Exercises: Proportional sharing

Divide the amount of $9\,000$ pesos in direct proportion to the numbers $2$, $3$ and $4$.

Divide the number $300$ into parts inversely proportional to $2$ and $3$.

In a distribution directly proportional to the quantities $3$, $5$ and $7$, the first person receives $2\,400$ pesos. Calculate the total amount shared and what the other two people receive.

Divide $330$ points in inverse proportion to $1$, $2$ and $3$.

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

An amount of $2\,600$ pesos is divided in inverse proportion to the numbers $2$, $3$ and $4$. How much corresponds to the part associated with the number $4$?

A grandfather decides to divide $2\,400$ pesos among his three grandchildren aged $8$, $10$ and $12$ in direct proportion to their ages. Determine the amount of money each receives.

In a school competition, $190$ bonus points are awarded in inverse proportion to the number of mistakes made. If the three finalists made $1$, $3$ and $4$ mistakes respectively, calculate the points obtained by each student.

Three partners, Nerea, Beatriz and Santiago, invest $15000$ pesos, $20000$ pesos and $25000$ pesos respectively to open a business. After one year, the company generates a net profit of $14400$ pesos. Calculate how much of the profit each partner receives if it is shared in direct proportion to the capital invested.

In an athletics competition, the organisers share out a prize of $68\,000$ pesos in inverse proportion to the time taken by the top three finishers to cross the finish line. If they crossed the finish line in $12$, $15$ and $20$ minutes respectively, determine the cash prize awarded to each runner.

Four workers paint a building. The first works $5$ days at a rate of $6$ hours per day; the second, $4$ days at $8$ hours per day; the third, $6$ days at $5$ hours per day and the fourth, $3$ days at $10$ hours per day. If they earn a total of $73\,600$ pesos and decide to share it in direct proportion to the total hours worked by each, find how much each worker will receive.

An amount of money has been divided in inverse proportion to the ages of three siblings, who are $10$, $15$ and $20$ years old. If it is known that the $15$-year-old sibling received $8\,000$ pesos, find the total amount shared and what the other two siblings receive.

An inheritance is divided in direct proportion to the numbers $4$, $7$ and $9$. Knowing that the heir who receives the largest share gets $70\,000$ pesos more than the one who receives the smallest share, calculate the total value of the inheritance and the amount assigned to each of the three shares.

In a laboratory experiment, a $145$-gram sample of a chemical compound must be divided among three containers in inverse proportion to their temperatures. The temperatures are $\frac{1}{2}$ ºC, $\frac{2}{3}$ ºC and $\frac{3}{4}$ ºC. Calculate the exact amount of compound that will be placed in each container.

To reforest a natural park, three municipalities provide different groups of volunteers. Municipality A provides $45$ volunteers; B, $60$ volunteers; and C, $75$ volunteers. The regional government awards them a total grant of $216\,000$ pesos for purchasing equipment, to be shared in direct proportion to the number of volunteers provided. What amount does each municipality receive?

A productivity bonus of $50\,600$ pesos is shared among three factory employees. The distribution is carried out in direct proportion to their years of service in the company ($4$, $5$ and $8$ years) and in inverse proportion to the number of days absent during the year ($2$, $3$ and $4$ days). Calculate the bonus received by each employee.

A total amount of $74$ is divided inversely proportional to three consecutive positive integers: $x$, $x+1$, and $x+2$. Knowing that the constant of inverse proportionality of the division is $k = 120$:

1. a) Set up the corresponding equation and find the three integers.
2. b) Determine the value of each of the three shares obtained in the division.

A prize of $120\,000$ pesos is to be shared among three cyclists at the end of a grand tour. The rules state that the sharing will be directly proportional to the number of stages won ($3$, $4$ and $5$ stages) and inversely proportional to the number of crashes suffered ($1$, $2$ and $x$ crashes). If the cyclist who won $4$ stages and suffered $2$ crashes received exactly $40\,000$ pesos, determine the number of crashes $x$ of the third cyclist and the amount of money they received.

A triangular plot of land is divided into three parcels by drawing straight lines from an interior point $P$ to the three vertices. The areas of the three resulting parcels ($S_1$, $S_2$ and $S_3$) are directly proportional to the numbers $3$, $4$ and $5$. If the plot of land has a total area of $2400$ m$^2$ and is sold at a price of $15$ pesos/m$^2$, calculate the monetary value of each of the three parcels.

A grandmother wants to divide $84\,000$ pesos among her three grandchildren aged $10$, $12$ and $15$ years old. She hesitates between doing it directly proportional to their ages or inversely proportional to them. Calculate how much each grandchild would receive under both options and state which grandchild experiences the greatest absolute difference in money between the two methods.

An amount of $174\,000$ pesos has been divided inversely proportional to three unknown numbers $A$, $B$ and $C$. The amounts received were $80\,000$ pesos, $60\,000$ pesos and $34\,000$ pesos respectively. Knowing that the smallest of the three original numbers is $51$, determine the value of the other two numbers.

A technology company shares $1\,000\,000$ pesos of profit among three departments in two phases. In the first phase, $40$% of the total is shared directly proportional to the overtime hours worked: $120$, $150$ and $130$ hours. In the second phase, the remaining $60$% is shared inversely proportional to the number of technical incidents reported: $2$, $4$ and $5$ incidents. Calculate the total final profit received by each of the three departments by combining both phases.

Divide the algebraic expression $N = 6x^2 + 8x$ directly proportional to the expressions $x$, $x+1$ and $x+3$. Express the three resulting parts in terms of $x$ and fully simplify the polynomials obtained.

Three different pipes fill an industrial tank. The first takes $a$ hours, the second $b$ hours and the third $c$ hours. It is known that the times $a$, $b$ and $c$ are directly proportional to the numbers $2$, $3$ and $4$. The maintenance costs of the pipes amount to $26\,000$ pesos and it is decided to split them inversely proportional to the time each one takes to fill the tank. Find how much maintenance must be paid for each pipe.