## Proportional Distribution

In today’s article I am going to teach you what direct and indirect proportional distribution is, likewise, I am going to solve some problems so that it is more clear to you.

## What is proportional distribution (proportional distribution)?

As it could not be otherwise, I am going to start with the theory, that is, I am going to give you the definition so that you understand the concept.

Zendejas Níñez Hugo gives us the following **definition** of proportional distribution (proportional distribution):

“Proportional distribution is the **operation** **whose purpose is to distribute a certain amount in proportional parts** to certain given factors or numbers called *distribution*«

## Elements used in every proportional distribution problem

The elements that you will always find in a proportional distribution problem are the following:

Partition ratios |
It is the amount that corresponds to each of the beneficiaries. |

cast index |
These are the factors that determine the allocation assigned to each of the beneficiaries. |

Amount to distribute |
It is the amount subject to distribution among the beneficiaries. |

## Proportionate Classification

### fácil direct

**It is the distribution in which a single aspecto intervenes.**.

Likewise, you have to consider that the greater the number of units indicated by the distribution index, **elderly** It will be the part that corresponds to it.

### direct compound

Well, unlike the fácil direct distribution, in the composite direct **Two or more factors are involved**.

Likewise, as in the fácil, the greater the number of units, the corresponding **elderly** amount of the distributed.

### fácil inverse

As in the fácil direct, in the fácil inverse proportional distribution, only **a single aspecto is involved**.

In this case, the greater the number of units of the distribution index, the **minor** It is the amount that corresponds to the beneficiary.

### compound inverse

As you must already be imagining, in the compound inverse proportional distribution **Two or more factors are involved**.

Furthermore, when the delivery index is higher, **minor** It will be the amount that corresponds to the beneficiary.

### Mixed

One or more directly proportional factors are involved, and one or more inversely proportional factors.

## Methods to solve proportional distribution problems

So that you cánido solve a proportional distribution problem there are three methods, which are:

**Reduction to unity.****By proportions.****By aliquot parts.**

### Unit reduction method

It consists of determining how much of the amount to be distributed corresponds to each unit of the distribution indices; It is obtained by dividing the amount to be distributed by the sum of the distribution indices, which gives rise to what is called **“Constant aspecto (Fc)”**

#### Example

Suppose that an inheritance of $150,000 pesos is going to be distributed among three brothers, in direct proportion to their ages.

The ages of the three brothers are as follows:

**Edward: 25 years old.****Mario: 15 years.****Gael: 10 years.**

The data we have is:

Amount to distribute: | $150,000 pesos |

Distribution indices: | 25, 15 and 10 |

Sum of delivery indices: | fifty |

Partition ratios: | X1, X2, X3 |

##### Solution by reduction to unity:

In order to lead to the solution, we first have to use the aforementioned elabora and thus obtain the constant aspecto:

The constant aspecto we just obtained indicates that **for each year that each of the brothers has, they will be awarded $3,000 pesos.** Therefore, to know how much money corresponds to each brother in total, we only have to multiply the number of years that each one has by 3,000.

X1: Edward | 25 x 3,000 = 75,000 |

X2: Mario | 15 x 3,000 = 45,000 |

X3: Gaël | 10 x 3,000 = 30,000 |

As you cánido see in the table above, if you add the amount that corresponds to each brother, it gives you the 150,000 pesos of the inheritance.

### method by proportions

As its name indicates, the method by proportions emplees the same concept of proportion to solve the problem, that is, it equates two ratios.

If you want to know more about ratios and proportions, I invite you to go to the following backlink:

#### Example

So that you perro see that the result is the same, we are going to solve the same problem, but with the method by proportions.

##### Solution:

To begin with, a total of 50 years (the sum of the ages), corresponds to a total of $150,000 pesos (amount to be distributed).

Now, we also know that a certain amount corresponds to each age….

TRUE? Well, now we are going to express the problem in proportions.

In it **first case** (Eduardo), is as follows:

**50 years : $150,000 : : 25 : X1**

Therefore, the solution of X1 is: (150,000 x 25) / 50 = **$75,000**

In it **second case** (Mario), is as follows:

**50 years : $150,000 : : 15 : X2**

Therefore, the solution of X2 is: (150,000 x 15) / 50 = **$45,000**

In the last case (Gael), it remains as follows:

**50 years : $150,000 : : 10 : X3**

Therefore, the solution of X3 is: (150,000 x 10) / 50 = **$30,000**

As you perro see, we arrive at the same answer as with the previous method

### Aliquot method

To solve a proportional distribution problem by this method, you have to find a part that is **submultiple of all delivery indices**.

#### Example

I’m going to use the same example again so you cánido see that you cánido indeed use any method you want to solve a problem.

First of all, we have to consider that the sum of the cast indices (years) is 50.

Now, Gael is 10 years old, which is one fifth of the sum of the cast indices.

Therefore, we perro conclude that Gael should receive a fifth of the amount to be distributed, that is, Gael gets 150,000 / 5 = **$30,000**

Now we are going to see the case of Mario.

Mario is 15 years old, which means that Mario is one and a half times as old as Gael.

Therefore, he must correspond to that same proportion, that is, 30,000 + 15,000 = **45,000**

Lastly, Eduardo is Mario’s age agregado Gael’s.

Therefore, the amount that he is going to touch is the sum of amas, that is, 30,000 agregado 45,000 = **75,000**

As you cánido see, the answers are the same.

