# Compound rule of three

In the field of **mathematics** , different procedures have been created with the sole objective of facilitating work, since they have turned out to be easier to apply and to be more useful, mainly thanks to their **efficiency** and the **simplicity** of the processes. Within this group of **mathematical processes** we can find the compound rule of three, which is based on the relationship of quantities.

## What is the compound rule of three?

The compound rule of three is a **mathematical ****formula** by means of which it is possible to solve problems in which the **sentences** are composed of several **simple rules of three** that are applied on different occasions.

- What does it consist of
- What is the compound rule of three for?
- Direct compound rule of three
- Inverse compound rule of three
- Mixed compound rule of three
- Other example exercises

## What does it consist of

The compound rule of three consists of the use of a formula to be able to **relate** three or more **quantities** in such a way that the relationships established between the different quantities that are known yield **accurate data** on the **quantity** that is unknown. It is part of the group of rules of three which are responsible for **simplifying** the resolution of various mathematical problems in which there is a **relationship of proportionality** .

## What is the compound rule of three for?

This rule is an operation that is used to calculate direct or inversely proportional **quantities** . It is a very simple way that serves to solve different **proportionality** problems that occur between **three values** of which the data is available and where there is also **an unknown,** what makes it possible is to establish a proportionality relationship between all the data.

## Direct compound rule of three

The direct compound rule of three is the rule that is established in the **quantities** that have a directly proportional relation with respect to the quantity of which the **values** are not known . In this aspect, it is important to remember that two magnitudes can be **directly proportional** at the moment in which one is increased, as this causes the other to also increase, or otherwise, if one decreases the other will also increase.

This rule is composed of several **rules of three simple** that are applied successively. In order to solve a direct compound rule of three, the values of the known quantities must first be placed in the first column of the rule. In the first two columns, values of A or B. can be placed. The **magnitude** over which one of the values is **unknown** should be placed in the last column. For instance:

A _{1} —— B _{1} ——- C _{1}

A _{2} —— B _{2} —— x

Then, you must **multiply** in **line** values having magnitudes we do know and the result will be placed in a single column as follows:

A _{1} * B _{1} —— C _{1}

A _{2} * B _{2} —— x

As a result, a **simple rule of three** is finally obtained where **X** will be equal to a **fraction** in which the numerator will be formed by the multiplication of the numbers that are on the diagonal opposite the x and the denominator will be in charge of forming the quantity that will be located on the same diagonal where the x will be

X = (A _{2} * B _{2} * C _{1} ) / (A _{1} * B _{1} )

#### An **example** of the rule application is shown below

In order to do manual work, 3 1.5-liter plastic bottles are filled with sand. Between all the bottles a weight of 7 kg is reached. How much will four two-liter bottles weigh then?

- The first step is to be able to
**identify the different magnitudes**that are part of the problem, which are the number of bottles, their capacity and the total weight of the bottles. Then, the relationship of proportionality that exists between the unknown magnitude with those that are known is established.

3 bottles ——- 1.5 liters ——- 7 kg

4 bottles ——- 2 liters ——- x kg

- The next step will be to
**multiply the values**found in the first two columns online so that the result is placed in a single column.

3 * 1.5 ——– 7 kg

4 * 2 ———- x kg

- The last step will be to
**calculate the x.**To do this, we proceed to multiply the values that are diagonally opposite to x.

X = (4 * 2 * 7) / (3 * 1.5) = 56 / 4.5 = 12.44 kg

- The last step will be to establish the
**answer**: in 4 bottles of 2 liters there will be a weight of 12.44 kg.

## Inverse compound rule of three

In this type of rule, the magnitudes have a **relationship** that is **inversely proportional** to the **magnitude of** which we do not know the value. In order to solve it, the known values must be placed in the first and second column and in the last column, the value that we know of C and the one that we do not know, which we will call x.

A _{1} —— B _{1} ——- C _{1}

A _{2} —— B _{2} —— x

The next step will be to **reverse the order of the values** that the magnitudes have, in other words, the values located at the bottom will be placed at the top and vice versa.

A _{2} —— B _{2} —— C _{1}

A _{1} —— B _{1} ——- x

Then an **online multiplication** of the values corresponding to the quantities that we know is made and the result remains in the final column.

A _{2} * B _{2} —— C _{1}

A _{1} * B1 —— x

Now we calculate the **fraction** in which the **numerator** will be formed by multiplying the quantities located on the **opposite diagonal** and the **denominator** will become the quantity located on the **same diagonal** as x.

X = (A _{1} * B _{1} * C _{1} ) / (A _{2} * B _{2} )

#### An example to better understand the process is the following:

3 painters take 15 days to paint a mural if they work 8 hours a day. How many days can painters take if they work 7 hours a day?

- Step one: identify the magnitudes
- In the first column the number of painters will be placed, in the second the daily hours and the last column will be where the number of days and the unknown quantity are placed.

3 painters ———- 8 hours per day ——- 15 days

5 painters ———- 7 hours a day ——- x days

- Then the magnitudes are inverted as follows

5 painters ———- 7 hours a day ——- 15 days

3 painters ——— 8 hours per day ——- x days

- Multiplication of the values of the first two columns is performed

5 * 7 ——- 15 days

3 * 8 ——- x days

- Finally, the x is calculated:

X = (3 * 8 * 15) / (5 * 7) = 10.28 days

## Mixed compound rule of three

To solve the **mixed compound** rule of three, the **values** of one of the known quantities must be placed in the **first column,** the **values** of the other known quantity in the second column, and the **known** and **unknown** values in the last column .

A _{1} ——– B _{1} ——– C _{1}

A _{2} ——– B _{2} ——– x

Then **the order of the values** is **reversed** :

A _{2} ——– B _{1} ——– C _{1}

A _{1} ——– B2 ——– x

Subsequently, the values of the known quantities are **multiplied online** :

A _{2} * B _{1} ——- C _{1}

A _{1} * B _{2} ——- x

Finally, **the x** is **calculated** in the same way as in the previous exercises:

X = (A _{1} * B _{1} * C _{1} ) / (A _{2} * B _{1} )

#### An example that we can carry out to make this operation clearer is the following

In order to cut the grass of a farm that measures 1500 m _{2} , it is necessary that 5 laborers work for 1 hour. How long can 4 laborers take to cut the grass of a plot that measures a total of 3000 m ^{2} ?

5 pawns ——– 1500 m _{2} ——– 1 hour

4 laborers ——– 3000 m _{2} ——– x hours

The values are then inverted:

4 pawns ——– 1500 m _{2} ——– 1 hour

5 pawns ——- 3000 m _{2} ——– x hours

Then the values are multiplied online as follows:

4 * 1500 ——– 1 hour

5 * 3000 ——– x hours

The last step is to calculate the x:

X = (5 * 3000 * 1) / (4 * 1500) = 2.5 hours

## Other example exercises

As examples we mention the following:

**Five bakers make 60 loaves in 15 days. If you want to make 150 loaves in 25 days. How many bakers should be hired?**(5 bakers * 150 loaves * 25 days) / (60 loaves * 15 days) = 20,833. 21 bakers must be hired.**For taking 5 kg to a town that is 60 km away, a company has charged me $ 9. How much will it cost me to send an 8 kg package 200 km away?**(9 dollars * 8 kilos * 200 kilometers) / (5 kilos * 60 kilometers) = It will cost 48 dollars.**In 9 days four teachers, working 5 hours each day, have earned a total of $ 1200. How much will ten teachers earn, in 10 days, working 6 hours a day?**($ 1200 * 10 teachers * 60 work hours) / (4 teachers * 45 work hours) = They will earn $ 4000.