Ratio and Proportion: Concept, formula and Examples

 Ratio and proportion are fundamental concepts in mathematics that describe the relationship between quantities. Here are the formulas for ratio and proportion:

Ratio Formula

The ratio of two quantities \(a\) and \(b\) is represented as:

\[ \text{Ratio} = \frac{a}{b} \]

In this formula, \(a\) and \(b\) are the quantities you're comparing. The ratio expresses how many times one quantity is contained in another. Ratios can be written in different ways, such as in the form of fractions, with a colon (e.g., \(a:b\)), or as a decimal or percentage.

Proportion Formula

A proportion is an equation that states two ratios are equal. It's often written in the form:

\[a:b::c:d\]

This can be written as 

\[ \frac{a}{b} = \frac{c}{d} \]

In this formula, \(a\) and \(b\) form one ratio, and \(c\) and \(d\) form another ratio. The two ratios are equal to each other, meaning that the relationship between the quantities represented by \(a\) and \(b\) is the same as the relationship between the quantities represented by \(c\) and \(d\).

Cross-Multiplication Method

To solve proportions, you can use the cross-multiplication method. Given the proportion \(\frac{a}{b} = \frac{c}{d}\), you can cross-multiply and set the products equal to each other:

\[ ad = bc \]

This gives -  The product of Means = The product of Extreme.

This equation allows you to find the value of one unknown quantity when the values of the other three are known.

For example, if you have the proportion \(\frac{2}{x} = \frac{4}{6}\), you can cross-multiply:

\[ 2 \cdot 6 = x \cdot 4 \]

Solving for \(x\) gives \(x = 3\), which is the missing quantity in the proportion.

Some workout Exercises:

1. Check whether the below numbers are in proportion or not.

i). 12, 14 , 24, 28

ii). 2,8,5,7

iii). 123,124,246,248

iv). 34, 44 , 54, 64

2. Write in ratio form.

i). 1.5km to 600 meters

ii). 30 rupees to 500 Paisa

iii). 4 hours to 550 minutes

These formulas and methods are fundamental in solving a wide range of mathematical problems involving ratios and proportions. They find applications in various fields, including finance, science, and engineering.