Maharashtra Board Class 9 Sample Papers

March 31, 202339 Insightful Publications

**Introduction to Ratios** : If you have observed your mom while preparing your favourite cake, you may know that she uses the ingredients in specific proportions. She needs 1 part egg, 1 part sugar and 1 part flour. This means the ingredients are taken in 1:1:1 ratio. This is the easiest example that you could think of for an introduction to ratios.

Similarly, if you love more strawberry pieces in your fruit cake than pineapples, you can use twice the amount of strawberries. When you say there are thrice the number of girls in your class than boys, you get a sense of ‘ratio’ of boys v/s girls. These are just a few examples of ratios that we come across in our daily lives. Let us learn more about the definition of ratios, with some examples, in this article.

A ratio can be defined as the relationship between two quantities that are similar, obtained when we divide one quantity by another. To calculate a ratio, the unit of measurement for both quantities must be the same. In the example about the number of boys and girls in a class, the number can be expressed as a ratio because we use numbers (like 30, 50 and so on) to express the number of both, boys and girls.

Let us understand two important points you need to remember while comparing two or more quantities.

- All the quantities that are being compared must be of the same kind. Comparing the length of a table to the weight of a television has no meaning.
- Three quantities 20 kg, 5 kg and 1000 g can be compared only when gram (g) is converted to kg.

We represent a ratio between two quantities using a colon (:) symbol. 3: 5 , 7: 6, 9: 3 are some of the examples of ratios. As you can see, the first number (before the colon) can be either smaller or larger than the second number (after the column).

Before we see some examples of ratios, let us get familiar with terms used in a ratio.

**Example 1**

In the ratio 9 : 3, number 9 is called the **antecedent **and the number 3 is called the **consequent**.

**Example 2**

In the ratio 7:6, 7 is the antecedent and 6 is the consequent.

As mentioned at the beginning, a ratio is obtained when we divide one quantity by the other, provided the unit of measurement is the same. Although the individual quantities have units of measurement like kg, metres, ml and so on, the ratio does not have a unit. Have you ever wondered why? Let’s find out.

Let us understand this with the help of an example. In finding out the ratio of 10 kg of rice to 15 kg 15 kg of wheat, we write this as 10 kg / 15 kg.

After simplifying, this can be represented as 2 kg / 3 kg.

When we are writing the same ratio, it can be written as 2:3. The unit ‘kg’ disappears as the unit in the numerator and denominator cancel out and the final answer, which is nothing but a ration, is without a unit.

A ratio has no unit and can be expressed as \(\frac{x}{y}.\) Since \(\frac{x}{y}\) is a fraction, it can have equivalent fractions like \(\frac{{\frac{x}{m}}}{{\frac{y}{m}}}\) and \(\frac{{mx}}{{my}},\) for any integer \(m,\) where \(m \ne 0.\)

When we equate all three fractions, we get,

\(\frac{{\frac{x}{m}}}{{\frac{y}{m}}} = \frac{{mx}}{{my}} = \frac{x}{y}\)

Rewriting the fractions using the ratio notation, we get,

\(\frac{x}{m}:\frac{y}{m} = mx:my = x:y,\) for any integer \(m,\) where \(m \ne 0.\)

We call \(\frac{x}{m} = \frac{y}{m}\) and \(mx:my\) as equivalent ratios of \(x:y,\) where \(x\) and \(y\) can be any rational numbers and \(y \ne 0.\)

The simplest form of a ratio \(x:y\) is where the terms \(x\) and \(y\) are integers and have no common factors other than \(1.\)

There are 4 types of ratios as listed below:

**Simple Ratio**: A ratio is called a simple ratio if both the terms are prime among themselves. For example: x : y or 4 : 5.**Inverse Ratio**: When the antecedent and consequent of a ration are interchanged, it results in an inverse ratio. For example, when 6:7 becomes 7:6 after interchanging the position of 6 and 7, it is called an inverse ratio.**Compound Ratio**: When product of antecedents and consequents- of two rations become the antecedent and consequent of a new ratio, the resulting ratio is called the compound ratio. For example: The mixed ratio of two ratios (a : b) and (c : d) will be (ac : bd).**Square Ratio**: If a ratio is mixed with the same to make a new ratio, then it is called square ratio. For example: Square Ratio of X : Y X × X : Y × Y i.e. X² : Y²

The *concept of ratios finds its application in business mathematics, finance and chemistry to name some. Hope you found this article on introduction of ratios very useful. For more such informative articles, follow Embibe.*