Exercises: Proportional sharing

Divide the number $4500$ in direct proportion to $2$, $3$ and $4$.

Divide the number $3900$ in inverse proportion to $2$, $3$ and $4$.

Three siblings aged $8$, $12$ and $15$ receive a gift of $1400$ € to be shared in direct proportion to their ages. How much money does each receive?

Divide $620$ in direct proportion to the fractions $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{5}$.

In a competition, a prize of $5700$ € is shared in inverse proportion to the number of penalties incurred by the three finalists: $1$, $2$ and $5$ penalties. Calculate the prize that each one receives.

When dividing a certain amount in direct proportion to the numbers $5$, $7$ and $9$, the second share is $840$. Calculate the total amount divided and the value of the other two shares.

Divide the number $2480$ in inverse proportion to $\frac{1}{3}$ and $\frac{1}{5}$.

Four partners invest $2000$ €, $3000$ €, $4000$ € and $6000$ € in a company. If the annual profit is $45000$ €, how much does each partner receive if profits are shared in direct proportion to the initial investment?

To distribute a bonus of $950$ € among three employees of a local shop in inverse proportion to the workdays they were absent this quarter ($2$, $4$, and $5$ days), Mateo proposes the following solution: "Since the distribution is inverse, we assign the days in reverse: the one who missed $5$ days receives $2$ shares, the one who missed $4$ days receives $4$ shares, and the one who missed $2$ days receives $5$ shares. We add $2 + 4 + 5 = 11$ shares and divide $950$ by $11$".

1. a) Identify Mateo's conceptual error. Explain why "reversing the list of numbers" does not represent inverse proportionality, looking at the relationship that should exist between the employee who missed $2$ days and the one who missed $4$ days.
2. b) Carry out the correct distribution by calculating the constant of proportionality using the corresponding inverse fractions, and verify that the sum of the three allocations is exactly $950$ €.

Three adjacent municipalities in a rural district form an association (mancomunidad) to jointly manage the municipal waste collection and treatment service. According to the municipal register of the Instituto Nacional de Estadística (INE), the three localities have $1200$, $1800$, and $3000$ inhabitants, respectively. The annual cost of the service amounts to $84\,000$ €, and the district statutes establish that it must be funded in direct proportion to the population of each municipality.

1. a) Model the situation by identifying the direct proportionality constant (annual cost per inhabitant) and interpret its practical meaning.
2. b) Calculate the annual financial contribution that each town council must make.
3. c) Verify that the sum of the three contributions equals the total budget and determine what percentage of the overall cost is borne by the municipality with the largest population.

A grandfather decides to share $8600$ € among his three grandchildren in direct proportion to their average grades ($7,5$, $8,0$ and $6,0$) and in inverse proportion to their ages ($15$, $16$ and $12$ years). How much does each receive?

An inheritance is divided in inverse proportion to the ages of three heirs, who are $20$, $24$ and $32$ years old. If the youngest receives $16000$ € more than the oldest, what was the total inheritance and how much does each heir receive?

In a company, a productivity bonus of $12400$ € is shared among three employees in direct proportion to their years of service ($4$, $5$ and $8$ years) and in inverse proportion to their days of absence ($2$, $3$ and $4$ days). Calculate the share allocated to each employee.

Divide the number $3420$ into parts in direct proportion to $1,2$, $0,8$ and $2,5$, and in inverse proportion to $\frac{3}{4}$, $\frac{1}{2}$ and $\frac{5}{2}$.

Three farmers hire a tractor for $1300$ €. The first uses it for $3$ days at a rate of $4$ hours a day; the second, for $5$ days at $6$ hours a day; and the third, for $4$ days at $5$ hours a day. How much must each farmer pay for the hire?

When dividing an amount $C$ in inverse proportion to $2$, $5$ and $8$, it is observed that the sum of the largest and smallest shares obtained is $2500$. Calculate the total amount $C$ and the value of each of the three shares.

We want to divide $9300$ € into parts in inverse proportion to the decimals $0,4$, $0,5$ and $0,8$. Calculate the amount corresponding to each share.

An alloy of $420$ grams is composed of gold, silver and copper. The masses of these metals in the alloy are directly proportional to the numbers $3$, $4$ and $5$, and inversely proportional to their relative densities, which are approximately $19$, $10$ and $9$ respectively. Set up the division and determine how many grams of each metal are in the alloy (express the result rounded to two decimal places).

During a prolonged drought recorded by the Agencia Estatal de Meteorología (AEMET), the Hydrographic Confederation imposes a joint fine of $12\,600$ € on three agricultural farms for overexploiting an aquifer. It is agreed to distribute the amount of the fine in inverse proportion to the volume of rainwater that each farm managed to store and reuse in its reservoirs: El Olivar farm stored $2000$ $\text{m}^3$, La Vega farm stored $3000$ $\text{m}^3$, and Los Llanos farm stored $6000$ $\text{m}^3$.

1. a) Before carrying out exact calculations, reasonably estimate which farm should pay exactly half of the total fine and which should pay the least amount. Is it reasonable to expect El Olivar farm to pay three times as much as Los Llanos farm? Justify your conjecture based on the data.
2. b) Calculate the exact distribution of the fine using the corresponding fractions.
3. c) Assess whether the amounts obtained match your initial estimation and verify that the total sum is exactly $12\,600$ €.

Three entrepreneurs (Alba, Bruno, and Clara) start a cooperative sustainable bicycle courier business and agree to distribute the first quarter's profits in direct proportion to the initial capital invested by each of them. Alba contributed $4000$ € and Bruno contributed $6000$ €. Due to an error in the accounting records, Clara's exact initial contribution is unknown, but it is documented that the total profit obtained was $13\,500$ € and that Clara received exactly $4500$ €.

