The world of mathematics of numbers is vast and varied. When your child was introduced to numbers, their main focus was on understanding value and place. Then the focus shifted to internalising operations such as addition, subtraction, multiplication and division. From there they journeyed through the lands of factors, multiples and prime numbers. Now, they have arrived here, in the domain of fractions, decimals and ratios.

## The Foundational Nature of Ratio, Proportions and Percentages

## The Big Idea: What are Ratios, Proportions and Percentages?

These three topics are a natural extension of the concept of **fractions** and **decimals**. Therefore, make sure you check out those pages and thoroughly understand them before attempting to start on ratios.

## What is Ratio?

When 3 litres of yellow paint is mixed with 2 litres of red paint, the result is perfectly orange paint.

Will mixing 3 litres of yellow paint with 3 litres of red paint also result in perfectly orange (PO) paint? The answer is no.

If you want to get more orange paint, how much yellow paint and how much red paint must you mix? You will get orange paint when you have 2 portions of red paint and 3 portions of yellow paint.

This means orange paint is created when for every 3 litres of yellow paint, 2 litres of red paint are added. We say that yellow paint and red paint are in the ratio **3 is to 2**, written as **3:2**.

A ratio tells us how much of the proportion of one thing to the other.

The **sequence (or order)** in which the numbers in a ratio are written matters. For example, the ratio of the yellow paint to the red paint paint is 3:2 and not 2:3. Why? Because yellow paint will always be 1 portion more than the red paint.

Ratio of 3 quantities can be expressed as,

If \(a:b = 2:3\) and \(b:c = 3:5,\) then \(a : b : c = 2 : 3 : 5\)

### Equivalent ratios and the simplest form of a ratio

Nina's fruit-basket has 20 apples and 15 oranges in it.

\(\begin{align} \text{The number of apples : the number of oranges} = 20 : 15 = 4 : 3\end{align}\)

Because of this equality, we say that the ratio \(20 :15\) is **equivalent to** the ratio \(4 : 3.\) The ratio \(4 : 3\) is a ratio in its **simplest** or **lowest form**. A ratio is in its lowest form when the two numbers in the ratio **have no factors in common**.

### Comparison of ratios

### Division of a quantity in the given ratio

Let’s try and understand this using an example.

Example - Sita has 200. She splits the amount between two of her friends, Mia and Isha, in the ratio \(3 : 7.\)

How much money does Mia receive?

\(\begin{align} \text{Mia’s share} = {3\over 3 + 7} \text{ of } 200 = {3\over 10} \times 200 = 60\end{align}\)

How much money does Isha receive?

\(\begin{align} \text{Isha’s share} = {7\over 3 + 7} \text{ of } 200 = {7\over 10} \times 200 = 140\end{align}\)

In the example given we calculated Mia’a and Sita’s share from the total.

## Tips and Tricks

- Ratios are best thought of as equivalent fractions. Using the trick of finding equivalent fractions is the best way to find equivalent ratios.

E.g. To get a particular shade of paint the two colours need to be mixed in the ratio \(3:4.\) If I have 5 litres of the first paint, how much of the second will I need?

Here we have to find a fraction equivalent to \(\begin{align} 3\over4 \end{align}\) but with numerator 5.

One way to do this… multiply by 10 and then divide by 6.

\(\begin{align} {3\over 4} = {30 \over 40} = {5\over 6.67} \end{align}\)

So I’ll need 6.67 litres of the second paint.

Build your intuition this way so that you follow the correct procedure.

## Test your knowledge

Q1. Answer the following questions

- In a class, the ratio of the number of boys to the number of girls is \(3 : 4.\) If the class has 24 boys, what is the number of girls?
- In a bus, the ratio of seats allotted to men and women are in the ratio \(7 : 5.\) If the bus has 60 seats, how many women can sit on the bus?