Inverse Proportion
Four students clean the school gym in 30 minutes. If 6 students help, it will take less time, but how much less? Open more faucets and a tank fills sooner. Pedal with a smaller sprocket and the rear wheel spins more. In all of these, when one quantity gets multiplied, the other one gets divided. That is inverse proportion.
What Is Inverse Proportion?
In direct proportion, doubling one quantity doubles the other. In inverse proportion, doubling one quantity cuts the other in half.
Here is a picture. The graph shows $x$ people sharing a job that takes $12$ person-hours in total. More people means fewer hours, and the points curve down without ever touching the axes.
Before using inverse proportion, ask two questions: does more of one mean less of the other, and is the total (the whole job, the whole distance, the whole prize) staying the same?
State, giving reasons, whether the following pairs of quantities have an inverse proportional relationship: a) The number of people painting a fence and the time they…
State, giving reasons, whether the following pairs of quantities have an inverse proportional relationship:
- a) The number of people painting a fence and the time they take to finish it.
- b) The number of pounds of apples you buy and the total price you pay.
- c) The speed at which a train travels and the time it takes to cover the distance between two cities.
- d) The number of pages in a book and its weight.
A group of friends wants to rent a van to go on a trip. If $6$ friends go, each one must pay $\$15$. Calculate how much each one would have to pay if $10…
A group of friends wants to rent a van to go on a trip. If $6$ friends go, each one must pay $\$15$. Calculate how much each one would have to pay if $10$ people finally go on the trip.
The Constant Product
Direct proportion has a constant ratio $y/x$. Inverse proportion has a constant product $x \cdot y$. We call it $k$.
The balance below shows the idea for $6x = 120$: whatever number you put on one side, dividing by $6$ gives the other side.
Complete the following table in your notebook knowing that the quantities $A$ and $B$ are inversely proportional. Also calculate the constant of proportionality $k$. |…
Complete the following table in your notebook knowing that the 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 prize of $\$300…
A group of friends has won a prize of $\$300$ in a robotics competition 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 ($y$) | $150$ | ? | ? | ? | ? |
A technology classroom has several rectangular plywood boards that all occupy the same area: $1200$ sq in. Complete the table of possible dimensions (base and height)…
A technology classroom has several rectangular plywood boards that all occupy the same area: $1200$ sq in. Complete the table of possible dimensions (base and height) for these boards and explain what relationship exists between them.
| Base ($cm$) | $10$ | $20$ | $30$ | $40$ | $60$ |
|---|---|---|---|---|---|
| Height ($cm$) | ? | $60$ | ? | ? | ? |
Solving Problems with $k$
Most word problems give you one complete pair and ask for a missing value in a second pair. Here is the plan.
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Decide which two quantities are inversely proportional.
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Multiply the first pair to get $k$.
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Set the second pair's product equal to $k$ and solve the equation.
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Check that the answer makes sense: more workers should mean less time.
Some people prefer a flipped proportion: $\dfrac{x_1}{x_2} = \dfrac{y_2}{y_1}$. It gives the same answer, because it is just $x_1 y_1 = x_2 y_2$ rearranged.
Think of each problem as a rectangle: the width is the number of workers (or faucets), the height is the time, and the area $k$ never changes. A wider rectangle must be shorter.
To set up the stage for a music festival, $4$ workers have worked for $12$ hours. If the festival decides to hire a total of $8$ workers working at the same rate, how…
To set up the stage for a music festival, $4$ workers have 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?
To set up the stage for a music festival, $8$ workers have worked for $15$ hours. If the city council needs the setup to be finished in just $10$ hours, how many…
To set up the stage for a music festival, $8$ workers have worked for $15$ hours. If the city council needs the setup to be finished in just $10$ hours, how many additional workers must they hire, assuming they all work at the same rate?
An animal shelter has rations of dog food to feed its $45$ dogs for $20$ days. If $15$ more dogs rescued from a litter arrive tomorrow, how many days will the stock of…
An animal shelter has rations of dog food to feed its $45$ dogs for $20$ days. If $15$ more dogs rescued from a litter arrive tomorrow, how many days will the stock of dog food last if the daily ration per dog is not changed?
Speed, Time, and Flow Rate
For a trip of a fixed distance, speed × time = distance, so speed and time are inversely proportional. In the same way, for a fixed tank, flow rate × time = volume. The constant $k$ is the distance or the volume.
Ashley has made a car trip at a constant speed of $90$ mph and has taken $4$ hours. If for the return trip she decides to go slower, at a speed of $60$ mph, how much…
Ashley has made a car trip at a constant speed of $90$ mph and has taken $4$ hours. If for the return trip she decides to go slower, at a speed of $60$ mph, how much time will she take to make the same trip?
