## Overview of Ratio and Proportion

Children find it difficult to learn ratios, but going through this article will make the process of understanding ratios much simpler. A ratio is an ordered pair of numbers a and b, written as, $\dfrac{a}{b}$, where b is not equal to 0. The equation in which two ratios are set equal to each other is called a proportion. You will learn in this article, about the ratio, the parts of a ratio, and the way of solving, when a string is cut into three parts, problems. Now, let's get started.

## What is a Ratio?

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity. For Example, the ratio of boys to girls in a class is 12: 15.

We use the ratio formula while comparing the relationship between two numbers or quantities. The general form of representing a ratio of between two quantities say 'a' and 'b' is a: b, which is read as 'a is to b'.

The fraction form that represents this ratio is $\dfrac{a}{b}$. To further simplify a ratio, we follow the same procedure that we use for simplifying a fraction. a:b = $\dfrac{a}{b}$.

## Parts of a Ratio

Based on the characteristics of the ratio, it can be classified into two parts:

Part-to-Part Ratio

The part-to-part ratio denotes how two distinct entities or groups are related to each other. For example, the ratio of boys to girls in a class is 12:15, which can be written as 4:5.

Part-to-Whole Ratio

The part-to-whole ratio is used to denote the relationship between a specific group and a whole quantity. An example of a part-to-whole ratio is; If there are 8 females in a class of 20 students the ratio of girls to students can be expressed as 8:20 (females to students).

## How to Divide a Number in Ratio?

The way how to divide a number in the ratio is given below:

Add all the ratios together.

Divide the quantity by the total sum of the parts.

Multiply the given value by each part in the ratio.

## Divide a Number into Three Parts in a Given Ratio

To divide a number into three parts in a given ratio, follow the given steps:

Add the three parts of the ratio together.

Divide the ratios by the sum of these three parts.

Multiply the total value by each of the three parts in the ratio.

Example: Divide 99 into three parts in the ratio 2 : 4 : 5.

Solution: Given ratio is 2 : 4 : 5

Since 2 + 4 + 5 = 11.

Therefore, the first part = $\dfrac{2}{11} \times 99 = 18$.

Second part = $\dfrac{4}{11} \times 99 = 36$.

And, the third part = $\dfrac{5}{11} \times 99 = 45$.

Thus, the division of a number into three parts in a given ratio is 18, 36, and 45.

Example: When a string is divided into three segments, the given ratios are obtained: $\dfrac{1}{2}$ is to $\dfrac{1}{3}$ is to $\dfrac{5}{12}$. Which is the longest part?

Ans: According to the given ratios, the first piece has the longest length, which is calculated as

$\dfrac{\left(\dfrac{1}{2}\right)}{\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{5}{12}\right)}$

= $\dfrac{\left(\dfrac{1}{2}\right)}{\left(\dfrac{6}{12}+\dfrac{4}{12}+\dfrac{5}{12}\right)}$

= $\dfrac{\left(\dfrac{1}{2}\right)}{\left(\dfrac{15}{12}\right)}$

= $\dfrac{6}{15}$

= $\dfrac{2}{5}$

So, the longest part obtained, when a string is cut into three parts, is $\dfrac{2}{5}$th of the total length of the string.

## Solved Examples

Q 1. Six years ago, the ratio of the ages of Kunal and Sagar was $6: 5$. Four years hence, the ratio of their ages will be $11: 10$. What is Sagar's age at present?

Ans: Let the ages of Kunal and Sugar 6 years ago be $6 x$ and $5 x$ years, respectively.

Then, $\dfrac{(6 x+6)+4}{(5 x+6)+4}=\dfrac{11}{10}$

$\Rightarrow 10(6 \mathrm{x}+10)=11(5 \mathrm{x}+10)$

$\Rightarrow 5 \mathrm{x}=10$

$\Rightarrow \mathrm{x}=1$

$\therefore$ Sagar's present age $=(5 x+6)=16$ years.

Q 2. Arrange the following ratios in decreasing order.

$2: 3,3: 4,5: 6$ and $1: 5$

Ans: Given ratios are $\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{5}{6}, \dfrac{1}{5}$

The L.C.M of $3,4,6,5$ is $2 \times 2 \times 3 \times 5=60$

Now, $\dfrac{2}{3}=\dfrac{2 \times 20}{3 \times 20}=\dfrac{40}{60}$

$\dfrac{3}{4}=\dfrac{3 \times 15}{4 \times 15}=\dfrac{45}{60}$

$\dfrac{5}{6}=\dfrac{5 \times 10}{6 \times 10}=\dfrac{50}{60}$

$\dfrac{1}{5}=\dfrac{1 \times 12}{5 \times 12}=\dfrac{12}{60}$

Clearly, $50 / 60>45 / 60>40 / 60>12 / 60$

Therefore, $\dfrac{5}{6}>\dfrac{3}{4}>\dfrac{2}{3}>\dfrac{1}{5}$

So, $5: 6>3: 4>2: 3>1: 5$

## Practice Questions

Q 1. Radha wants to mix orange and melon juice at a ratio of 4:7. If Radha wants to make 22 litres of juice. How much orange juice will she need for the drink?

Ans: 8 litres.

Q 2. Two numbers are in a ratio of 3:4. If the sum of numbers is 63. Find the numbers?

Ans: Two numbers are 27 and 36.

Q 3. The length of a room is 6 metres and the breadth is 4 metres. Give the ratio of length to the breadth of the room.

Ans: $\dfrac{3}{2}$ or 3:2.

## Summary

In mathematics, a ratio is a comparison between two numbers of the same kind. Ratios are typically expressed either as fractions or by using colons (e.g. 6/8 or 34:100). So we have understood the concept of ratio and proportions, along with the explanation of the situation, when a string is cut into three parts. Some practice problems are given, that is to be solved by the students. If you find any difficulty related to the topic do ask in the comments below.

## FAQs on Parts of a Ratio

1. What is the unit of a ratio?

There are no units used to express the ratio.

2. What is the first term in a ratio called?

The first term of a ratio is called the antecedent.

3. Why is proportion important in our daily life?

The proportion has a wide application in one's daily life, especially in cooking. It helps us to determine the quantity of a substance needed to prepare a dish. It helps in increasing or decreasing the ratio.