Exercises: Inverse proportion

State whether each pair of quantities is directly proportional, inversely proportional, or neither:

1. a) The speed of a cyclist and the time taken to arrive.
2. b) The kilograms of strawberries bought and the price paid.
3. c) The number of people harvesting a garden and the time they take.
4. d) The age of a tree and its height.

Complete the table knowing that the quantities are inversely proportional:

Workers	$2$	$4$	$6$	$3$
Days	$12$	$6$

1. a) How many days do $6$ workers take?
2. b) How many days do $3$ workers take?
3. c) How many days would $1$ worker take?

Five identical taps fill a tank in $8$ hours. Let's calculate it step by step:

1. a) If there were only $1$ tap, how many hours would it take? (Think: a single tap works $5$ times slower.)
2. b) What if there were $10$ taps?

Check whether the quantities in these tables are inversely proportional by calculating the product of each pair of values:

a)

$x$	$2$	$3$	$4$	$6$
$y$	$18$	$12$	$9$	$6$

b)

$x$	$2$	$3$	$4$	$5$
$y$	$30$	$20$	$15$	$10$

A team of $4$ bricklayers takes $18$ days to build the perimeter wall of a municipal sports centre. To finish earlier, the site manager decides to bring in $2$ more bricklayers with the same work rate, making a total of $6$ bricklayers.

Cian, one of the labourers, sets up the following rule of three to calculate the duration of the work:

$$4\ \text{bricklayers} \longrightarrow 18\ \text{days}$$ $$6\ \text{bricklayers} \longrightarrow x\ \text{days}$$ $$x = \frac{6 \times 18}{4} = 27\ \text{days}$$

1. a) Without doing complex numerical operations, explain why Cian's answer makes no sense from a real-world perspective.
2. b) What conceptual mistake did Cian make when setting up the proportion?
3. c) Correct the calculation and find the exact number of days that the $6$ bricklayers will take using the inverse proportionality constant (the constant product).

The municipal swimming pool of a town is usually filled in $12\ \text{hours}$ using $3$ identical hydrants connected to the water mains. Due to pre-summer maintenance work, one of the hydrants is broken, so the filling must be done using only $2$ hydrants with the same flow rate.

1. a) Before calculating: explain whether the required time will be greater or less than $12\ \text{hours}$. Do you think it will double and take $24\ \text{hours}$ or more? Write a rough, reasoned estimate of the time interval in which you expect to find the solution.
2. b) Perform the formal calculation applying inverse proportionality to find out the exact time it will take to fill the pool with the $2$ available hydrants.
3. c) Compare the obtained value with your initial estimate and explain whether the final result is consistent.

A team of $6$ workers takes $10$ days to repair a park track.

1. a) How many days would $4$ workers take working at the same rate?
2. b) How many workers are needed to finish the track in $5$ days?

A tap delivering $120$ litres per minute fills a swimming pool in $15$ hours.

1. a) How long will another tap delivering $200$ litres per minute take to fill it?
2. b) What is the volume of the pool, in litres? Explain why it is not necessary to know this to answer part a).

A bus travelling at $60\ \text{km/h}$ takes $45$ minutes to complete its route.

1. a) How long would it take travelling at $90\ \text{km/h}$?
2. b) At what speed should it travel to complete the route in $30$ minutes?

The graph shows the number of days taken by different teams of painters to paint the same house according to the number of painters in the team.

1. a) How many days does a team of $4$ painters take? And a team of $2$?
2. b) How many painters are needed to paint the house in $6$ days?
3. c) For each point on the graph, multiply the number of painters by the number of days taken. What do you notice? What type of proportion is there between the two quantities?

On a farm there is enough feed to feed $20$ horses for $30$ days.

1. a) For how many horses would the same feed last $25$ days?
2. b) If the farm buys $10$ more horses, how many days will the feed last?

At a campsite there is enough food prepared for $12$ campers to eat for $15$ days.

1. a) On the first day, $3$ more campers arrive. For how many days will the group have food?
2. b) If instead of people arriving, $2$ campers left before starting, how many days would the food last?

At a pilgrim hostel on the Camino de Santiago, there are provisions in the pantry to feed $30$ people for $12$ days at a rate of one full ration per person per day. Unexpectedly, a group of $10$ more pilgrims arrives (becoming $40$ people in total). The hostel manager wants to know how many days the provisions will last if the size of the rations is kept the same.

Two volunteers propose different ways to calculate it:

• Method 1 (Unitary method): First find out how many days that same food would last if there were only one person in the hostel, and from there calculate the days for $40$ people.

• Method 2 (Inverse proportionality constant): Calculate the accumulated total of rations ($30 \times 12$) and set up the equation $40 \times x = 360$.

1. a) Solve the problem applying Method 1. What is the real-world meaning of the quantity you obtain in the intermediate step when calculating the duration for a single pilgrim?
2. b) Solve the problem applying Method 2 and verify that the same result is obtained.
3. c) If instead of $10$ people arriving, only $3$ had arrived (making $33$ pilgrims in total), which of the two methods would be more convenient for working with simplified fractions and avoiding infinite decimals? Justify your choice.

