
A group of $4$ Year 10 students has decided to clean a stretch of beach after a music festival. They calculate that it will take them $6$ hours to leave everything…
A group of $4$ Year 10 students has decided to clean a stretch of beach after a music festival. They calculate that it will take them $6$ hours to leave everything spotless. If $4$ more friends join them in the end (making $8$ in total), how long will it take them to finish the clean-up if everyone works at the same pace?
For the end-of-year trip, the students have hired a bus. If $40$ students go, each one must pay $\$300$. At the last minute, $10…
For the end-of-year trip, the students have hired a bus. If $40$ students go, each one must pay $\$300$. At the last minute, $10$ students fall ill and cannot go. How much will each of the remaining students have to pay to cover the total cost of the bus?
A farmer has fodder stored to feed his $20$ cows for $30$ days. If he decides to sell $5$ of his cows to reduce expenses, for how many days will he be able to feed the…
A farmer has fodder stored to feed his $20$ cows for $30$ days. If he decides to sell $5$ of his cows to reduce expenses, for how many days will he be able to feed the remaining cows with the same amount of fodder?
Compound proportionality
When three or more quantities are involved, you compare each one with the unknown separately (assuming the others don't change) to see whether it is direct or inverse.
Three identical taps take $10$ hours to fill a water tank intended for watering a school vegetable garden. The goal is to reduce the filling time so that the water is…
Three identical taps take $10$ hours to fill a water tank intended for watering a school vegetable garden. The goal is to reduce the filling time so that the water is ready before classes start.
How many additional taps of the same type would need to be added to fill the tank in just $6$ hours? Express the answer as the number of additional taps.
Five state-of-the-art 3D printers take $4$ hours to print the parts for $20$ school robot models. Calculate how many printers with the same specifications would be…
Five state-of-the-art 3D printers take $4$ hours to print the parts for $20$ school robot models.
Calculate how many printers with the same specifications would be needed to manufacture those same $20$ models in just $100$ minutes. (Be careful with the units of time before doing the calculations).