Inverse Proportionality

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 $k$.

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 $X = 4$.

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

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?

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 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, $420$ jugs of $5$ liters capacity are needed. If it is finally decided to use glass bottles of $1.5$ liters, how many bottles will be needed? Furthermore, if each empty bottle costs $4$ pesos, 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 pesos) 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?

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.

Divide the amount of $6\,000$ pesos into parts inversely proportional to the numbers $2$ and $3$.

A grandfather decides to divide $90\,000$ pesos 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.

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.

A company distributes a bonus of $144\,000$ pesos 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 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.

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.

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 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 $28\,000$ pesos and it is known that the first partner (the one with the $2$-year-old vehicle) had to pay $16\,000$ pesos, what is the age of the third partner's car and how much will the second partner pay?

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 color illustrations (which contain no text), how many pages in total will the new paperback edition have?