Exercises: Inverse proportion

State which of the following pairs of quantities are inversely proportional:

• The speed at which a train travels and the time taken to cover a fixed distance.

• The number of kilograms of apples you buy and the total price you pay.

• The number of workers on a building site and the days taken to finish it.

• A person's age and their height.

• The flow rate of water from a tap and the time needed to fill a swimming pool.

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

A team of $4$ painters takes $6$ days to paint the facade of a building. If the company decides to send only $3$ painters, how many days will it take them to do exactly the same job?

A car travels at a constant speed of $90\text{ km/h}$ and takes $4$ hours to reach its destination. If on the return journey it wants to cover the same route in $3$ hours, at what constant speed must it travel?

A tap discharges a flow rate of $15$ litres per minute and takes $20$ minutes to fill a tank. How long will it take to fill the same tank using another tap whose flow rate is $25$ litres per minute?

Divide the amount of $300$ € into parts inversely proportional to the numbers $2$ and $3$.

A gear with $20$ teeth is connected to a motor and makes $30$ revolutions per minute. If it is coupled to another smaller gear that has $15$ teeth, how many revolutions per minute will this second gear make?

The following graph represents an inverse proportion relationship between quantities $X$ and $Y$. Look at the highlighted point and determine the constant of proportionality $k$. Then, calculate what value quantity $Y$ will take when $X = 4$.

[SOMEONE ELSE'S ERROR] In a school charity race, $260\text{ euros}$ are to be distributed among the top three finishers in inverse proportion to the number of unexcused absences accumulated during the term: Alicia had $2$ absences, Bruno had $3$ absences, and Carlos had $4$ absences.

Marcos, the class representative, proposes solving it like this:

“Since the sum of absences is $2 + 3 + 4 = 9$, so that whoever has fewer absences gets more money, we allocate $\frac{4}{9}$ of the prize to Alicia, $\frac{3}{9}$ to Bruno, and $\frac{2}{9}$ to Carlos”.

1. a) Explain accurately why Marcos's method is incorrect. To prove it, calculate the product of the absences and the money each person would receive according to Marcos, and check whether the constant of inverse proportionality ($x \cdot y = k$) is preserved.
2. b) Set up the correct distribution by calculating the share that legitimately corresponds to each participant.

[ESTIMATE AND CHECK] To fill the irrigation pond of the school garden, two water intakes can be used: the main hose $A$, which fills the empty tank on its own in $3\text{ hours}$, and an auxiliary intake $B$, which takes $6\text{ hours}$ alone. It is decided to open both at the same time.

1. a) Before doing any formal calculations, make a reasoned estimate: why must the joint time necessarily be less than $3\text{ hours}$ and greater than $1,5\text{ hours}$? Justify both bounds by thinking about the filling rate.
2. b) Calculate exactly the time it will take them to fill the pond working together by adding their hourly work rates. Express the result in hours and minutes, and check whether it confirms your initial estimate.

A team of $12$ workers commits to finishing the renovation of a park in $15$ days. After $5$ days of work, $2$ workers go on sick leave and cannot be replaced. How many days will it take the remaining workers to finish the rest of the work?

A grandfather decides to divide $4500$ € among his three grandchildren in inverse proportion to their ages, which are $10$, $12$ and $15$ years old. Calculate the exact amount of money each grandchild receives.

A tap $A$ can fill a tank in $6$ hours, while tap $B$ would fill it in $4$ hours. If both taps are opened simultaneously starting with an empty tank, how long will it take to fill it? Express the result in hours and minutes.

A freight train makes a journey at an average speed of $120\text{ km/h}$ and takes $2$ hours and $45$ minutes. If on the return journey, with a heavier load, the journey takes $3$ hours and $18$ minutes, what was its average speed on the return journey?

To bottle a cooperative's oil harvest, $420$ flagons of $5$ litres capacity are needed. If it is finally decided to use glass bottles of $1,5$ litres, how many bottles will be needed? Furthermore, if each empty bottle costs $0,20$ €, what will be the total cost of the containers?

A group of friends wants to buy a birthday present together. The following graph shows how much each person must pay (in euros) depending on the number of friends who finally participate. Look at the highlighted points to deduce the total price of the present. If in the end only $3$ friends participate, how much will each pay?

An animal protection foundation needs to distribute $1550\text{ kg}$ of animal feed among three shelters. They decide to do so in inverse proportion to the fractions $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{5}$ (which represent indices of own resources for each shelter). Calculate how many kilograms of feed each will receive.

In a charity relay race, a team of runners maintains an average speed of $12\text{ km/h}$ and takes $40$ minutes to complete their route. If in the next edition they train to maintain an average speed of $15\text{ km/h}$, how much time will they save compared to their previous mark? Express the difference in minutes.

[MODEL] The Directorate-General for Traffic (DGT) monitors speed in the Guadarrama tunnel (AP-6 motorway) using a section speed camera whose exact length is $3,3\text{ km}$. The maximum speed limit on this section is $100\text{ km/h}$.

1. a) Model the function $t(v)$ that determines the time $t$ (in seconds) it takes a vehicle to travel the section as a function of its average speed $v$ (in $\text{km/h}$). What type of function is it, and what are the value and units of its constant of proportionality?
2. b) What is the minimum legal time, measured in seconds to one decimal place, that the speed camera must record so that the driver is not fined?
3. c) If a driver travels at $80\text{ km/h}$ ($20\,\%$ below the legal speed limit), does their travel time increase by exactly $20\,\%$? Prove algebraically what percentage it actually increases by, and justify this behavior using the properties of inverse proportionality.

[CHOOSE THE METHOD] A regional grant awards an extraordinary aid of $4500\text{ euros}$ for digitalization to three high schools in the same district. The amount is distributed in inverse proportion to each school's student-to-computer ratio: school $A$ has a ratio of $6$ students/PC, school $B$ has $8$ students/PC, and school $C$ has $12$ students/PC.

