Next we are going to explain step by step how to make a direct rule of three and an inverse rule of three, with exercises solved step by step

Índice de Contenidos

- 1 Directly proportional values. Direct proportionality.
- 2 What is the direct rule of three and when is it used
- 3 How to make a direct rule of three step by step
- 4 Solved problems of direct rule of three
- 5 Inversely proportional values. Inverse proportionality.
- 6 What is the inverse three rule and when is it used
- 7 How to Make a Inverse Three Rule Step by Step
- 8 Resolved exercises of inverse rule of three

## Directly proportional values. Direct proportionality.

Two values are **directly proportiona**l when:

- As one value
**increases**, the other**increases in the same proportion**. - As one value
**decreases**, the other**decreases in the same proportion**.

Whenever this happens, we speak of **direct proportionality**.

The proportion with which the value increases or decreases is **constant.** This constant is called **direct** **proportionality ratio**.

Let’s look at some examples of direct proportionality:

- A car takes 1 hour to travel 100 km. If it is 2 hours, it will cover 200 km.
- Time and distance travelled are two
**directly proportional**magnitudes, because if time increases, the distance travelled increases and if time decreases, the distance travelled decreases.

- Time and distance travelled are two
- 1 kilo of lemons costs 1 euro. 3 kilos of lemons will cost 3 euros
- The weight of the lemons and the price are two directly proportional magnitudes, because if the weight increases the price increases and if the weight decreases, the price decreases.

## What is the direct rule of three and when is it used

The direct rule of three is a method for calculating an unknown value that is directly proportional to another value we know.

It is used when the magnitudes we are dealing with are directly proportional, that is, they have the following relationship:

- If one magnitude increases, the other also increases in the same proportion
- If one magnitude decreases, the other also decreases in the same proportion.

It is also used to change units (from meters to kilometers, from minutes to hours…) or to calculate percentages.

## How to make a direct rule of three step by step

If for a value A of one magnitude, we have a value B of the other magnitude, for a value of C of the first magnitude, the second magnitude will have a value of X:

What is the value of that X?

In a direct rule of three, the X is calculated by multiplying the two values that are in the diagonal where the X is not, divided by the value that is in the same diagonal as the X. To remind us, it is said that X is solved in cross:

The formula to calculate the x is:

## Solved problems of direct rule of three

To solve rule of three problems, we must always work with the same units between the two magnitudes. One of the difficulties that there can be is to pass everything to the same unit

**If 3 kilos of oranges cost $4.00, how many kilos of oranges can you buy with $32.00?**

For more kilos, more money, then you have to use a direct rule of three:

**A motorcycle travels 30 km in 15 minutes, how many kilometers will it travel in 2 hours?**

For more time, more distance, so you have to use a direct rule of three.

Here you have to pass all the time in minutes. To pass the hours to minutes we can use another rule of three direct :

Now let’s go with the problem:

**If 50% of an amount is 60, how much is 25% of that same amount? What is the amount?**

The less the percentage the less the amount, so you have to use a direct rule of three.

Let’s calculate 25%:

To calculate the amount, we must calculate 100%:

**A worker earns $60 in 1 day, how much will he earn in a month?**

The more days, the more money, then you have to use a direct rule of three.

We consider that a month has 30 days.

Now we continue with the inverse three rule. How is it done?

But first you have to understand that they are inversely proportional values.

## Inversely proportional values. Inverse proportionality.

Two values are **inversely proportional** when:

- As one value
**increases**, the other**decreases in the same proportion** - When one value
**decreases**, the other**increases in the same proportion**

Whenever this happens, **we speak of inverse proportionality**.

The proportion with which the value increases or decreases is **constant**. This constant is called **inverse proportionality ratio**.

Let’s look at some examples of inverse proportionality:

- 3 workers take 4 hours to open a ditch. If you want to open it in less time, you will need more workers.
- The number of workers and the time to open the ditch are two inversely proportional magnitudes, because if the number of workers increases, the time decreases and if the number of workers decreases, the time increases.

- A bus takes 1 hour to finish its journey at a speed of 80 km/h. If you increase the speed to 100 km/h, it will take less time.
- The time taken by the bus and the speed are two inversely proportional magnitudes, because if the speed increases, the time taken decreases and if the speed decreases, the time taken increases.

## What is the inverse three rule and when is it used

The inverse rule of three is a method for **calculating an unknown value** that is **inversely proportional** to another value we know.

It is used when the magnitudes we are dealing with are **inversely proportional**, that is, they have the following relationship:

- If one magnitude increases, the other decreases in the same proportion
- If one magnitude decreases, the other increases in the same proportion

## How to Make a Inverse Three Rule Step by Step

If for a value A of one magnitude, we have a value B of the other magnitude, for a value of C of the first magnitude, the second magnitude will have a value of X:

What is the value of that X?

In an inverse rule of three, X is calculated by multiplying the two values that are on the line where X is not, divided by the value that is on the same line as X. To remind us, it is said that X is solved in line (unlike the direct rule three which is a cross):

The formula to calculate the x is:

## Resolved exercises of inverse rule of three

To solve rule of three problems, we must always work with the **same units** between the two magnitudes. One of the difficulties that there can be is to pass everything to the same unit

**10 workers take 2 months to build a house. How many days would it take 15 workers?**

The more workers, the less time they will take, then you have to use an inverse rule of three.

First of all we have to pass the months to days, by means of a direct rule of three:

And now we work with the data of the problem in days, which is what they ask us:

**1 tap with a certain flow takes 30 minutes to fill a tank. How many minutes would it take to fill the tank with 3 taps with the same flow?**

The more taps (or more flow) the less time, then you have to use an inverse rule of three:

**A bus takes 1 hour to finish its journey at a speed of 80 km/h. If the speed increases to 100 km/h, how long will it take to finish the journey?**

The faster you speed, the less time you take, then you have to use an inverse rule of three:

If we want to know the time in minutes, we use a direct rule of three: