Sharing in a Given Ratio
Three partners have invested different amounts of capital in a business. When sharing out the profit, it is not enough to divide by three: it has to be done fairly, according to what each one has put in. That is a proportional sharing, and here you will meet it in its direct and inverse forms, and even with two criteria at once.
Direct and inverse sharing
Divide the amount of $450$ € in direct proportion to the numbers $2$, $3$ and $4$.
Divide the amount of $450$ € in direct proportion to the numbers $2$, $3$ and $4$.
Divide the number $300$ into parts inversely proportional to $2$ and $3$.
Divide the number $300$ into parts inversely proportional to $2$ and $3$.
Divide $330$ points in inverse proportion to $1$, $2$ and $3$.
Divide $330$ points in inverse proportion to $1$, $2$ and $3$.
Sharing with fractions or decimals
When the parts are given as fractions, it is a good idea to rewrite them over a common denominator before adding: the sharing does not change if you work with the resulting numerators.
Share $620$ shares in direct proportion to the fractions $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{5}$.
Share $620$ shares in direct proportion to the fractions $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{5}$.
In a laboratory experiment, a $145$-gram sample of a chemical compound must be divided among three containers in inverse proportion to their temperatures. The…
In a laboratory experiment, a $145$-gram sample of a chemical compound must be divided among three containers in inverse proportion to their temperatures. The temperatures are $\frac{1}{2}$ ºC, $\frac{2}{3}$ ºC and $\frac{3}{4}$ ºC. Calculate the exact amount of compound that will be placed in each container.
Two criteria combined
Sometimes the sharing depends on two different things: one directly proportional and one inversely proportional, or even a sharing in two stages with different criteria.
A productivity bonus of $2530$ € is shared among three factory employees. The distribution is carried out in direct proportion to their years of service in the company…
A productivity bonus of $2530$ € is shared among three factory employees. The distribution is carried out in direct proportion to their years of service in the company ($4$, $5$ and $8$ years) and in inverse proportion to the number of days absent during the year ($2$, $3$ and $4$ days). Calculate the bonus received by each employee.
A technology company shares $50000$ € of profit among three departments in two phases. In the first phase, $40$% of the total is shared directly proportional to the…
A technology company shares $50000$ € of profit among three departments in two phases. In the first phase, $40$% of the total is shared directly proportional to the overtime hours worked: $120$, $150$ and $130$ hours. In the second phase, the remaining $60$% is shared inversely proportional to the number of technical incidents reported: $2$, $4$ and $5$ incidents. Calculate the total final profit received by each of the three departments by combining both phases.
The algebraic approach
In 4th ESO the sharing often comes with a hidden piece of data (one known share, an unknown total, or the quantities themselves expressed with an unknown): you have to set up an equation using the constant $k$.
In a distribution directly proportional to the quantities $3$, $5$ and $7$, the first person receives $120$ €. Calculate the total amount shared and what the other two…
In a distribution directly proportional to the quantities $3$, $5$ and $7$, the first person receives $120$ €. Calculate the total amount shared and what the other two people receive.
An inheritance is divided in direct proportion to the numbers $4$, $7$ and $9$. Knowing that the heir who receives the largest share gets $3500$ € more than the one who…
An inheritance is divided in direct proportion to the numbers $4$, $7$ and $9$. Knowing that the heir who receives the largest share gets $3500$ € more than the one who receives the smallest share, calculate the total value of the inheritance and the amount assigned to each of the three shares.
A total amount of $74$ is divided inversely proportional to three consecutive positive integers: $x$, $x+1$, and $x+2$. Knowing that the constant of inverse…
A total amount of $74$ is divided inversely proportional to three consecutive positive integers: $x$, $x+1$, and $x+2$. Knowing that the constant of inverse proportionality of the division is $k = 120$:
- a) Set up the corresponding equation and find the three integers.
- b) Determine the value of each of the three shares obtained in the division.
Back to the start: To share the profit fairly among the three partners, make a sharing directly proportional to the capital each contributed: each one receives $k \cdot c_i$, with $k = \dfrac{\text{beneficio}}{\text{suma de capitales}}}.