Inverse Proportionality
For the end-of-year class trip, the students rent a bus. If $40$ students go, each one pays $\text{RD}\$900$. If only $20$ go, each one would have to pay $\text{RD}\$1{,}800$. Half as many students, double the fare per person. When one quantity goes up and the other goes down in the same proportion, they are inversely proportional. How do you calculate those values without guessing? That is what you will see here.
Inversely proportional quantities
Example: a bus is rented for $\text{RD}\$36{,}000$ and the cost is shared equally.
| Students ($x$) | 10 | 20 | 30 | 40 | 60 |
|---|---|---|---|---|---|
| Pesos per student ($y$) | 3,600 | 1,800 | 1,200 | 900 | 600 |
| $x \times y$ | 36,000 | 36,000 | 36,000 | 36,000 | 36,000 |
The constant $k = 36{,}000$ is the total cost of the bus, which does not 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 $\text{RD\$}\ 3\,600$ 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 | $12$ | ... | ... | $3$ | ... |
A group of 9th-grade students (3.º de ESO) is organizing an end-of-year dinner. The restaurant has given them a fixed budget for the rental of the hall and the special…
A group of 9th-grade students (3.º de ESO) is organizing an end-of-year dinner. The restaurant has given them a fixed budget for the rental of the hall and the special menu, which will be split equally among all the attendees.
Copy the following table into your notebook and complete it to analyze how the price per person changes depending on how many students finally sign up:
| No. of students ($x$) | $15$ | $20$ | $24$ | $30$ | $40$ |
|---|---|---|---|---|---|
| Price per student ($y$) | $80$ pesos |
Calculate the constant of inverse proportionality $k'$ and explain what it means in this context.
The graph: a hyperbola
The points of an inverse proportionality do not form a line: they form a curve called a hyperbola, which gets close to the axes but never touches them (neither $x$ nor $y$ equals $0$).
Notice, as you move the slider, that at any point $x \times y$ is always $k$.
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?
To complete a workout app challenge, Lucía needs to travel a distance of $12$ km on her electric scooter. If she maintains a constant speed of $15$ km/h, it takes her…
To complete a workout app challenge, Lucía needs to travel a distance of $12$ km on her electric scooter. If she maintains a constant speed of $15$ km/h, it takes her exactly $48$ minutes.
If the next day she decides to take it easier at a speed of $10$ km/h to save battery, how many minutes will she take to complete the same route? And if she wanted to take only half an hour, at what speed should she travel?
A group of marine biologists is studying the relationship between the pressure and volume of a gas inside a sampling device (Boyle's Law). At sea level ($1$ atm…
A group of marine biologists is studying the relationship between the pressure and volume of a gas inside a sampling device (Boyle's Law). At sea level ($1$ atm pressure), the gas occupies a volume of $24$ liters. Upon diving to a certain depth, the pressure increases to $3.2$ atm. What will the new volume of the gas be? If the device can only withstand a maximum compression leaving the volume at $5$ liters, what is the maximum pressure it can withstand before breaking?
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 $\text{RD\$}\ 900$. 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 $\text{RD\$}\ 900$. 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?
An animal shelter has received a donation of bags of dry food. With the $45$ dogs currently in the shelter, there is enough food for $20$ days. However, after a rescue…
An animal shelter has received a donation of bags of dry food. With the $45$ dogs currently in the shelter, there is enough food for $20$ days.
However, after a rescue campaign, the number of dogs increases to $60$. How many days will the same amount of dry food last now? If they wanted the food to last $25$ days, what is the maximum number of dogs they could keep in the shelter?
Inversely proportional sharing
Sometimes you have to share an amount so that whoever has more of something gets less. For example, a prize shared according to the mistakes made: fewer mistakes, more prize. That is an inversely proportional sharing.
The idea: sharing inversely proportionally to $a, b, c$ is the same as sharing directly proportionally to their inverses $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$. To avoid fractions, you multiply the inverses by a common denominator and share as in direct proportionality.
In a video game design competition, the panel awards a prize of $168\,000$ pesos inversely proportional to the number of errors detected in the testing phase. Team…
In a video game design competition, the panel awards a prize of $168\,000$ pesos inversely proportional to the number of errors detected in the testing phase.
Team "Pixel" had $2$ errors, team "Voxel" had $4$ errors and team "Sprite" had $5$ errors. How much money will each team receive?
Compound proportionality
When three or more quantities are involved, you compare each one with the unknown separately (assuming the others do not 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).
A local municipality hires a team of $15$ workers to build a skatepark in $20$ days, working $6$ hours daily. After $8$ days of work, the municipality requests that the…
A local municipality hires a team of $15$ workers to build a skatepark in $20$ days, working $6$ hours daily. After $8$ days of work, the municipality requests that the work be finished $4$ days earlier than planned to coincide with local festivities. At that exact moment, $3$ workers go on leave. How many hours a day will the remaining workers need to work to meet the new deadline?