Exercises: Inverse proportion

State, giving reasons, whether the following pairs of quantities have an inverse proportional relationship:

1. a) The number of people painting a fence and the time they take to finish it.
2. b) The number of pounds of apples you buy and the total price you pay.
3. c) The speed at which a train travels and the time it takes to cover the distance between two cities.
4. d) The number of pages in a book and its weight.

Complete the following table in your notebook knowing that the quantities $A$ and $B$ are inversely proportional. Also calculate the constant of proportionality $k$.

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?

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.

A farmer has enough feed to feed his $20$ cows for $30$ days. If he sells $5$ cows and keeps $15$, how many days will he be able to feed the remaining cows with the same amount of feed?

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?

A water tank is filled in $6$ hours using a faucet with a flow rate of $10$ gallons per minute. How much time will it take to fill the same tank if the flow rate of the faucet increases to $15$ gallons per minute?

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 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.

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?

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 dog food last if the daily ration per dog is not changed?

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?

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.

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?

Six identical faucets take $40$ minutes to fill a wading pool. If two of the faucets break down and are out of service, how much time will the remaining faucets take to fill the same pool? Express the result in hours and minutes.

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?

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?

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.

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.

• a) What will be the height of the new rectangle?

• b) By what percentage has the height decreased compared to the original rectangle?

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?

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?

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.

• a) How much time will it take to complete the return journey?

• b) Express the result in hours, minutes and seconds.

Look at the following table which represents the relationship between two quantities $M$ and $N$. Determine if it is an inverse proportional relationship by calculating its constant $k$. Once found, complete the missing values, expressing them as integers or exact decimals.