Fractions

Fractions are a fundamental concept in mathematics that play a crucial role in our daily lives. They allow us to represent and work with parts of a whole and help us solve various real-world problems. In this chapter, we'll dive into the world of fractions, understand what they are, how to work with them and why they are essential.

Fractions

A fraction is a numerical expression that represents a part of a whole. It is a ratio of two numbers. It consists of two parts: the numerator and the denominator. The numerator represents the number of equal parts we have and the denominator represents the total number of equal parts that make up a whole. Examples of fractions are as follows:

A fraction is expressed in the form of x/y, where y is a number that must not be equal to 0. In this representation, "x" is referred to as the numerator which signifies the number of equal parts we have and "y" is known as the denominator which signifies the total number of equal parts that make up the whole.

Example:

→ In the fraction113, "11" is the numerator (you have one part), and "3" is the denominator (the whole is divided into two equal parts).

Types of Fractions

Types of Fractions are as follows:

1. Unit Fractions: Unit fractions are fractions where the numerator is always 1.
   Examples include 1/2, 1/3, 1/4, 1/5, 1/9, 1/10, 1/11, 1/13, 1/17, etc.
2. Proper Fractions: Proper fractions are those in which the numerator is smaller than the denominator.
   Examples include 1/2, 2/3, 3/4, 4/5, 7/9, 11/13, 21/23, 51/57, 60/62, etc.
3. Improper Fractions: In contrast, improper fractions feature a numerator that is larger than the denominator.
   Examples include 3/2, 4/3, 5/4, 7/5, 9/8, 19/18, 39/35, 69/61, etc.
4. Mixed Numbers: Mixed numbers represent a combination of a whole number and a proper fraction.

Fractions on the Number Line

Much like whole numbers, we can also illustrate fractions on a number line. To do this, we follow a specific set of steps:

Step 1: Begin by dividing the number line between two numbers that enclose the given fraction. The division should create the same number of segments as the denominator of the fraction.

Step 2: Identify the portion on the number line that lies between the integer value just less than the fraction and the numerator part of the fraction. This particular section of the number line accurately represents the given fraction.

For visual clarity, consider the example below:

In the first number line, you can see a whole represented as one unit. The second number line is divided into two equal parts, each representing a half. Likewise, the third, fourth and fifth number lines are divided into 4, 8 and 16 equal parts, respectively to show fractions with different denominators. This method allows us to effectively visualize and locate fractions on a number line.

Fraction as a Part of Collection

Out of 15 balls, there are 4 green balls, 5 blue balls and 6 red balls.

We can express the fractions of each colour as follows:

→ The fraction of green balls is 4/15.
→ The fraction of blue balls is 5/15.
→ The fraction of red balls is 6/15.

These fractions represent the proportion of each coloured ball relative to the total number of balls which is 15.

Converting a Mixed Fraction into an Improper Fraction

To convert a mixed fraction into an improper fraction, you can achieve this by multiplying the denominator of the fractional part by the whole number and subsequently adding the resulting product to the numerator.

Conversion is shown as:

Converting an Improper Fraction into a Mixed Fraction

To change an improper fraction into a mixed fraction, perform division on the numerator by the denominator. This has both the quotient and the remainder. Following this, you can express the mixed fraction using the obtained quotient and remainder.

Mixed Fraction on the Number Line

Mixed fractions are represented on the number line by combining a whole number with a proper fraction.

Examples of mixed fractions on the number is shown as:

Equivalent Fractions

Equivalent fractions are fractions that have different numerators and denominators but have the same value or proportion of the whole.

Finding Equivalent Fractions

To find equivalent fractions for a given fraction, one can achieve this by multiplying or dividing both the numerator and denominator of the fraction by the same number.

Example: Equivalent fractions for a fraction 4/12 is:

Multiplying both the numerator and denominator of the fraction by the same number 5.

Multiplying both the numerator and denominator of the fraction by the same number 7.

