Proportional Distributions

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?

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?

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?

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?

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?

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 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?

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 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?

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 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?