Inverse proportion
4 students set up the school sports hall for the Christmas concert in 30 minutes. If 6 students work, they will take less time, but not by as much as you might guess by eye. The more taps you open to fill a tank, the fewer hours it takes. In these problems, when you multiply one quantity by a number, the other is divided. This is inverse proportion.
What is inverse proportion?
State with reasons whether the following pairs of quantities have an inversely proportional relationship: a) The number of people painting a fence and the time taken to…
State with reasons whether the following pairs of quantities have an inversely proportional relationship:
- a) The number of people painting a fence and the time taken to finish it.
- b) The number of kilograms of apples you buy and the total price you pay.
- c) The speed at which a train travels and the time taken to cover the distance between two cities.
- d) The number of pages in a book and its weight.
To set up the stage for a music festival, $4$ workers worked for $12$ hours. If the festival decides to hire a total of $8$ workers working at the same rate, how many…
To set up the stage for a music festival, $4$ workers worked for $12$ hours. If the festival decides to hire a total of $8$ workers working at the same rate, how many hours will they take to set up the same stage?
The constant product
A group of friends wants to hire a van to go on a trip. If $6$ friends go, each must pay $15$ euros. Calculate how much each would have to pay if $10$ people end up…
A group of friends wants to hire a van to go on a trip. If $6$ friends go, each must pay $15$ euros. Calculate how much each would have to pay if $10$ people end up going on the trip.
To set up the stage for a music festival, $8$ workers worked for $15$ hours. If the local council needs the setup to be finished in just $10$ hours, how many additional…
To set up the stage for a music festival, $8$ workers worked for $15$ hours. If the local council needs the setup to be finished in just $10$ hours, how many additional workers will they need to hire, assuming all work at the same rate?
The unitary method and equal products
Aoife took a car journey at a constant speed of $90$ km/h and it took $4$ hours. If for the return journey she decides to go slower, at a speed of $60$ km/h, how long…
Aoife took a car journey at a constant speed of $90$ km/h and it took $4$ hours. If for the return journey she decides to go slower, at a speed of $60$ km/h, how long will it take her to complete the same journey?
A delivery drone takes $12$ minutes to deliver a parcel travelling at a speed of $40$ km/h. If the control centre decides to increase its speed to $60$ km/h for the next…
A delivery drone takes $12$ minutes to deliver a parcel travelling at a speed of $40$ km/h. If the control centre decides to increase its speed to $60$ km/h for the next delivery to the same location, how long will it take to arrive?
A freight train travels between two cities at an average speed of $80$ km/h, taking $3$ hours and $45$ minutes. On the return journey, due to a track fault, it must…
A freight train travels between two cities at an average speed of $80$ km/h, taking $3$ hours and $45$ minutes. On the return journey, due to a track fault, it must reduce its average speed to $60$ km/h.
-
a) How long will it take to complete the return journey?
-
b) Express the result in hours, minutes, and seconds.
Back to the start: If 4 students take 30 minutes, the work constant is $4 \cdot 30 = 120$ student-minutes. With 6 students they will take $120 \div 6 = 20$ minutes.
Formula and tables
When $x$ and $y$ are inversely proportional, $x \cdot y = k$ for a constant $k$. So you can write $y = \dfrac{k}{x}$. The constant $k$ is the "total": the whole prize, the whole cost, the total area.
Look at a table for $k = 12$:
| $x$ | 1 | 2 | 3 | 4 | 6 | 12 |
|---|---|---|---|---|---|---|
| $y$ | 12 | 6 | 4 | 3 | 2 | 1 |
Every column gives $x \cdot y = 12$. As $x$ gets bigger, $y$ gets smaller, but the product never changes.
Complete the following table in your notebook knowing that quantities $A$ and $B$ are inversely proportional. Also calculate the constant of proportionality $k$. |…
Complete the following table in your notebook knowing that quantities $A$ and $B$ are inversely proportional. Also calculate the constant of proportionality $k$.
| Quantity $A$ | $2$ | $4$ | $8$ | $1$ |
|---|---|---|---|---|
| Quantity $B$ | $12$ | $6$ | ... | ... |
A group of friends has won a $300$ € prize in a robotics contest and they have decided to share it equally. Complete the following table to find out how much money each…
A group of friends has won a $300$ € prize in a robotics contest and they have decided to share it equally. Complete the following table to find out how much money each one would receive depending on the number of people in the group. Also state what the constant of inverse proportionality is.
| Number of friends ($x$) | $2$ | $3$ | $4$ | $5$ | $6$ |
|---|---|---|---|---|---|
| Money per person ($y$) | $150$ | ? | ? | ? | ? |
In the graph you can see the case of $2$ friends marked.
A youth association from a neighbourhood in Cork hires a bus for an excursion to the Burren for a fixed cost of $720$ €, which is split equally among the people…
A youth association from a neighbourhood in Cork hires a bus for an excursion to the Burren for a fixed cost of $720$ €, which is split equally among the people attending.\na) Write the mathematical formula that relates the price per person ($y$, in euros) as a function of the number of attendees ($x$). State the value of the constant of inverse proportionality and explain what it represents in this context.\nb) The organisation establishes that the trip will only take place if the price per person stays between $15$ € and $30$ € (both inclusive). Propose three possible values for the number of attendees ($x$) and calculate the price per person ($y$) that would correspond to each of them.
More situations
Two gear wheels are meshed together. The first has $48$ teeth and turns at $150$ revolutions per minute (rpm). At what speed will the second wheel turn if it has $30$…
Two gear wheels are meshed together. The first has $48$ teeth and turns at $150$ revolutions per minute (rpm). At what speed will the second wheel turn if it has $30$ teeth?
Three friends decide to share out the $282$ € prize won in a robotics competition in inverse proportion to the number of minutes they took to solve their final test. If…
Three friends decide to share out the $282$ € prize won in a robotics competition in inverse proportion to the number of minutes they took to solve their final test. If the first friend took $12$ minutes, the second $15$ minutes, and the third $20$ minutes, how much money does each person get?
A mountain expedition has supplies stored to feed $20$ hikers for $12$ days. However, after the first $4$ days of the trek, they meet another group of $5$ people who…
A mountain expedition has supplies stored to feed $20$ hikers for $12$ days. However, after the first $4$ days of the trek, they meet another group of $5$ people who have run out of supplies and decide to share their food. If they keep the same daily ration per person, how many days fewer than planned will the remaining food last?