Exercises: Proportional division

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, Aoife, Seán and Ciara, collaborated at a charity market selling crafts. They decide to share the $\text{€}\ 300$ profit directly proportional to the number of hours each worked: Aoife worked $2$ hours, Seán $3$ hours and Ciara $5$ hours. Calculate how much money each receives.

In a video game competition, $100$ bonus points are shared between two players inversely proportional to the number of mistakes they made. If the first player made $2$ mistakes and the second made $4$ mistakes, how many points does each receive?

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 Galway, $5$ days in Cork and $9$ days in Dublin. How many photos will she put in each city's section?

A carpenter has a wooden strip of $240\text{ cm}$ and needs to cut it into three pieces whose lengths are proportional to the numbers $1$, $2$ and $3$. We call $k$ the length of the shortest piece, so that the other two measure $2k$ and $3k$. What will be the length of each piece?

A prize of $\text{€}\ 770$ is to be shared among the top three runners in a school race inversely proportional to the position obtained (1st, 2nd and 3rd place). How much money does each of the three winners receive?

A group of three YouTubers has received a bonus of $\text{€}\ 450$ for a joint video. They have decided to share the money directly proportional to the number of videos each uploaded to the channel that month: $4$, $5$ and $6$ videos respectively. How much money will each receive?

Two students, Aoife and Seán, have to share $10$ hours of games console time over the weekend. Since Julia got better grades, they decide to share the time inversely proportional to the number of failed subjects: Aoife failed $4$ subjects and Seán failed $6$. How many hours of game time does each get?

To share a prize of $180$ € among three sisters directly proportional to the hours they spent cleaning the garden ($2$ hours Lucía, $3$ hours Elena, and $4$ hours Marta), their older brother Pablo reasons as follows: "Since there are three people, I divide $180$ by $3$, which gives $60$ € per share. Then I multiply by their hours: Lucía gets $2 \times 60 = 120$ €, Elena $3 \times 60 = 180$ €, and Marta $4 \times 60 = 240$ €."

1. a) Explain in your own words what conceptual error Pablo made when calculating the value of "one share" ($k$).
2. b) Check the sum of the amounts obtained by Pablo to show why his solution is impossible in practice.
3. c) Correct the exercise by properly carrying out the directly proportional distribution and calculate how much money each sister actually receives.

Three 2nd Year classes from a secondary school take part in a charity food drive and have collected a total of $450\text{ kg}$. The school leadership will reward their effort by distributing $90$ cinema tickets directly proportional to the kilograms contributed by each class: 2nd Year A collected $100\text{ kg}$, 2nd Year B collected $150\text{ kg}$, and 2nd Year C collected $200\text{ kg}$.

1. a) Before calculating anything, estimate: will group 2nd Year C receive more or less than half of the available tickets? Justify your estimate by comparing the kilograms of 2nd Year C with the total collected.
2. b) Calculate the value of each share ($k$) and determine exactly how many tickets correspond to each classroom.
3. c) Check whether your initial estimate was correct and confirm that the sum of the three amounts matches the total number of tickets.

Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rental is $\text{€}\ 153$. 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?

An environmental association has received a grant of $\text{€}\ 2\ 160$ to reward three volunteers who participated in cleaning the beaches. The money will be shared directly proportional to the number of rubbish bags collected by each. If the first volunteer collected $24$ bags, the second $30$ bags and the third $36$ bags, calculate the amount of money due to each volunteer.

In a drone race, $310$ bonus points are shared among the top three finishers inversely proportional to the number of technical fouls they committed during the circuit. The first-place finisher committed $2$ fouls, the second committed $5$ fouls and the third committed $10$ fouls. How many points does each receive?

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. Aoife forgot $2$ times, Seán $3$ times and Niamh $6$ times. How much money will each receive?

A small graphic design studio has made a profit of $\text{€}\ 5\ 400$ from an advertising project. The director decides to divide the profit among their three employees directly proportional to their years of experience at the company. Employee A has $2$ years, employee B has $4$ years and employee C has $6$ years. Calculate the share of the profit that each worker receives.

In an archery competition, $135$ handcrafted arrows are shared among the top three archers inversely proportional to the distance (in centimetres) from the centre of the target on their last shot. Archer A was $3\ cm$ away, archer B $4\ cm$ away and archer C $6\ cm$ away. How many arrows does each get?

A group of three friends, Aoife, Cian and Niamh, set up a lemonade stand at the neighbourhood festival. To start, Aoife contributed €10, Cian €15 and Niamh €25. At the end of the day, they made a net profit of €160. If they decide to share the profits directly proportional to the money each put in initially, how much money will each of the three friends receive?

A box containing $120$ tickets for a music festival is to be shared among three local secondary schools. The distribution will be made inversely proportional to the number of students who arrived late on the last cultural day. School "A" had $10$ late arrivals, School "B" had $15$ late arrivals and School "C" had $30$ late arrivals. How many tickets will each school receive?

In a school robotics competition, $270$ points are distributed among the three finalist teams inversely proportional to the number of technical penalties incurred during the event: team Alfa incurred $3$ penalties, team Beta incurred $4$, and team Gamma incurred $6$.

