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, Alba, Beto and Caio, collaborated at a charity market selling crafts. They decide to share the $\text{RD\$}\ 300$ 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.

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 organize her album by dividing the photos directly proportional to the days she spent in each city: $4$ days in Santo Domingo, $5$ days in Santiago, and $9$ days in Puerto Plata. 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 $\text{RD\$}\ 770$ 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 $\text{RD\$}\ 27\,000$ 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, Julia and Marcos, 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: Julia failed $4$ subjects and Marcos failed $6$. How many hours of game time does each get?

Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rental is $\text{RD\$}\ 153$. 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 $\text{RD\$}\ 129\,600$ 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 $\text{RD\$}\ 6\,840$ 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 small graphic design studio has made a profit of $\text{RD\$}\ 324\,000$ 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 centimeters) from the center 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, Sara, Luis and Alba, set up a lemonade stand at the neighbourhood festival. To start, Sara contributed $\text{RD\$}\ 600$, Luis $\text{RD\$}\ 900$ and Alba $\text{RD\$}\ 1\,500$. At the end of the day, they made a net profit of $\text{RD\$}\ 9\,600$. 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?

Three friends from an e-sports club have won a regional tournament and are going to share out a $1\,900$ pesos 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 $1\,160$ pesos 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 $90\,000$ pesos. They decide to share the expense in direct proportion to the number of nights that each family stayed in the house. The Sánchez family stayed $5$ nights, the Pérez family stayed twice as many nights as the García family, and the García family stayed $3$ nights fewer than the Sánchez 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 $1\,200$ pesos for having $300\,000$ streams. If artist $Y$ had $450\,000$ streams and artist $Z$ received $2\,400$ pesos, 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 $110\,000$ pesos 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 $60\,000$ pesos, how much money does each of the other two charities receive?

In a recycling competition, a municipality distributes a fund of $156\,000$ pesos among three schools in inverse proportion to the number of tons of non-recycled waste they generated. School $1$ generated $\frac{1}{2}$ ton, School $2$ generated $\frac{1}{3}$ of a ton, and School $3$ generated $\frac{1}{4}$ of a ton. What cash prize does each school receive?

A rectangular plot of $1\,200$ $m^2$ is divided into three school vegetable gardens for 9th grade classes. The area is divided in direct proportion to the number of students in each class. Class $A$ has $24$ students, class $B$ has $28$ students, and class $C$ has a number of students such that it received $450$ $m^2$ of garden.

1. How many students does class $C$ have?

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