Inverse Proportionality
Four people clean the middle school courtyard in 30 minutes. If 2 more people show up, will it take the same time, more, or less? And how long exactly? The more people work, the less time it takes. In these problems, when you multiply one quantity by a number, the other is divided by that same number. That is inverse proportionality.
What is inverse proportionality?
For example: if the number of people doubles, the time is cut in half; if it triples, the time is cut to a third.
The graph of an inverse proportionality is a curve called a hyperbola. Notice that it is not a straight line, which is why it doesn't look like the graph of direct proportionality.
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 direct proportionality the constant is the quotient $y/x$. In inverse proportionality, the constant is the product $x \times y$.
Completing tables
In an inverse proportionality table, first calculate $k$ with a complete pair and then divide $k$ by the other value. For example, if $k = 12$:
| $x$ | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| $y$ | 12 | 6 | 4 | 3 | 2 |
If you see that the product $x \times y$ is not always the same, the relationship is not inverse proportionality.
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 $300$ 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 $300$ pesos. Calculate how much each would have to pay if $10$ people end up going on the trip.
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.
Inverse rule of three
Watch out for units and changes in rate
Sometimes the problem isn't about people, but about speed and time (fixed distance), flow rate and time (fixed capacity), or base and height (fixed area). In all of them, the product is the fixed quantity. For example, if a 4-hour trip at 90 km/h is made at 60 km/h: $90 \times 4 = 360$ km, and $360 \div 60 = 6$ hours.
If you have two changes (people and rate), first calculate the total amount of work and then divide by the new quantity.
Marta 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…
Marta 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 package traveling at a speed of $40$ km/h. If the control center decides to increase its speed to $60$ km/h for the next…
A delivery drone takes $12$ minutes to deliver a package traveling at a speed of $40$ km/h. If the control center decides to increase its speed to $60$ km/h for the next delivery to the same location, how long will it take to arrive?
Six high-speed printers take $4$ hours and $30$ minutes to print $5000$ advertising leaflets. If two of the printers break down before starting the job, how many hours…
Six high-speed printers take $4$ hours and $30$ minutes to print $5000$ advertising leaflets. If two of the printers break down before starting the job, how many hours and minutes will the remaining printers take to carry out the same order?
Gears and inverse sharing
Two connected gears: the one with more teeth turns more slowly. Teeth × revolutions per minute is constant.
You can also share inversely: whoever took less time receives more. The amount is shared in parts proportional to $\dfrac{1}{\text{time}}$.
Two gears are connected. The first has $12$ teeth and turns at $20$ revolutions per minute. How many revolutions will the second gear make if it has $24$ teeth?
Two gears are connected. The first has $12$ teeth and turns at $20$ revolutions per minute. How many revolutions will the second gear make if it has $24$ teeth?
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 total work is $4 \times 30 = 120$ person-minutes. With 6 people it takes $120 \div 6 = 20$ minutes.