1. a) Set up a linear equation where the unknown $x$ is the capital contributed by Clara, relying on the definition of the direct proportional distribution constant.
2. b) Solve the equation to find out how much money Clara invested when starting the business.
3. c) Without needing to recalculate Clara's capital, set up and solve a proportion to directly calculate the profit corresponding to Alba and that corresponding to Bruno, and verify that the sum of the three profits is $13\,500$ €.

Prove mathematically that dividing an amount $S$ in direct proportion to the numbers $a$, $b$ and $c$ is equivalent to dividing it in inverse proportion to the numbers $\frac{1}{a}$, $\frac{1}{b}$ and $\frac{1}{c}$.

A sum of money of $30000$ € is divided into three parts in direct proportion to $x$, $x+2$ and $x+4$. If the largest share obtained is $12000$ €, set up an equation to determine the value of $x$ and calculate the other two shares.

Three partners, A, B and C, invest capital in a business. Partner A invests twice as much capital as B, and B invests three times as much as C. The business lasts for $2$ years, but A withdraws their capital after $8$ months and B withdraws theirs after $18$ months (C keeps theirs in for the full $24$ months). If the total profit is $68000$ €, how should it be shared in direct proportion to the capital invested and the time it was kept in the business?

When a number $N$ is divided in inverse proportion to $2$, $3$ and $6$, the first share exceeds the second by $400$ units. If that same number $N$ were divided in direct proportion to $2$, $3$ and $6$, what would be the exact difference between the largest and the smallest shares?

An amount of money is divided in direct proportion to $3$, $5$ and $k$. Knowing that the second share corresponds to $1500$ € and the total amount shared is $4200$ €, find the value of the parameter $k$ and calculate the amounts of the first and third shares.

In a tournament, a prize of $P$ euros is divided among the top three competitors in inverse proportion to the times they took to complete the event: $t_1$, $t_2$ and $t_3$. Knowing that $t_1 = \frac{t_2}{2}$ and $t_3 = \frac{4t_2}{3}$, and that the difference between the first and third prize is $4500$ €, find the total value of the prize $P$.

A parent divides an amount $N$ among their three children in direct proportion to their ages $a$, $b$ and $c$. It is known that the sum of their ages is $36$ years. The $a$-year-old child receives $1200$ €, the $b$-year-old receives $1600$ € and the $c$-year-old receives $2000$ €. Set up a system of equations to determine the total amount shared $N$ and the age of each child.

Two towns A and B decide to build a joint hospital. The cost of $15$ million euros will be shared in direct proportion to their populations and inverse proportion to their distance to the hospital. Town A has $12000$ inhabitants and is $4$ km from the hospital, while town B has $18000$ inhabitants and is $9$ km away. Calculate how much each town will pay for the construction of the hospital.

The Ministry of Agriculture, Fisheries and Food (MAPA) announces an extraordinary grant of $77\,000$ € to modernize the machinery fleet of three agricultural cooperatives ($A$, $B$, and $C$) located in areas facing demographic decline. The regulatory terms establish a distribution based on a dual criterion: directly proportional to the cultivated agricultural area (in hectares, ha) and inversely proportional to the average reported income per hectare (in thousands of euros per ha) over the last three years:

• Cooperative $A$: manages $120$ ha with an income of $4$ thousand €/ha.

• Cooperative $B$: manages $200$ ha with an income of $5$ thousand €/ha.

• Cooperative $C$: manages $80$ ha with an income of $2$ thousand €/ha.

1. a) Model the allocation by calculating the combined weighted coefficient of each cooperative (multiplying the direct factor by the inverse factor) and simplify the results to whole numbers.
2. b) Calculate the grant amount that corresponds to each cooperative from the available budget of $77\,000$ €.
3. c) Justify mathematically why cooperatives $B$ and $C$ receive exactly the same amount, even though cooperative $B$ cultivates $120$ hectares more than cooperative $C$ ($150\,\%$ more area). Explain the socioeconomic reasoning behind this type of distribution sought by the administration.

A tech company distributes a bonus of $6300$ € among three IT support technicians in inverse proportion to the number of user complaints received on their projects: Rocío had $3$ complaints, Sergio had $4$ complaints, and Alba had $6$ complaints. Three classmates propose different methods to calculate the distribution:

• Eva's method: Reduces the inverse fractions $\frac{1}{3}$, $\frac{1}{4}$, and $\frac{1}{6}$ to a common denominator ($12$) to transform the situation into a distribution directly proportional to the integer numerators ($4$, $3$, and $2$).

• Marcos's method: Defines the inverse proportionality constant $k$ by setting up the equation $\frac{k}{3} + \frac{k}{4} + \frac{k}{6} = 6300$, solves for the overall value of $k$, and then calculates each person's allocation by dividing $k$ by their complaints.

• Laura's method: Converts each inverse into a decimal number rounded to two decimal places ($\frac{1}{3} \approx 0,33$; $\frac{1}{4} = 0,25$; $\frac{1}{6} \approx 0,17$) and carries out the directly proportional distribution using these decimals.

1. a) Apply Eva's method and calculate the money received by each technician, verifying that they add up to $6300$ €.
2. b) Apply Marcos's method and verify that exactly the same amounts are obtained as with Eva's method.
3. c) Critically analyze Laura's method: perform the calculation with her rounded decimals and explain what numerical issue and economic unfairness arises in the amounts received by each technician.
4. d) Compare Eva's and Marcos's methods: explain which one you find more convenient when a calculator is not allowed, and which one makes the task easier if we had to add a fourth person to the distribution once the constant is known.