A delivery drone takes $12$ minutes to deliver a parcel traveling at a speed of $40$ mph. If the control center decides to increase its speed to $60$ mph for the next…
A delivery drone takes $12$ minutes to deliver a parcel traveling at a speed of $40$ mph. If the control center decides to increase its speed to $60$ mph for the next delivery to the same place, how much time will it take to arrive?
A farmer uses a hose with a flow rate of $5$ gallons per minute to fill a water tank in $12$ hours. What flow rate should a new hose have if he wants to fill the same…
A farmer uses a hose with a flow rate of $5$ gallons per minute to fill a water tank in $12$ hours. What flow rate should a new hose have if he wants to fill the same tank in just $3$ hours?
Gears and Bike Chains
When two gears mesh, each tooth that passes one gear passes the other too. So the number of teeth moved is equal for both:
$$\text{teeth}_1 \times \text{rpm}_1 = \text{teeth}_2 \times \text{rpm}_2$$A gear with more teeth turns slower. Teeth and rpm (revolutions per minute) are inversely proportional.
On a bike, the chain connects the chainring (pedals) to the rear sprocket. If you pedal the same amount, the same length of chain goes around the sprocket. So sprocket teeth × wheel turns stays constant.
Two gears are connected. The first has $12$ teeth and makes $20$ 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 makes $20$ revolutions per minute. How many revolutions will the second gear make if it has $24$ teeth?
Two cogwheels 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 cogwheels 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?
On a mountain bike, the ratio between the number of teeth on the chainring and the sprocket determines the distance traveled. If with a $48$-tooth chainring and a…
On a mountain bike, the ratio between the number of teeth on the chainring and the sprocket determines the distance traveled. If with a $48$-tooth chainring and a $24$-tooth sprocket the rear wheel turns exactly $200$ times on a stretch, how many turns would the wheel make if we changed to a $16$-tooth sprocket while maintaining the same pedaling (same chainring turns)? Justify the type of proportion.
When the Situation Changes Midway
Some problems change partway through. The trick is to work with what is left: how much food, work, or distance remains, and what is the new group size.
Sometimes the answer is a decimal number of hours. To convert to hours, minutes and seconds, convert each decimal part by multiplying by $60$. For example, $2.505$ hours: $0.505 \times 60 = 30.3$ minutes, and $0.3 \times 60 = 18$ seconds, so $2$ h $30$ min $18$ s.
A mountain expedition has provisions 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 provisions 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 provisions. If they keep the same daily ration per person, how many fewer days than planned will the remaining food last?
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?
A freight train makes the journey between two cities at an average speed of $80$ mph, taking $3$ hours and $45$ minutes. On the return journey, due to a fault on the…
A freight train makes the journey between two cities at an average speed of $80$ mph, taking $3$ hours and $45$ minutes. On the return journey, due to a fault on the track, it must reduce its average speed to $60$ mph.
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a) How much time will it take to complete the return journey?
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b) Express the result in hours, minutes and seconds.
Sharing in Inverse Proportion
Sometimes a prize is divided so that the person with the smaller number gets more. For example, the fastest runner gets the biggest share. Then the shares are proportional to the reciprocals (the fractions $\tfrac{1}{\text{number}}$) of the given numbers.
Use the sharing tool to see how a prize splits when the numbers change.
Three friends decide to share the $\$282…
Three friends decide to share the $\$282$ won as a prize in a robotics competition in inverse proportion to the 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 one receive?
Same Area, Same Total
Inverse proportion shows up whenever two numbers must multiply to a fixed total. Two more useful cases follow.
Rectangles with the same area. If the base is multiplied by a number, the height is divided by that number.
Three factors. Sometimes the fixed total is a product of three numbers, such as (number of faucets) × (flow rate) × (time) = total volume.
A rectangle has a base of $12$ in and a height of $8$ in. We want to construct another rectangle that has exactly the same area, but whose base is $25$ % longer than…
A rectangle has a base of $12$ in and a height of $8$ in. We want to construct another rectangle that has exactly the same area, but whose base is $25$ % longer than that of the first one.
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a) What will be the height of the new rectangle?
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b) By what percentage has the height decreased compared to the original rectangle?
To fill an Olympic swimming pool, $4$ faucets are used, each delivering a flow rate of $12$ gallons per minute, taking a total of $30$ hours. If we want the pool to be…
To fill an Olympic swimming pool, $4$ faucets are used, each delivering a flow rate of $12$ gallons per minute, taking a total of $30$ hours. If we want the pool to be filled in just $18$ hours using faucets with a higher flow rate of $16$ gallons per minute, how many faucets of this new model will we need to open?
Back to the start: 4 students clean the gym in 30 minutes, so $k = 4 \cdot 30 = 120$ student-minutes of work. With 6 students, $6t = 120$, so $t = 20$ minutes. Six is $1.5$ times as many students, so the time is divided by $1.5$.