Following a severe storm on the Cantabrian coast, a municipal crew of $6$ workers takes $8\ \text{hours}$ full time to remove the algae and plastics accumulated on a beach.

The town hall's safety and tide regulations require that continuous cleaning work in that area must under no circumstances exceed $5\ \text{hours}$ per day.

1. a) Justify why the number of workers and the cleaning time are inversely proportional quantities and calculate the inverse proportionality constant, stating its units and what it represents in this real-world context.
2. b) To finish the work in a single working day strictly complying with the maximum limit of $5\ \text{hours}$, what is the minimum number of workers that the council must hire? Keep in mind that the workforce must consist of whole persons.
3. c) If in the end the town hall can only deploy $9$ workers, how much time will they spend on the clean-up exactly? Express the result broken down into whole hours and minutes.

State whether the following statements are true or false, giving reasons for your answer:

1. a) If as one quantity increases the other decreases, the two quantities are inversely proportional.
2. b) If two quantities are inversely proportional, doubling one of them halves the other.
3. c) A mother's age and her daughter's age are inversely proportional quantities.
4. d) If two inversely proportional quantities take the values $(3, 20)$, they can also take the values $(5, 12)$.

A team of $12$ workers can finish a building project in $20$ days, all working at the same rate.

1. a) How many days would $8$ workers take?
2. b) The work starts with the $12$ workers, but after $5$ days of work, $4$ workers leave for another project. How many more days will the remaining $8$ workers need to finish it?
3. c) How many days did the work take in total?

Two cogwheels are meshed together: when one turns, it forces the other to turn. The large wheel has $36$ teeth and makes $12$ complete turns.

1. a) If the small wheel has $16$ teeth, how many turns does it make in the same time?
2. b) Why are the number of teeth and the number of turns inversely proportional quantities? What product remains constant?
3. c) If we wanted the small wheel to make $72$ turns, how many teeth should it have?

A car travelling at $90\ \text{km/h}$ takes $4$ hours to go from one city to another. To find out how long it would take travelling at $120\ \text{km/h}$, a pupil set up a direct proportion and got $5,\widehat{3}$ hours.

1. a) Explain what mistake they made.
2. b) Calculate the correct time at $120\ \text{km/h}$.
3. c) What speed should it travel at to take only $3$ hours?

A tank fills in $6$ hours if $3$ identical taps are turned on at the same time.

1. a) How long would it take to fill with just one tap?
2. b) All $3$ taps are turned on, but after $2$ hours one of them breaks down and stops working. How many more hours will the remaining $2$ taps take to finish filling the tank?
3. c) How many hours in total did the tank take to fill?

A grandmother wants to share $\text{€}\ 180$ between her three grandchildren, aged $2$, $3$ and $6$ years old, in inverse proportion to their ages (so that the youngest receives more).

1. a) Sharing in inverse proportion to their ages is the same as sharing in direct proportion to $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. Calculate the sum of these three fractions.
2. b) What fraction of the total corresponds to each grandchild?
3. c) Calculate how many euros each one receives and check that they add up to $\text{€}\ 180$.

A sports foundation distributes a financial scholarship of $\text{€}\ 1300$ among the top three athletes of a school race. To reward merit, the money is distributed in inverse proportion to the time in minutes that each runner took to cross the finish line.

The first-place finisher took $20\ \text{minutes}$ and the second-place finisher took $30\ \text{minutes}$. However, the bib with the chip of the third-place finisher fell off during the race, and their exact time was not recorded in the computer system. The official prize distribution certificate shows that the first-place finisher received exactly $\text{€}\ 600$.

1. a) From the prize obtained by the first-place finisher, calculate the distribution constant ($k$) and find how much money the second-place finisher received.
2. b) Deduce how much money was left over from the total fund of $\text{€}\ 1300$ to be awarded to the third-place finisher.
3. c) Set up the corresponding equation and reconstruct the exact time (in minutes) that the third-place finisher took to complete the race.

In a wind turbine at a wind farm in Castilla-La Mancha, the outer blades rotate slowly, so a gearbox with two coupled gear wheels is used to drive the electric alternator. The driving wheel has $90\ \text{teeth}$ and rotates at a constant speed of $16\ \text{rpm}$ (revolutions per minute). The driven wheel has fewer teeth, so it rotates faster.

1. a) Mathematically justify why the number of teeth on a wheel and its rotational speed (in $\text{rpm}$) are inversely proportional quantities, explaining what physical quantity remains constant at the point of contact between both wheels during one minute.
2. b) For the alternator to start supplying electricity to the power grid, the driven wheel must rotate at a minimum of $60\ \text{rpm}$. What is the maximum number of teeth that this driven wheel can have?
3. c) For safety and thermal resistance reasons, the driven wheel must never exceed $100\ \text{rpm}$. The wind farm's spare parts catalogue has standard gear wheels with $12$, $18$, and $24$ teeth. Analyse which of these three models simultaneously meet both technical requirements (minimum speed of $60\ \text{rpm}$ and maximum of $100\ \text{rpm}$) and justify which one should be installed if the goal is to obtain the highest possible speed within safe margins.