Proportional sharing
Three friends win 300 € in a video game tournament. Alex has played 2 games, Bea 3 and Cris 5. Is it fair to give 100 € to each of them? It seems fairer that whoever has played more should receive more. That is a proportional share.
Directly proportional sharing
The amount is shared in parts proportional to some numbers: whoever has more, receives more.
Change the total or the parts and see how the bar is shared out:
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…
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…
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.
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…
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?
Inversely proportional sharing
Now whoever has more receives less. For example, bonus points in a robotics competition are shared out inversely proportionally to the time each robot took: the fastest (least time) gets the most.
The trick: share directly proportionally to the reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$.
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…
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?
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…
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 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…
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?
Sharing with two criteria
Sometimes an amount is shared according to two things at once: for example, the money invested and the length of time it was invested. You multiply each person's numbers and share proportionally to those products.
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,…
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 $3$, $5$…
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?