Exercises: Proportional division

Three friends, Alex, Bea and Cris, took part in a team video game tournament and won a chest with $240$ virtual gold coins. They have decided to share the loot in direct proportion to the number of levels each one cleared: Alex cleared $2$ levels, Bea $3$ levels and Cris $5$ levels. How many coins does each person get?

A youth association has received a grant of $1200$ € to organise workshops. The money must be shared among three local groups in direct proportion to the number of volunteers each group has: Group A has $10$ volunteers, Group B has $15$ and Group C has $25$. Calculate how much money each group will receive.

In a robotics contest, $140$ extra points are shared out inversely proportionally to the number of mistakes made on the course. If team "Alfa" made $1$ mistake, team "Beta" made $2$ mistakes and team "Gamma" made $4$ mistakes, how many points will each team receive?

Three siblings help their grandparents harvest almonds. Their grandparents give them a total tip of $90$ € to share in direct proportion to the number of hours each one worked. If the eldest worked $5$ hours, the middle one $3$ hours and the youngest $1$ hour, how much money does each one get?

A local council wants to reward recycling and shares $3000$ € among three schools in direct proportion to the kilograms of paper recycled. The "El Olivo" school recycled $600$ kg, the "La Encina" school $900$ kg and the "El Pino" school $1500$ kg. Determine the amount of money each school receives.

Three friends hire a car for a trip and must pay a deposit of $180$ €. They decide that the return of the deposit (if there is no damage) will be shared in inverse proportion to the number of times each one had to be warned by the driver for not wearing a seatbelt. If Dani had $2$ warnings, Elena $3$ warnings and Fede $6$ warnings, what portion of the $180$ € will each person get back?

Three waiters at a café have decided to share the weekly tip jar, which totals $210$ €, in direct proportion to the number of days each one worked: Juan worked $2$ days, Marta $5$ days and Pedro worked the full week ($7$ days). How much tip money does each person receive?

Daniel wants to divide $180$ € among his three nephews aged $6$, $8$, and $12$ years in inverse proportion to their ages to reward the youngest ones. To solve it, Daniel states: «Since it is an inverse division, I assign each one the other's age: the $6$-year-old gets $12$ parts, the $12$-year-old gets $6$ parts, and the $8$-year-old keeps $8$ parts. I add $12 + 8 + 6 = 26$, calculate the constant by dividing $180$ by $26$, and multiply by each one's parts». a) Explain in detail why Daniel's reasoning is mathematically flawed when there are three participants and what property of fractions he did not take into account. b) Correct the exercise by performing the inversely proportional division properly and verify that the sum of the divided amounts matches the $180$ € available.

Three students share a flat in Granada during the month of July and agree to split the electricity and air conditioning bill ($90$ €) in direct proportion to the days each one has stayed in the flat: Lucía stayed $15$ days, Marcos stayed $20$ days, and Sara stayed $25$ days. a) Before calculating the exact result, reasonably estimate the approximate amount that Sara will pay: taking into account that the sum of the days is $60$, should her contribution be greater or less than one third of the total ($30$ €)? And greater or less than half ($45$ €)? Justify your prediction. b) Perform the exact directly proportional division by calculating the constant of proportionality $k$ (cost per day) and the amount that each student must pay. c) Check that the sum of the three contributions is $90$ € and analyze whether the result obtained for Sara confirms your initial estimate.

Three friends decide to enter a team eSports tournament. After winning first prize, the organisers give them a chest containing $2700$ virtual coins. To be fair, they decide to share the coins directly proportional to the experience points (XP) each contributed in the final game: Álex contributed $12$ points, Bea $15$ points and Cristian $18$ points. How many coins does each person get?

To encourage reading, a youth library is distributing a set of $155$ comics among the three users who have read the most books in the last term. The distribution will be directly proportional to the number of books read. If the users read $12$, $18$ and $32$ books respectively, how many comics will each one receive? (Note: If the result is not a whole number, round to the nearest unit and state whether the division is exact).

Three siblings have to share household chores. Their mother has decided that the number of hours each one will spend cleaning the garden will be inversely proportional to their average marks for the school year: Arnau has $9$, Amina $8$ and Jorge $6$. If they have to cover $74$ hours of work in total between the three of them, how many hours does each one get?

Three partners in an eco-friendly courier startup initially invested different amounts: $4000 \text{ €}$, $7000 \text{ €}$ and $9000 \text{ €}$. At the end of the year, they made a net profit of $12000 \text{ €}$. How should the profits be divided directly proportional to their initial investment?

A group of three farmers decides to hire machinery for $570 \text{ €}$. The cost will be shared directly proportional to the area of their plots, which measure $3$, $5$ and $7$ hectares respectively. However, the owner of the machinery gives them a $10 \%$ discount on the total because it is off-season. How much will each farmer have to pay after applying the discount?

