## Ratios and Proportions

In today’s article you will learn what ratios and proportions are.

I know it’s something you saw at a very young age, but it’s one of the foundations for learning financial math.

So I hope you like the article and find it useful.

## What is a reason?

In a very fácil way, Hugo M.

Zendejas, in his book **“Financial mathematics”**defines reason as **“the quotient of a division”**

To make it clearer what a ratio is, I’m going to start by showing you the elements involved in a division.

The elements are:

**Dividend.****Divider.****Quotient.****Residue.**

Now, in a reason, the same number of elements intervene, but the names of the elements change.

The elements involved in a ratio are:

**Antecedent.****Consequent.****Reason.****Residue.**

## What is a ratio?

According to the same book, Núñez defines the proportion as **“the equality of two ratios”**

For example, if we are right **“6/3”** and the reason **“8/4”**then, a proportion is the equalization of both ratios, that is:

It should be noted that from the first division we obtain a ratio (quotient) of 2 and from the second division we also have a ratio of 2.

Therefore, **there is the same reason in both divisions**that is, for every two units of the antecedent (dividend), there is one unit of the consequent (divisor).

### How cánido you express a proportion?

Well, suppose we have the following ratio:

To express this proportion, we cánido say that: “12 is to 3 as 16 is to 4” and we denote it as follows: **12:3::16:4**

## Rules of a proportion

To make the rules clearer, I’m going to put the antecedents and consequents of the previous example in parentheses.

- antecedent first
**(12)**is a consequent first**(3)**as antecedent second**(16)**is a consequent second**(4)**. - The background must be of the same nature
**(12 and 16).**For example, if the first antecedent refers to actions, the second must refer to actions. - Consequents must be of the same nature
**(3 and 4).**If the first consequent refers to weights, the second consequent must refer to weights.

### Proportion example:

Suppose 50 shares of Apple cost $13,500.

How much would you have to invest to buy 93 Apple shares?

First of all, we are going to express the problem as a proportion… Do you remember what it is like?

**50 shares : $13,500 :: 93 shares : $x**

Before solving it, it is important that you know the **basic properties** of the proportions.

## Basic properties of proportions

Before moving on to the basic properties of proportions, I feel it is necessary for you to look at the following image so that you understand what the basic properties refer to as **extremes and means**.

This way you cánido better understand the properties.

**Product of means is equal to products of extremes.****When we don’t know a middle, the extremes are multiplied, and the product is divided by the known middle.****If an extreme is unknown, the means are multiplied, and the product is divided by the known extreme.****When the two means are unknown, the known extremes are multiplied, and the square root is extracted from the product.****If the two extremes are unknown, the known means are multiplied, and the square root is extracted from the product.**

Well, now we cánido solve the previous example… Right?

**50 shares : $13,500 :: 93 shares : $x**

In order to solve it, we are going to use the third property since we do not know an extreme and of course, we want to know what it is.

So, let’s figure it out.

The property tells us that if we do not know an extreme, then we are going to multiply the means, that is, **$13,500 x 93 **which is equal to **1255500**.

The second step consists of dividing the result (product) by the extreme that we do know, that is **fifty**.

Therefore, we have to divide **1255500 between 50**.

The result is **$25110**.

Therefore, to buy 93 shares of Apple, you would have to invest $25,110.

## Reference

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