Proportional Sharing
Three friends win $\$300$ in a video game tournament. Alex played 2 games, Bea 3 and Cris 5. Is it fair to give $\$100$ to each one? It seems fairer for the person who played the most to get more. That is a proportional sharing.
Directly proportional sharing
The amount is shared in parts that are proportional to some numbers: the more you have, the more you receive.
Change the total or the parts and see how the bar is shared:
Other methods for the same sharing
There is no single way to do it. With the same tournament you can reason in other ways, and they all give the same result:
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With fractions of the total: Alex played 2 of the 10 games, so he gets $\frac{2}{10}$ of the prize: $\frac{2}{10} \times 300 = 60$ pesos. Bea gets $\frac{3}{10} \times 300 = 90$; Cris gets $\frac{5}{10} \times 300 = 150$.
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With percentages: 2 out of 10 is 20%, 3 out of 10 is 30% and 5 out of 10 is 50%. 20% of 300 is 60, and so on for the others.
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With a proportionality table: if 10 games are worth 300 pesos, complete the table of games and pesos (1 game → 30 pesos) and read off each value.
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 $24\,000$ pesos to organise workshops. The money must be shared among three local groups in direct proportion to the number…
A youth association has received a grant of $24\,000$ pesos 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 $180$ pesos. 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 $180$ 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?
Inversely proportional sharing
Now the one who has more receives less. For example, extra points in a robotics contest are shared inversely proportionally to the time each robot took: the fastest one (least time) gets more.
The trick: share directly proportionally to the inverses $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$.
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 municipal government wants to reward recycling and shares $60{,}000$ pesos among three schools in direct proportion to the kilograms of paper recycled. The "Benito…
A municipal government wants to reward recycling and shares $60{,}000$ pesos among three schools in direct proportion to the kilograms of paper recycled. The "Benito Juárez" school recycled $600$ kg, the "Sor Juana Inés de la Cruz" school $900$ kg and the "Niños Héroes" 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, money invested and the time it was invested. You multiply each person's numbers and share in proportion to those products.
Three partners in an eco-friendly courier startup initially invested different amounts: $\$4000$, $\$7000$ and $\$9000…
Three partners in an eco-friendly courier startup initially invested different amounts: $\$4000$, $\$7000$ and $\$9000$. At the end of the year, they made a net profit of $\$12000$. How should the profits be divided directly proportional to their initial investment?
A group of three farmers decides to hire machinery for $\$570$. The cost will be shared directly proportional to the area of their plots, which measure $3$, $5$ and $7…
A group of three farmers decides to hire machinery for $\$570$. 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?