To solve the distribution, two jury members propose different approaches:

• Method 1: Add the reciprocal fractions $\dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{6}$, divide the total points by that fractional sum to find the constant $k$, and multiply each fraction by $k$.

• Method 2: Multiply the reciprocals by the lowest common multiple of the denominators (LCM$(3, 4, 6) = 12$) to obtain equivalent integers ($4$, $3$, and $2$), and perform a directly proportional distribution on those integers.

1. a) Apply both methods step by step and verify that both yield exactly the same score for each team.
2. b) Justify which of the two procedures seems more convenient and reliable to you when working by hand in an exam, explaining what difficulties the other approach presents.

Three farmers from an irrigation community in the Valencian huerta share the electricity bill for the well pump that supplies their fields. In July, the total cost of electricity was $1\,500$ €. However, the community benefits from a regional subsidy for efficient water use that covers $20\%$ of the total amount before issuing charges. The resulting net amount must be shared directly proportional to the volume of water consumed by each farm:

• Vicente: $1\,200\text{ m}^3$

• Amparo: $1\,800\text{ m}^3$

• Carles: $3\,000\text{ m}^3$

1. a) Calculate the net amount remaining to be paid after applying the $20\%$ subsidy.
2. b) Find the price per cubic metre ($k$) for this distribution and determine how much each of the three farmers must pay.
3. c) Why must the $20\%$ subsidy be applied to the total before making the proportional distribution, instead of first dividing the $1\,500$ € and then subtracting one third of the aid from each person? Justify your answer based on the fairness of proportional distribution.

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 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.

Three families have rented a country house for the Easter holidays. The total cost of the rent is $1\,500$ €. They decide to share the expense in direct proportion to the number of nights that each family stayed in the house. The Murphy family stayed $5$ nights, the Walsh family stayed twice as many nights as the O'Brien family, and the O'Brien family stayed $3$ nights fewer than the Murphy family. How much must each family pay?

A music app has distributed its quarterly profits among three emerging artists in direct proportion to the number of streams of their tracks. We know that artist $X$ received $1\,200$ € for having $300\,000$ streams. If artist $Y$ had $450\,000$ streams and artist $Z$ received $2\,400$ €, calculate:

1. The number of streams artist $Z$ had.

2. The amount of money artist $Y$ received.

3. The total profit distributed by the app among the three.

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?

A donation of $110\,000$ € is shared among three charities in direct proportion to the places they offer. Charity $P$ has $x$ places, charity $Q$ has three times as many places as $P$, and charity $R$ has half as many places as $Q$. Knowing that charity $Q$ received $60\,000$ €, how much money does each of the other two charities receive?

In a recycling competition, a local council shares out a fund of $2\,600$ € among three schools in inverse proportion to the number of tonnes of non-recycled waste they generated. School $1$ generated $\frac{1}{2}$ tonne, School $2$ generated $\frac{1}{3}$ of a tonne, and School $3$ generated $\frac{1}{4}$ of a tonne. What cash prize does each school receive?

A rectangular plot of $1\,200$ $m^2$ is divided into three school gardens for 2nd Year classes. The division of the area is directly proportional to the number of students in each class. Class $A$ has $24$ students, class $B$ has $26$ students, and class $C$ has a number of students such that it received $450$ $m^2$ of garden.

1. How many students are there in class $C$?

2. What area corresponds to classes $A$ and $B$?

Three founding partners of a mobile app development startup decide to share the net profits from the first quarter directly proportional to the capital contributed by each when incorporating the company. It is known that Aoife contributed $12\,000$ € and that Seán contributed $18\,000$ €. At the accounting firm, the record of the capital contributed by Niamh has been misplaced, but the following settlement records are available:

1. Seán was allocated exactly $5\,400$ € in profit.

2. Niamh received $1\,800$ € more profit than Aoife.

1. a) Find the constant of direct proportionality $k$ (euros of profit per euro contributed).
2. b) Calculate how much profit corresponded to Aoife and how much to Niamh.
3. c) Reconstruct, with reasoning, the exact amount of capital that Niamh contributed at the inception of the company.
4. d) What was the total profit shared among the three partners? Verify that the sum of the three individual profits matches the product of the total capital and the constant $k$.

A town council allocates a budget item of $3\,700$ € to three neighbourhood associations for sustainable mobility projects. Under municipal regulations, each association first receives a fixed individual grant of $200$ € for secretarial and management expenses. The remaining money from the budget item is distributed inversely proportional to the number of resident complaints about noise and traffic accumulated by each neighbourhood during the year: Barrio Norte ($2$ complaints), Barrio Centro ($3$ complaints), and Barrio Sur ($5$ complaints).

1. a) Determine what amount of the municipal budget item is distributed through inversely proportional allocation and calculate the share corresponding to each neighbourhood for this concept.
2. b) Add the fixed secretarial grant and calculate the final total amount that each of the three associations will receive.
3. c) The spokesperson for Barrio Sur complains, stating: "We only have three more complaints than Barrio Norte, but they receive more than twice as much money as we do in the proportional distribution; this distribution is unfair." Refute her complaint mathematically: analyse the multiplicative relationship between the complaints ($5$ versus $2$) and explain how inverse proportionality operates in the allocation of shares.