# Ratio

## Understanding Ratios: The Relationship Between Quantities

When it comes to sharing a box of chocolates or cookies among siblings or friends, the concept of ratio can come in handy. In this article, we will delve into what ratio is and how it can help determine fair shares.

## Defining Ratio

Ratio is a comparison between two or more quantities, showing their relationship in terms of size. It helps us understand how much of one quantity is present in another. Ratios are particularly useful in scenarios where things are divided or shared among a group.

Ratios can be expressed in their simplest form or simplified by dividing them by the highest common factor. It's important to note that ratios can compare whole quantities or a part of a whole with the whole itself.

## Notation for Ratios

There are three ways to express ratios: number notation, word notation, and fraction notation.

### Number Notation

In number notation, ratios are written with numbers and a colon (:) or a slash (/) in between. For example: 3:4 or 3/4.

### Word Notation

Word notation uses the phrase "is to" to convey ratios. For example: 3 is to 4 or 5 is to 6 is to 1 or 2 is to 7 or 7 is to 2 is to 11 is to 15.

### Fraction Notation

Ratios involving only two quantities are expressed as fractions in fraction notation. For example: 3/4.

## The Formula for Ratios

The fundamental principle behind calculating ratios is division. Depending on the scenario, different formulas are used to determine the ratio.

### Ratio Between Two Whole Quantities

To find the ratio between two whole quantities, we divide the first quantity by the second. The first quantity is referred to as the antecedent, while the second is the consequent. For example: if the first quantity is 5 and the second is 7, the ratio would be 5:7.

#### Example:

Henderson and Robinson received 5 and 7 oranges, respectively. What is the ratio of oranges between them?

Solution: The ratio of oranges between Henderson and Robinson is 5:7.

### Ratio Between a Part and a Whole

To find the ratio between a part and a whole, we divide the part by the whole. It's important to note that sometimes the total quantities are given, while other times, we need to calculate it by finding the sum of the parts.

For instance: if m is a part of t, where t is the total of all quantities, the ratio of m to t can be expressed as m:t.

The ratio of m to the sum of the quantities m, n, and o is m:(m+n+o).

#### Example:

Out of 6 packs of sweets in a box, Doyle received 5. What is the ratio of Doyle's share to the total number of sweets in the box?

Solution: The ratio of Doyle's share to the total number of sweets in the box is 5:6.

#### Example:

A bag contains 3 black balls, 2 red balls, and 7 white balls. What is the ratio of white balls to all the balls in the bag?

Solution: First, we identify the ratio we are calculating - in this case, it is the ratio of white balls to all balls. Then, we find the total number of white balls (7) and the total number of balls in the bag (12). The ratio of white balls to all balls is 7:12.

## Ratio Scaling

Ratio scaling involves obtaining equivalent ratios by multiplying or dividing with constants. This allows us to increase or decrease the measurements of geometric shapes while maintaining the same ratio.

In the example below, the length of the rectangle is 4 units and the width is 2 units, so the ratio is 4:2 or 2:1.

<