A relay competition awards $900 \text{ €}$ among the three teams that crossed the finish line first. The prize is inversely proportional to the total time taken. Team "Rayo" took $10$ minutes, team "Gacela" took $12$ minutes and team "Flecha" took $15$ minutes. What amount of money does each team receive? Explain why the team that took the least time receives the largest amount.

In a school athletics day organized by a city council, $420$ classification points are divided among three schools in direct proportion to the number of medals obtained. Colegio San Viator obtained $5$ medals and Colegio Parque de las Avenidas achieved $7$ medals. When the official results were published, the third school, Colegio Mayorazgo, was awarded exactly $180$ points. a) From the definition of the constant of proportionality or by setting up an equation, deduce how many medals Colegio Mayorazgo obtained. b) Calculate how many points each of the other two schools received and verify that the sum of the scores of the three schools amounts to the total $420$ points.

An irrigation community in the Murcian orchard receives a regional grant of $38\,000$ € aimed at modernizing its irrigation channels against drought. The assembly agrees to distribute the money combining two criteria: in direct proportion to the cultivated area (in hectares) and in inverse proportion to the water consumption recorded in the previous season (in thousands of $\text{m}^3$), rewarding those who have more land but have demonstrated greater savings. The data for three plots are: Plot A ($12$ ha and consumption of $4$ thousand $\text{m}^3$), Plot B ($10$ ha and consumption of $2$ thousand $\text{m}^3$), and Plot C ($9$ ha and consumption of $6$ thousand $\text{m}^3$). a) Model the situation by calculating the relative distribution index of each plot when combining both criteria. b) Find the grant amount corresponding to each of the plots and verify that the total sum covers exactly the assigned $38\,000$ €.

An Escape Room company offers a monthly voucher of $450 €$ to be shared among the three teams that manage to escape their hardest room in under an hour. The prize is divided inversely proportional to the time taken. Team "Alpha" escaped in $40$ minutes, team "Beta" in $45$ minutes and team "Gamma" in $50$ minutes. Calculate the share of the prize that each team receives.

Three local councils decide to subsidise a reforestation project with a total of $120000 €$. The contribution of each council will be inversely proportional to the number of hectares of forest burned in its municipal area the previous year. Council A had $200$ ha burned, Council B had $500$ ha and Council C had $800$ ha. If the council with the most fires appeals the decision and it is decided that the distribution should be directly proportional to those hectares to help those most affected, what would be the difference in money for Council B between the first method and the second?

A grandfather wishes to divide $44400 €$ among his three grandchildren aged $15, 20$ and $30$ years so that the division is inversely proportional to their ages. Before the division is made, the middle grandchild decides to donate $25\%$ of their share to an NGO. What net amount will the $20$-year-old grandchild end up receiving?

A cycling club has a prize of $1860$ € to divide among three riders in inverse proportion to the accumulated time penalties in the season: Brais committed $2$ penalty minutes, Carmen accumulated $3$ minutes, and Diego had $5$ minutes. To solve it, two board members propose different paths: Method A (directly use the fractional reciprocals $\dfrac{1}{2}$, $\dfrac{1}{3}$, and $\dfrac{1}{5}$, add them together, and calculate $k = \dfrac{1860}{\sum \text{reciprocals}}$) and Method B (reduce the reciprocals to a common denominator using $\text{l.c.m.}(2, 3, 5) = 30$, take the resulting integer numerators $15$, $10$, and $6$, and divide directly according to those numbers). a) Fully work out both methods step by step and check that the amounts allocated to each cyclist match in both cases. b) Formally justify why multiplying all the fractions by the l.c.m. in Method B does not alter the proportion of the division. c) Argue which of the two methods is more efficient and carries less risk of error in an exam if the division were among four or more people with large denominators.

A courier company in Zaragoza distributes a quarterly bonus of $6600$ € among three of its delivery drivers (Alba, Bruno, and Clara) in inverse proportion to the number of packages delivered late. The records show that Alba had $2$ delays and Bruno accumulated $6$ delays, but Clara's data ($c$ delays) was erased due to a computer glitch. However, the accounting report confirms that Alba received on her payslip exactly $2200$ € more than Bruno. a) Model the constant of proportionality $k$ and the financial allocation of each worker using algebraic expressions in terms of the unknown value $c$. b) From the difference in money between Alba and Bruno, set up and solve a linear equation to determine exactly how many delays $c$ Clara had. c) Perform the final division with the number of delays found and verify that both conditions are simultaneously satisfied: that the sum of the three payments is $6600$ € and that the difference between Alba and Bruno is $2200$ €.