412 = 4 × 712 × 7 = 2884

Dividing both the numerator and denominator of the fraction by the same number 2.

Test for Equivalent Fractions

For two equivalent fractions, the result of multiplying the numerator of the first fraction with the denominator of the second fraction is equivalent to the product of the denominator of the first fraction and the numerator of the second fraction.

→ Look for whether the fractions are equivalent or not.

Products are:

7 × 12 = 84

4 × 21 = 84

→ Look for whether the fractions are equivalent or not.

Products are:

7 × 13 = 91

4 × 20 = 80

Simplest/Lowest Form of a Fractions

A fraction is considered to be in its simplest (or lowest) form when its numerator and denominator share no common factors other than 1.

Example: As there are no common factors of 3 and 7 other than 1, the simplest form of ²⁴⁄₅₆ is ³⁄₇

Like and Unlike Fractions

Like Fractions: Fractions that share the same denominator are termed like fractions.

Unlike Fractions: Fractions that share different denominators are termed unlike fractions.

Unlike Fractions to Like Fraction

To convert unlike fractions into like fractions, it is necessary to find the Least Common Multiple (LCM) of their denominators. Then, you can multiply both the numerators and denominators of the fractions by specific numbers to make the denominators equal to the LCM of the given fractions' denominators.

Example: To convert unlike fractions ³⁄₅ and ⁷⁄₉ into like fractions, it is necessary to find the Least Common Multiple (LCM) of 5 and 9. LCM of 5 and 9 is 45.

Certainly, here's a simplified and rephrased version of the content suitable for a Class 6 audience:

Comparing Fractions

When we compare fractions, we need to consider whether they have the same or different denominators.

Comparing Like Fractions (Same Denominator)

When fractions have the same denominator, the one with the larger numerator is greater.

Comparing Unlike Fractions (Different Denominators)

To compare fractions with different denominators, follow these steps:

1. Find the least common multiple (LCM) of all the denominators.
2. Change each fraction so they all have the same denominator, which is the LCM.
3. Now that they have the same denominator, you can easily compare them.
4. Finally, give your answer in terms of the original fractions.

Comparing Unlike Fractions with Equal Numerators

If the numerators of two fractions are the same, the one with the larger denominator is actually smaller.

Adding and Subtracting Like Fractions

When you want to add or subtract fractions that have the same denominator, follow these steps:

Step 1: Add or subtract the numerators.
Step 2: Keep the common denominator the same.
Step 3: The result is the sum or difference of the numerators over the common denominator.

Adding and Subtracting Unlike Fractions

When you need to add or subtract fractions with different denominators, use these steps:

Step 1: Convert them into like fractions with the same denominator.
Step 2: Add or subtract their numerators while keeping the denominators the same.
Step 3: Simplify the result if necessary.

Adding and Subtracting Mixed Fractions

When you want to add or subtract two mixed fractions, follow these steps:

Step 1: Start by converting the mixed fractions into improper fractions.
Step 2: Perform the addition or subtraction with the improper fractions.
Step 3: Express the result as an equivalent mixed fraction, if necessary.

This way, you can work with mixed fractions more easily by first converting them to improper fractions, performing the operation, and then simplifying the result back to a mixed fraction if needed.

Examples:

Example 1: Which of the following number lines represent 4³⁄₅?

a)
b)
c)
d)

Answer: c)

Explanation: This number lines represent 4³⁄₅. 

Example 2: Henry leaves $36250 to his wife and four children such that two-fifths of this money is given to his wife and the remaining is distributed equally among the children. How much money does each child get?

a) $5437.5
b) $5467.5
c) $5473.5
d) $5743.5

Answer: a) $5437.5

Explanation: Total amount = $36250

Amount given to his wife = 2⁄5 of $36250
                                      = 2⁄5 × $36250
                                        = $14500

Remaining amount = $36250 − $14500 = $21750

This amount is distributed among four children equally.

Each's share = $21750 ÷ 4 = $5437.5