Fractions

Fractions are perhaps the most fundamental concept in mathematics, serving as the connecting link between whole numbers and decimals. Fractions enable us to represent part of a whole, compare amounts, and make precise calculations in day-to-day life. In this chapter, we will dissect the basics of fractions and discuss their various forms, operations, and applications in real life to provide you with a deeper understanding of the topic.

1.0What is a Fraction?

A fraction represents a part of a whole or a ratio between any two numbers. Simply put, a fraction tells us about how many parts of something we have in relation to the total number of parts of that thing. A fraction is represented using a horizontal line between the two numbers, like this: 

$\text{ Fraction }=\frac{\text{ Numerator }}{\text{ Denominator }}$

Here, 

• The top number, or the numerator, tells us how many parts are being considered or taken.
• The bottom number, or the denominator, tells us how many equal parts the total is divided into.

2.0Types of Fractions 

Fractions are classified into several types based on the values of numerators and denominators. Here are some major types of fractions: 

1. Proper Fraction: A proper fraction is identified as the numerator being smaller than the denominator of the fraction. This results in the value of the fraction being less than 1. Examples of proper fractions are $\frac{3}{5},\frac{2}{7}\text{, and }\frac{6}{11}.$
2. Improper Fraction: The numerator of the improper fraction is equal to or greater than the denominator, resulting in the value of the simplified fraction equal to or greater than 1. For example $\frac{7}{4},\frac{9}{9},\text{ and }\frac{12}{5}.$
3. Mixed Fractions: Mixed fractions are the combination of a whole number and a proper fraction. It is a way to represent quantities greater than one in practical situations. Examples of mixed fractions include $1\frac{1}{2},3\frac{2}{3},\text{ and }7\frac{5}{8}.$
4. Like Fractions: Two or more fractions with the same denominator are known as like fractions. These types of fractions make it easier to compare, add, or subtract fractions. Examples of like fractions include $\frac{2}{9},\frac{5}{9},\text{ and }\frac{7}{9}.$
5. Unlike Fractions: Fractions with different values of denominators are known as unlike fractions. For performing operations on such fractions, conversion of like denominators is required. For example $\frac{1}{4},\frac{3}{7},\text{ and }\frac{2}{5}.$
6. Equivalent Fractions: Equivalent fractions refer to different fractions that represent the same value or proportion after simplification. These fractions are formed by either multiplying or dividing both the numerator and the denominator by the same non-zero number. Examples of equivalent fractions include $\frac{1}{2}=\frac{2}{4}=\frac{4}{8}.$
7. Unit Fraction: It is a special type of fraction where the numerator always remains 1, while the denominator is a positive whole number. Generally, it is the one equal part of a whole that divides the whole into several parts, like this: 

$\text{ Unit Fraction }=\frac{1}{n}\text{, where }n\in N,n\ge 1$

For Example: 

• ½ → One out of two equal parts.
• ¼ ​→ One out of four equal parts.

3.0Simplifying Fractions: 

To simplify a fraction, divide the numerator as well as the denominator by their highest common factor (HCF). For this, simply find the HCF of the numerator and the denominator and then divide both by the HCF, like in this example: 

$\Rightarrow \frac{18}{24}=\frac{3}{4}\text{ (Here, HCF of }18\text{ and }24\text{ is }6\text{ ). }$

4.0Operations on Fractions: 

Like any other mathematical quantities, fractions can also be added, subtracted, multiplied, or divided. These operations can be performed as:  

1. Addition/Subtraction: To add or subtract any two or more fractions, equalise the denominators of the fractions using the LCM method, and then add or subtract numerators, as per the requirement of the question. For example: 

$\begin{array}{rl}& \frac{2}{5}+\frac{1}{2}=\frac{4+5}{10}=\frac{9}{10}\\ & \frac{3}{4}-\frac{1}{2}=\frac{3-2}{4}=\frac{1}{4}\end{array}$

2. Multiplication: For multiplying fractions, simply multiply the numerator and denominator of one fraction by the numerator and denominator of another fraction, like this: 

$\frac{2}{3}\times \frac{4}{5}=\frac{8}{15}$

3. Division: To divide one fraction by another fraction, multiply the first fraction by the reciprocal of the second one. This is how it is done: 

$\frac{3}{4}÷\frac{2}{5}=\frac{3}{4}\times \frac{5}{2}=\frac{15}{8}$

4. Comparing Fractions: Comparing fractions helps identify the smallest fraction among different fractions. For like denominators, simply compare the numerators directly. However, for unlike denominators, first equalise the denominators or use cross-multiplication or decimal values, then compare the numerators of the fractions. For example: 

$\frac{2}{5}=0.4,\frac{3}{4}=0.75\Rightarrow \frac{3}{4}>\frac{2}{5}$

5.0Word Problems on Fractions

Problem 1: A watermelon is cut into 10 equal pieces. If Riya eats four pieces, what fraction has she eaten?

Solution: According to the question

Total number of pieces in the watermelon = 10

Watermelon pieces eaten by Riya = 4

Hence, the fraction eaten by her $=\frac{4}{10}\text{ or }\frac{2}{5}$

Problem 2: A container is $\frac{2}{3}$​ full. If its total capacity is 90 litres, how much water is inside?

Solution: According to the question

Total Capacity of the container = 90 litres

Fraction of the container that is filled with water = ⅔ 

Quantity of water = $90\times \frac{2}{3}=60\text{ litres }$

Problem 3: Rahul has a rope that is 18 meters long. He cuts ⅔ of it. How many meters did he cut?

Solution: According to the question

The length of the rope = 18 meters 

Fraction of the rope Rahul cut = ⅔ 

Length of the rope cut by Rahul = $\frac{2}{3}\times 18=12\text{ meters }$