Exercises: Inverse proportion

State with reasons whether the following pairs of quantities have an inversely proportional relationship:

1. a) The number of people painting a fence and the time taken to finish it.
2. b) The number of kilograms of apples you buy and the total price you pay.
3. c) The speed at which a train travels and the time taken 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 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 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 must pay $900$ pesos. Calculate how much each would have to pay if $10$ people end up going on the trip.

A farmer has enough feed to nourish his $20$ cows for $30$ days. If he sells $5$ cows and keeps $15$, for 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 turns at $20$ 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 $10$ liters per minute. How long will the same tank take to fill if the flow rate of the faucet increases to $15$ liters per minute?

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 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.

In the graph you can see the case of $2$ friends marked.

A delivery drone takes $12$ minutes to deliver a parcel 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?

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?

An animal shelter has food rations to feed its $45$ dogs for $20$ days. If $15$ more rescued dogs from a litter arrive tomorrow, how many days will the food supply last if the daily ration per dog remains unchanged?

A farmer uses a hose with a flow rate of $5$ liters 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 sheets that all have the same area: $1200$ cm². Complete the table of possible dimensions (base and height) for these sheets and explain what relationship exists between them.

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?

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

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?

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?

On a mountain bike, the relationship between the number of teeth on the chainring and on the sprocket determines how far the bike advances. If, with a chainring of $48$ teeth and a sprocket of $24$ teeth, the rear wheel makes exactly $200$ turns over a stretch, how many turns would the wheel make if we switched to a sprocket of $16$ teeth while keeping the same pedaling (same number of chainring turns)? Justify the type of proportionality.

A rectangle has a base of $12$ cm and a height of $8$ cm. We want to construct another rectangle that has exactly the same area, but whose base is $25$% longer than that of the first.

• 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, $4$ taps are used, each delivering a flow rate of $12$ liters per minute, taking a total of $30$ hours. If we want the pool to fill in just $18$ hours using taps with a higher flow rate of $16$ liters per minute, how many taps of this new model will we need to turn on?

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.

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