Inverse Proportionality
Four people clean the school playground in 30 minutes. If 2 more people arrive and 6 are working, it will take less time, but how much less? The more taps you open to fill a tank, the fewer hours you need. In these problems, when you multiply one quantity by a number, the other one is divided. That is inverse proportionality.
What Is Inverse Proportionality?
Examples: the number of workers and the days they take on a construction job; the speed of a bus and the travel time from Santiago to Santo Domingo; the number of friends and what each one pays when splitting a bill.
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
In a table of inverse proportionality, the product of each column is the same:
| People | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Minutes | 60 | 30 | 20 | 15 |
Here $k = 2 \times 60 = 4 \times 30 = 6 \times 20 = 8 \times 15 = 120$. If the product changes in any column, the relationship is not inverse.
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 wants to rent a van to go on a trip. If $6$ friends go, each must pay $900$ pesos. Calculate how much each would have to pay if $10$ people end up…
A group of friends wants to rent a van to go on a trip. If $6$ friends go, each must pay $900$ pesos. 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 municipality 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 municipality 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?
Inverse Rule of Three
For problems with speed and time, or with flow rate and time, the same idea works: the distance (or volume) is fixed, so speed × time is constant. If you travel at $90$ km/h for $4$ hours, the distance is $360$ km; at $60$ km/h you take $360 : 60 = 6$ hours.
Gears are also inverse: the wheel with more teeth turns more slowly. Teeth × revolutions per minute is constant.
When a problem asks you to share something inversely proportionally, you share in proportion to the inverses: for example, if they take $2$ and $4$ minutes, they get shares in the proportion $\frac12$ and $\frac14$, that is, $2 : 1$. Convert the fractions to a common denominator so the proportion is in whole numbers.
A group of friends has won a $300$ pesos prize in a robotics contest and they have decided to share it equally. Complete the following table to find out how much money…
A group of friends has won a $300$ pesos 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.
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 $282$ pesos prize won in a robotics competition in inverse proportion to the number of minutes they took to solve their final test.…
Three friends decide to share out the $282$ pesos 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?
Back to the beginning: if 4 people take 30 minutes, the constant is $4 \times 30 = 120$ person-minutes. With 6 people it will take $120 : 6 = 20$ minutes.