Three friends from an e-sports club win a tournament and share a $\$1{,}900$ prize. In the final, the first player scored 150 points, the second 120 and the third 110. Is it fair to split the money equally? It seems fairer that the player with more points gets more money. That is a proportional share, and by the end of this page you will know exactly how much each player gets.
Direct Proportional Sharing
In a directly proportional share, the amount each person gets is proportional to a number that measures their effort or contribution: points, hours worked, money invested. More contribution means a bigger share.
The shares are in the same ratio as the numbers. Here the ratio is $150 : 120 : 110$. The money per point is the same for everyone. That constant is the constant of proportionality, also called the unit rate.
Change the total or the parts and watch how the bar is divided:
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Three siblings decide to share a bag of $120$ candies directly proportionally to their ages, which are $2$, $3$ and $7$ years old. How many candies does each one receive?
Three siblings decide to share a bag of $120$ candies directly proportionally to their ages, which are $2$, $3$ and $7$ years old. How many candies does each one receive?
Three friends, Ava, Ben and Cal, have collaborated in a charity market selling crafts. They decide to share the $300 \ \$…
Three friends, Ava, Ben and Cal, have collaborated in a charity market selling crafts. They decide to share the $300 \ \$$ profit directly proportionally to the number of hours each one has worked: Ava worked $2$ hours, Ben $3$ hours and Cal $5$ hours. Calculate how much money each one receives.
Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rent is $153\ \$…
Three friends decide to rent a private server to play an online video game during the holidays. The total cost of the rent is $153\ \$$. They have agreed to share the payment directly proportionally to the time each one has been connected: Alex was connected for $12$ hours, Bea for $18$ hours and Christian for $21$ hours. How much must each one contribute to pay the bill?
Now the opposite situation: the person with more of something gets less. For example, a prize is shared inversely proportionally to the number of mistakes: the fewer mistakes you make, the more you receive.
Here is the key idea. If Emma has half as many mistakes as Liam, she should get twice as much. So the shares are directly proportional to the reciprocals $\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c}$ of the numbers.
See how the bar changes when the numbers are inverse:
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In a video game competition, $100$ bonus points are shared between two players inversely proportionally to the number of mistakes they have made. If the first player has…
In a video game competition, $100$ bonus points are shared between two players inversely proportionally to the number of mistakes they have made. If the first player has made $2$ mistakes and the second has made $4$ mistakes, how many points does each one receive?
In a drone race, $310$ bonus points are shared among the top three finishers inversely proportionally to the number of technical faults they have committed during the…
In a drone race, $310$ bonus points are shared among the top three finishers inversely proportionally to the number of technical faults they have committed during the circuit. The first finisher had $2$ faults, the second had $5$ faults and the third had $10$ faults. How many points does each one receive?
A sustainable courier company wants to reward its three best delivery riders from the bicycle section. The total prize is $\$1\,160…
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 inversely proportionally to the number of packages that arrived late during the last month. Courier $A$ had $2$ late deliveries, courier $B$ had $3$ and courier $C$ had $4$. Work out how much money each one receives.
What if the numbers you are given are already fractions? Then the reciprocals are easy: the reciprocal of $\frac{1}{2}$ is $2$, of $\frac{1}{3}$ is $3$, and so on.
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In a recycling competition, a city council shares a fund of $\$2\,600…
In a recycling competition, a city council shares a fund of $\$2\,600$ among three schools inversely proportionally to the number of tons of unrecycled waste they have 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 financial prize does each school receive?
Many problems do not end with the share. Sometimes you must subtract a fee first, or add a bonus to one person afterward. The picture shows the idea for a fee.
Extra steps can also be mixed with an inverse share: do the inverse share first, then add the bonus to the right person. The same order rules apply.
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Three friends from an e-sports club have won a regional tournament and are going to share a prize of $\$1\,900…
Three friends from an e-sports club have won a regional tournament and are going to share a prize of $\$1\,900$. The share must be directly proportional to the points obtained in the grand final. The first player got $150$ points, the second $120$ points and the third $110$ points. However, before collecting, the club deducts $5\%$ of the total prize for administration fees. What net amount will each of the three players receive after applying the deduction and calculating the proportional shares?
A grandfather wants to share a collection of $165$ old coins among his three grandchildren inversely proportionally to their ages, which are $12, 15$ and $20$ years old.…
A grandfather wants to share a collection of $165$ old coins among his three grandchildren inversely proportionally 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 that he takes from his own personal reserve. How many coins will each grandchild end up with?
Sometimes the problem does not give you the total or all the numbers. Instead, you know one share and the number that goes with it. That is enough to find the constant of proportionality $k$ (the unit rate):
$$k = \frac{\text{a known share}}{\text{its number}}$$
Once you know $k$, you can find any missing share ($\text{share} = k \times \text{number}$) or any missing number ($\text{number} = \text{share} \div k$).
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A music app has shared its quarterly profits among three emerging artists directly proportionally to the number of streams of their songs. We know that artist $X$…
A music app has shared its quarterly profits among three emerging artists directly proportionally to the number of streams of their songs. 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:
The number of streams that artist $Z$ had.
The amount of money that artist $Y$ received.
The total profit that the app shared among the three.
A rectangular plot of $1\,200$ $m^2$ is divided into three school garden plots for the 7th grade classes. The share of the area is done directly proportionally to the…
A rectangular plot of $1\,200$ $m^2$ is divided into three school garden plots for the 7th grade classes. The share of the area is done directly proportionally to the number of students in each class. In class $A$ there are $24$ students, in class $B$ there are $28$ students and class $C$ has a number of students such that they have been allocated $450$ $m^2$ of garden plot.
Sometimes the numbers are described by how they compare, such as "twice as many" or "3 fewer". Work out each number first, then share as usual.
If the numbers are unknown but related, write them in terms of one unknown and turn them into a ratio of whole numbers. Then use a known share to find the value of one part.
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Three families have rented a cabin to spend Spring Break. The total cost of the rent is $\$1\,500…
Three families have rented a cabin to spend Spring Break. The total cost of the rent is $\$1\,500$. They decide to share the cost directly proportionally to the number of nights each family has stayed in the cabin. The Miller family stayed $5$ nights, the Johnson family stayed twice as many nights as the Smith family, and the Smith family stayed $3$ nights fewer than the Miller family. What amount must each family pay?
An inheritance of $\$112\,000$ is shared among three charities directly proportionally to the places they offer. Charity $P$ has $x$ places, charity $Q…
An inheritance of $\$112\,000$ is shared among three charities directly proportionally 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 $Q$ has received $\$60\,000$, how much money does each of the other two charities receive?
Back to the start: the e-sports players get $\$750$, $\$600$ and $\$550$, because each point is worth $\$5$. The player with the most points gets the most money, and the shares add up to $\$1{,}900$.