Proportional Distributions

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 local municipality wants to reward recycling and shares $180\,000$ pesos among three schools in direct proportion to the kilograms of paper recycled. The "El Olivo" school recycled $600$ kg, the "La Caoba" school $900$ kg, and the "El Pino" school $1500$ kg. Determine the amount of money each school receives.

Three friends decide to enter a team eSports tournament. After winning first prize, the organizers 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?

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 friends hire a car for a trip and must pay a deposit of $180$ pesos. 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 $180$ pesos will each person get back?

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?

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).

A group of three farmers decides to hire machinery for $\text{RD\$}\ 570$. 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 grandfather wishes to divide $\text{RD\$}\ 2\,664\,000$ 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?

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

A relay competition awards $\text{RD\$}\ 54\,000$ 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.

An Escape Room company offers a monthly voucher of $\text{RD\$}\ 450$ 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 municipalities decide to subsidize a reforestation project with a total of $\text{RD\$}\ 7\,200\,000$. The contribution of each municipality will be inversely proportional to the number of hectares of forest burned in its municipal area the previous year. Municipality A had $200$ ha burned, Municipality B had $500$ ha, and Municipality C had $800$ ha. If the municipality 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 Municipality B between the first method and the second?