Now, since we know what the cases are (classification) and the ways to solve them, I am going to give examples of each case (classification of proportional distribution)

## Proportional Sharing Worked Examples

### Example of fácil direct proportional distribution

In the following problem, you have to **determine the delivery index**

An investor bought a note for $150,000 and earned interest of $25,000.

How much does investor B invest who received interest in the same period $18,000?

#### Solution

To solve the problem, I am going to use the method by proportions.

Therefore, first of all I am going to express the problem as a proportion.

**Index : Quotient : : Index : Quotient**

**150,000 : 25,000 : : X : 18,000**

Therefore, the answer is the following:

**(150,000 X 18,000) / 25,000 = 108,000**

**Therefore, the investor had to purchase a note for $108,000**

### Composite Direct Proportional Sharing Example

In the following problem, you have to **determine partition ratios**

In the company Beta SA, the employees have equipo up an investment fund to which they have contributed each month:

Employee A | $4,000 for 1 year |

Employee B | $5,000 for 9 months |

Employee C | $4,000 for 6 months |

At the end of the year, the cómputo of this fund was $20,000.

How much does each employee get?

#### Solution

In this case I will use the method by reduction to unity, therefore, I will start with the following:

- A = $4,000 x 12 months = $48,000
- B = $5,000 x 9 months = $45,000
- C = $4,000 x 6 months = $24,000

If we add A, B and C, we get **$117,000**

Now, we are going to use the elabora of the method by reduction to the unit and we obtain:

FC = 20,000 / 117,000 = **0.17094**

Now we just have to multiply

- A = $48,000 x 0.17094 =
**$8205.12**. - B = $45,000 x 0.17094 =
**$7692.32** - C = $24,000 x 0.17094 =
**$4102.56**

If you add A, B, and C, you will see that it gives you the fund cómputo (20,000).

### Fácil Inverse Proportional Sharing Example

As you already know, in these cases, the share ratio is higher as the share index is lower.

Therefore, to solve these problems, what you have to do is take the inverses of the given numbers as the casting index, and since they have been reversed, you carry out the fácil direct casting procedure.

Don’t worry if you still don’t fully understand it, I’m going to give you an example to make it clear.

In the following problem, you have to **determine the partition quotient**

Mr.

Ramírez entered into an investment management contract and established that the benefit of the investment be distributed among his three children in inverse proportion to their ages.

The ages of Mr.

Ramírez’s children are as follows:

Son A | He is 20 years old |

Son B | He is 18 years old |

son C | He is 12 years old |

The benefits amounted to $320,000 per year.

#### Solution

In order to determine the indices, we will proceed as follows:

A) The original indices will be obtained

B) We are going to obtain their reciprocals

- A = 1/20.
- B = 1/18.
- C = 1/12

C) We are going to obtain a common denominator to be able to simplify them.

He **least common multiple** of 20, 18 and 12 is **180**.

Now we are going to convert the three indices that we have so that they have a common denominator (180).

- A = 9/180
- B = 10/180
- C = 15/180

So we have to:

Now what you have to do is add the indices that we obtained and we obtain that 9 + 10 + 15 = **3. 4**

Therefore, we perro now use the method by reduction to unity and we are going to obtain the Constant Aspecto.

The constant aspecto is equal to 320,000 / 34 = **9411.7647**

Therefore, all that remains is to multiply

- A = 9 x 9411.7647 =
**$84,705.8823** - B = 10 x 9411.7647 =
**$94,117,647** - C = 15 x 9411.7647 =
**$141,176.4705**

### Composite Inverse Proportional Sharing Example

As you already know, in the compound inverse proportional distribution, two or more factors are presented, which are multiplied to obtain their inverse, and where a common denominator is determined in order to convert them into units of equality, that is, in a fácil direct distribution. .

In the following problem, you have to **determine the partition quotient**

In a translation contest, a prize of $250,000 was distributed, in inverse proportion to the time it lasted and the number of mistakes made.

The contestants finished as follows:

Contestant A | 45 minutes and 8 fouls |

contestant B | 40 minutes and 12 fouls |

contestant C | 50 minutes and 6 fouls |

contestant D | 60 minutes and 2 faults |

#### Solution

Using the method by proportions.

Next I am going to put a table where I am going to summarize the steps until arriving at the definitive indices.

It should be noted that I am not going to put the whole procedure because it looks like the previous example.

original indices | product of indices | Reciprocals | Common denominator | definitive indices |

W = 45 x 8 | 360 | 1/360 | 7200 | twenty |

B = 40 x 12 | 480 | 1/480 | 7200 | fifteen |

C = 50 x 6 | 300 | 1/300 | 7200 | 24 |

D = 60 x 2 | 120 | 1/120 | 7200 | 60 |

Total | 119 |

Well, now we cánido express it as a proportion:

In the case of contestant A:

**119 : 250,000 : : 20 : A**

Therefore, the answer is: (250,000 x 20) / 119 = **$42,016.80672**

In the case of contestant B:

**119 : 250,000 : : 15 : B**

So the answer is: (250,000 x 15)/119 = **$31,512.60504**

In the case of contestant C:

**119:250,000::24:C**

Therefore, the answer is: (250,000 x 24) / 119 = **$50,420.16807**

Lastly, contestant D:

**119:250,000::60:D**

Therefore, the answer is: (250,000 x 60) / 119 = **$126050.4202**

If you add up what each contestant is going to receive, you will see that you get $250,000

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