Proportional sharing
Three friends in an e-sports club win a tournament and share 1 900 €. The first scored 150 points in the final, the second 120 and the third 110. Is it fair to share it out equally? It seems fairer that whoever scored more points gets more money. That is a proportional share.
Directly proportional sharing
You share in parts that are proportional to some numbers: whoever has more receives more.
Change the total or the parts and see how the bar is shared out:
Three siblings decide to share a bag of $120$ sweets directly proportional to their ages, which are $2$, $3$ and $7$ years. How many sweets does each get?
Three siblings decide to share a bag of $120$ sweets directly proportional to their ages, which are $2$, $3$ and $7$ years. How many sweets does each get?
Three friends, Alba, Beto and Caio, collaborated at a charity market selling crafts. They decide to share the $300 \ €$ profit directly proportional to the number of…
Three friends, Alba, Beto and Caio, collaborated at a charity market selling crafts. They decide to share the $300 \ €$ profit directly proportional to the number of hours each worked: Alba worked $2$ hours, Beto $3$ hours and Caio $5$ hours. Calculate how much money each receives.
Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rental is $153\ €$. They have agreed to share the…
Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rental is $153\ €$. They have agreed to share the payment directly proportional to the time each was connected: Álex was on for $12$ hours, Bea for $18$ hours and Cristian for $21$ hours. How much should each contribute to pay the bill?
Inversely proportional sharing
Now whoever has more of something receives less. For example, a town council shares out a fund among several schools in inverse proportion to the tonnes of non-recycled waste: the less a school pollutes, the more it receives.
The trick: share out in direct proportion to the reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$.
A photographer took a total of $180$ photos during a trip. She wants to organise her album by dividing the photos directly proportional to the days she spent in each…
A photographer took a total of $180$ photos during a trip. She wants to organise her album by dividing the photos directly proportional to the days she spent in each city: $4$ days in Seville, $5$ days in Granada and $9$ days in Córdoba. How many photos will she put in each city's section?
Three siblings help with household chores and their grandfather decides to give them a reward of $114\ €$ for their savings. However, to encourage responsibility, he…
Three siblings help with household chores and their grandfather decides to give them a reward of $114\ €$ for their savings. However, to encourage responsibility, he shares the money inversely proportional to the number of times they forgot to tidy their room this month. Lucía forgot $2$ times, Jorge $3$ times and Marta $6$ times. How much money will each receive?
A sustainable courier company wants to reward its three best delivery riders from the bicycle section. The total prize is $1\,160$ € and will be shared out in inverse…
A sustainable courier company wants to reward its three best delivery riders from the bicycle section. The total prize is $1\,160$ € and will be shared out in inverse proportion to the number of late packages delivered during the last month. Courier $A$ had $2$ delays, courier $B$ had $3$, and courier $C$ had $4$. Calculate how much money each one receives.
Shares with an extra step
Many problems do not end with the sharing: first you have to deduct a cost, or afterwards you have to add a fixed amount to one of the shares.
Three friends from an e-sports club have won a regional tournament and are going to share out a $1\,900$ € prize. The distribution must be directly proportional to the…
Three friends from an e-sports club have won a regional tournament and are going to share out a $1\,900$ € prize. The distribution must be directly proportional to the points scored in the grand final. The first player got $150$ points, the second $120$ points, and the third $110$ points. However, before payout, the club deducts $5\%$ of the total prize for administration fees. What net amount will each of the three players receive after applying the discount and making the proportional distribution?
A grandfather wants to share a collection of $168$ antique coins among his three grandchildren in inverse proportion to their ages, which are $12$, $15$ and $20$ years…
A grandfather wants to share a collection of $168$ antique coins among his three grandchildren in inverse proportion to their ages, which are $12$, $15$ and $20$ years old. The youngest grandchild is so excited that the grandfather decides to give him, in addition, $10$ more coins taken from his own personal reserve. How many coins will each grandchild end up with?