Two solving procedures are proposed:

• Method 1 (Direct to reciprocals): Distribute directly in proportion to the fractions $\frac{1}{6}$, $\frac{1}{8}$, and $\frac{1}{12}$, reducing them to a common denominator and dividing the total according to the resulting integer numerators.

• Method 2 (Equation with constant $k$): Express each school's allocation as $\frac{k}{6}$, $\frac{k}{8}$, and $\frac{k}{12}$, solve the equation $\frac{k}{6} + \frac{k}{8} + \frac{k}{12} = 4500$, and calculate each allocation from $k$.

1. a) Solve the distribution using both methods step by step and verify that the same result is obtained.
2. b) Compare the computational workload of both methods and justify which of the two would be more efficient if you had to program a spreadsheet to distribute funds among the $85$ educational institutions across the entire province.

An irrigation water tank has two inlet taps and a drain. The first tap would fill it from empty in $4$ hours, and the second tap would fill it in $6$ hours. Meanwhile, the drain is capable of emptying the tank completely in $8$ hours. If the tank is empty and all three pipes are opened simultaneously, how long will it take to fill? Express the result in hours, minutes and seconds.

A company distributes a bonus of $7200$ € among three of its employees in inverse proportion to the number of absences they had during the year: $2$, $3$ and $6$ absences respectively. Before transferring the money, the company applies a tiered tax deduction: it retains $15\%$ from the highest earner, $10\%$ from the middle earner and $5\%$ from the lowest earner. What net amount of money does each employee finally receive?

A construction company calculates that a workforce of $20$ workers will finish a building in $40$ days. After $10$ days of uninterrupted work, a materials transport strike forces work to stop for $5$ days. To avoid breaching the contract, the company decides to hire more staff. How many additional workers will need to be hired to finish the building within the total timeframe originally planned?

An industrial mechanism consists of three gears $A$, $B$ and $C$ connected in line ($A$ drives $B$, and $B$ drives $C$). Gear $A$ has $24$ teeth, $B$ has $36$ teeth and $C$ has $16$ teeth. If the motor turns gear $A$ at a speed of $120$ revolutions per minute (rpm), calculate the rotational speed of gear $C$. Does the number of teeth on the intermediate gear $B$ affect the final speed of gear $C$? Justify your answer.

The two curves in the following graph show inverse proportionality relationships of the form $y = \frac{k}{x}$. Curve $A$ passes through the point $(2, 6)$ and curve $B$ passes through the point $(3, 8)$. Find the constants of proportionality $k_A$ and $k_B$. Next, calculate the exact value of $x$ for which curve $B$ takes the value $y = 6$.

Three business partners must split the repair costs for their premises in inverse proportion to the age (in years) of their company vehicles, which are $2$, $4$ and $x$ years old. If the total repair cost comes to $1400$ € and it is known that the first partner (the one with the $2$-year-old vehicle) had to pay $800$ €, what is the age of the third partner's car and how much will the second partner pay?

An amateur cyclist completes an out-and-back mountain route (there and back along the same path). On the uphill section, they maintain an average speed of $12\text{ km/h}$, and on the downhill section, their average speed is $36\text{ km/h}$. If the total pedalling time was exactly $4$ hours, what is the total distance covered by the cyclist, combining the outward and return journeys?

A novel has a total of $240$ pages. In its original format, each page contains $30$ lines and each line has an average of $60$ characters. The publisher decides to launch a paperback edition, reducing the font size so that there are now $40$ lines per page and $75$ characters per line. If they also decide to add $15$ extra pages of colour illustrations (which contain no text), how many pages in total will the new paperback edition have?

[INVERSE PROBLEM] A 4th-year ESO teacher wants to write an exam problem about combined machine work that has “clean” solutions. The draft says:

“Two industrial 3D printers working simultaneously take exactly $6\text{ hours}$ to complete an urban planning model. Working separately, the faster printer takes $x\text{ hours}$ and the slower one takes $y\text{ hours}$ (with $x \le y$)”.

The teacher needs both $x$ and $y$ to be integer numbers of hours.

1. a) Set up the equation relating the work rates and transform it algebraically to write it in the factored form $(x - a) \cdot (y - b) = c$, identifying the integers $a$, $b$, and $c$.
2. b) Using the positive integer divisors of the number $c$, find all possible combinations of integer times $(x, y)$ that the teacher could choose for the problem statement.
3. c) If the teacher also adds the constraint that the slow machine must not take more than three times as long as the fast machine ($y \le 3x$), which of the solutions found in the previous section remain valid?

[MODEL] A logistics company analyzes the round trip made by a delivery van between a central warehouse and a regional industrial park, separated by a fixed distance of $d = 120\text{ km}$ ($240\text{ km}$ in total).

On the outbound journey, due to morning traffic and stops, the van registers an average speed of $v_1 = 60\text{ km/h}$.

1. a) Prove algebraically that the overall average speed of the complete trip, $V_m$, as a function of the average speed on the return journey, $v_2$, is given by the rational function: $V_m(v_2) = \dfrac{120 \cdot v_2}{60 + v_2}$.
2. b) Management requires that the average speed of the complete round trip reaches $120\text{ km/h}$. Try to calculate what speed $v_2$ it should reach on the return: set up the corresponding equation, attempt to solve it, and interpret physically what happens to the available time.
3. c) Study the horizontal asymptote of the function by calculating the value that $V_m(v_2)$ approaches when $v_2$ takes extremely large values ($v_2 \to \infty$). Explain what insurmountable limit the physics of the problem imposes on the overall average speed, regardless of how fast the vehicle travels on the return journey.