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 hire a van to go on a trip. If $6$ friends go, each must pay $15$ euros. 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 tap with a flow rate of $10$ litres per minute. How long will the same tank take to fill if the flow rate of the tap increases to $15$ litres 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?

To fill the fire-fighting reservoir in a municipality of the Sierra de Cazorla, $3$ identical water hydrants are used, which take $12$ hours to fill it completely. Due to an emergency, the municipal team decides to connect $6$ identical hydrants at the same time. Marcos, one of the technicians, calculates the time they will take by setting up the following operation: $x = \dfrac{6 \cdot 12}{3} = 24\text{ hours}$.\na) Without doing complex calculations, reason by analysing an extreme or logical case why the result obtained by Marcos does not make sense in reality.\nb) Explain what conceptual error Marcos made when setting up his rule of three.\nc) Correct the approach using the constant of inverse proportionality and determine how many hours the $6$ hydrants will actually take.

A youth association from a neighbourhood in Zaragoza hires a bus for an excursion to the Monasterio de Piedra for a fixed cost of $720$ €, which is split equally among the people attending.\na) Write the mathematical formula that relates the price per person ($y$, in euros) as a function of the number of attendees ($x$). State the value of the constant of inverse proportionality and explain what it represents in this context.\nb) The organisation establishes that the trip will only take place if the price per person stays between $15$ € and $30$ € (both inclusive). Propose three possible values for the number of attendees ($x$) and calculate the price per person ($y$) that would correspond to each of them.

A group of friends has won a $300$ € 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 travelling at a speed of $40$ km/h. If the control centre 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 local council 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 $5$ litres 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.

A forestry brigade made up of $4$ workers takes $18$ hours of actual work to clear a firebreak strip in a communal woodland in Galicia. In anticipation of a heatwave, it is decided to bring in $2$ more workers, working at the same pace, forming a team of $6$ people.\na) Before calculating the exact value, make a reasoned estimate: knowing that double the number of workers ($8$) would take $9$ hours, will the team of $6$ workers take more or less than $10$ hours? Explain your mental deduction without solving the equation yet.\nb) Calculate the exact time that the $6$ workers will take using the constant product property ($a_1 \cdot b_1 = a_2 \cdot b_2$).\nc) Check whether the numerical result confirms your initial estimate and interpret what physical meaning the obtained proportionality constant has in terms of "hours of work for a single worker".

A regular line bus covers the $120\text{ km}$ journey connecting two towns in Castilla y León. The average speed of the journey ($v$, in $\text{km/h}$) and the time spent ($t$, in hours) are inversely proportional quantities.\na) Write the function that expresses the time $t$ (in hours) as a function of the average speed $v$.\nb) On a clear-traffic day the bus travels at an average speed of $80\text{ km/h}$, while on a day with dense fog its average speed drops to $50\text{ km/h}$. Calculate the travel time in each situation, expressing the result precisely in hours and minutes.\nc) If the transport consortium requires the journey to be completed in a maximum time of $1$ hour and $15$ minutes, at what minimum average speed must the bus travel? Justify your procedure.

Three friends decide to share out the $282$ € 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 $12$ litres 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 $16$ litres 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.

In a wine cooperative in La Rioja, a crew of $12$ grape pickers harvests the grapes from a plot in $20$ days. The management needs to anticipate how many days it would take to harvest the same plot under three possible hiring scenarios: with $24$ grape pickers, with $6$ grape pickers, and with $15$ grape pickers.\nTwo ways of solving this are proposed:\n- Method A: Set up three independent inverse rules of three for each case.\n- Method B: Calculate the constant of inverse proportionality $k$ (total person-days required), formulate the expression $y = \dfrac{k}{x}$, and use simple proportionality relationships (double/half) whenever possible.\na) Apply both methods to calculate the days corresponding to each of the three scenarios.\nb) Compare both procedures and justify with reasoning why Method B is more efficient and reliable when multiple staffing scenarios need to be evaluated.

In the Science laboratory of a secondary school, Boyle-Mariotte's law is tested experimentally: at constant temperature, the volume ($V$, in liters) occupied by a mass of gas and the pressure ($P$, in atmospheres) it is subjected to are inversely proportional ($P \cdot V = k$). At the start of the experiment, the gas occupies $12\text{ liters}$ under a pressure of $2\text{ atm}$.\na) A piston is operated to compress the gas so that its final volume is between $3\text{ liters}$ and $4\text{ liters}$ (both values included). Determine, giving your reasoning, the pressure interval ($P$) in which the compressor must work.\nb) A student puts forward the following hypothesis: «Since doubling the pressure reduces the volume by half, if we increase the pressure by $50\,\%$, the volume will also decrease by $50\,\%$». Prove algebraically whether his deduction is true or false, and calculate the exact percentage by which the volume actually decreases when the pressure increases by $50\,\%$.