Fractions Chapter Notes | Maths for Class 6 (Ganita Prakash) - New NCERT PDF Download

Introduction

Fractions are a way of representing parts of a whole. Imagine you have a pizza that is cut into equal slices. If you take one slice, you are taking a fraction of the whole pizza. Fractions help us understand and describe situations where something is divided into equal parts. They are used in everyday life, such as when sharing food, measuring ingredients in a recipe, or dividing time.
In this chapter, you will learn about different types of fractions, how to compare them, and how to add or subtract them. You will also explore how fractions are represented on a number line and discover the concept of equivalent fractions. 

Fractional Units and Equal Shares

A fractional unit is a part of a whole object or quantity that has been divided into equal parts. When we divide something into equal parts, each part is called a fractional unit.

Understanding with Example

Imagine you have a chocolate bar, and you want to share it equally with your friends. Let’s say you have 1 chocolate bar and 5 friends, including yourself, so you need to divide the chocolate into 5 equal pieces.

• Each piece of chocolate that you get is a fractional unit of the whole chocolate bar. In this case, since the chocolate is divided into 5 equal parts, each piece is 1/5 of the chocolate bar.
• Now, let’s consider another situation. You have the same 1 chocolate bar, but this time you need to share it with 9 friends.
• If you divide the chocolate bar into 9 equal parts, each piece will be 1/9 of the whole chocolate bar.

Comparison: Which piece is bigger, 1/5 or 1/9?
When you divide the chocolate into 5 parts, each piece is larger because fewer people are sharing it. So, 1/5  is bigger than 1/9.

This example shows that as the number of parts increases, each fractional unit (or piece) becomes smaller.

Knowledge from the Past: The History of Fractions in India

• Fractions have a long history in India, dating back to ancient times when they were used and named in various texts.
• In the Rig Veda, which is one of the oldest texts in Indian literature, fractions were referred to as "tripada."
• The term "tripada" has a similar meaning to how fractions are expressed in many Indian languages today. For example, in colloquial Hindi, people say "teen paav," and in Tamil, the word is "mukkaal."
• This shows that the words for fractions used in modern Indian languages have their roots in ancient times.
• Overall, the way fractions are understood and named in India today has a deep historical background.

Fractional Units as Parts of a Whole

A fractional unit is a single part of a whole object that has been divided into equal pieces. When we divide something into equal parts, each part is called a fractional unit of the whole.

Understanding with Example

Imagine you have a traditional sweet called "chikki," which is a rectangular bar made of jaggery and nuts. Now, let’s say you have one whole chikki and you want to divide it into equal parts.

Dividing the Chikki into 2 Parts:

• If you break the chikki into 2 equal pieces, each piece is 1/2 of the whole chikki.
• Here, the fractional unit is 1/2, meaning each piece represents half of the original chikki.

Dividing the Chikki into 6 Parts:

• Now, if you take the same chikki and break it into 6 equal pieces, each piece is 1/6 of the whole chikki.
• Here, the fractional unit is 1/6, meaning each piece is one-sixth of the original chikki.

Understanding Different Shapes
Sometimes, even if the pieces look different in shape, they can still be the same size. For example, if you cut the chikki into 6 equal pieces, each piece might look different depending on how you cut it, but each piece is still 1/6 of the whole chikki.
Example: Imagine cutting the chikki into 6 squares or 6 triangles. No matter the shape, each piece will still be 1/6 of the original chikki if the pieces are equal in size.

Measuring Using Fractional Units

Measuring Using Fractional Units means breaking down a whole object or quantity into smaller, equal parts and then using these parts to measure. When something is too large to be measured as a whole, we can divide it into fractions, which are smaller, equal pieces. This method helps us describe and understand how much of something we have, even when it's not a complete whole.

Understanding with an Example

Imagine you have a strip of paper that is one unit long. If you fold this strip into two equal parts, each part is now 1/2 (one-half) of the original strip. If you fold it again into four equal parts, each part becomes 1/4 (one-quarter) of the original strip.

Now, let's think about a whole roti (a round flatbread). If you cut the roti into two equal parts, each part is 1/2 of the whole roti. If you cut it into four equal parts, each part is 1/4 of the whole roti. If you eat two parts, you've eaten 2/4 or 1/2 of the roti.

This way, fractional units like 1/2, 1/4, and so on help us measure and describe parts of a whole object or quantity.

Reading Fractions

• Fractions can be read in different ways. For example, the fraction 3/4 can be read as "three quarters" or "three upon four." However, reading it as "3 times a" helps us understand the size of the fraction better. This way, we can see what the fractional unit is (1/4) and how many of these units there are (3).
• In a fraction, the top number is called the numerator, and the bottom number is called the denominator. For example, in the fraction 5/6, 5 is the numerator, and 6 is the denominator.

Marking Fraction Lengths on the Number Line

Marking Fraction Lengths on the Number Line means dividing a unit length on the number line into equal parts and then labeling these parts with fractions. This helps us visualize where different fractions lie between whole numbers on the number line.

Understanding with an Example

Imagine a number line with the numbers 0, 1, 2, and so on. The distance between 0 and 1 is one unit long. Now, if we divide this distance into two equal parts, each part represents 1/2 (one-half).

• If you mark a point halfway between 0 and 1, this point represents 1/2.
• If you divide the distance between 0 and 1 into four equal parts, each part represents 1/4 (one-quarter). The first mark would be 1/4, the second mark 2/4 (which is the same as 1/2), and so on.

So, if a blue line is drawn from 0 to the halfway point between 0 and 1, the length of that blue line is 1/2.
This process helps us understand where fractional values like 1/2, 1/4, and so on are located on the number line.

Mixed Fractions

A mixed number / mixed fraction contains a whole number (called the whole part) and a fraction that is less than 1 (called the fractional part).

Example: Imagine you have 1 whole pizza and another half of a pizza. You can describe this as a mixed fraction: 1 1/2. This means you have 1 whole pizza plus another 1/2 (half) of a pizza.

Fractions Greater Than One

On a number line, you can have fractions that are less than 1, like 1/2 or 3/4, which are called proper fractions. These fractions are smaller than one whole unit.

But when you combine a whole number with a fraction, like 1x1/2 or 2x1/4, the length on the number line becomes more than 1 unit. These are called mixed fractions because they include both a whole number and a fraction.

Writing fractions greater than one as mixed numbers

When we have a fraction greater than one, we can rewrite it as a mixed number. A mixed number is a combination of a whole number and a fraction. This is helpful because it shows how many whole parts we have and what fraction of a whole part is left over.

Example: Let's take the fraction 5/2.

• First, see how many whole parts are in 5/2. We know that 2/2 makes 1 whole.
• In 5/2, we can take out 2/2 two times (which makes 2 wholes), and we have 1/2 left over.
• So, 5/2 can be written as the mixed number 2 1/2.

This shows that 5/2 is the same as 2 whole parts plus another half, written as 2x1/2.

Equivalent Fractions

Using a fraction wall to find equal fractional lengths!

Equivalent fractions represent the same length but are expressed in different fractional units. To check if two fractions, like 1/2 and 2/4, are equivalent, we can use paper strips.

Understanding Equivalent Fractions using Equal Shares

Understanding with an Example

When one roti is shared equally among four children, each child receives a fraction of the whole roti. The diagram illustrates how the roti is divided into four equal shares.

The fraction of the roti that each child receives is 1/4. It's important that all four shares are equal.
This scenario can also be represented using division, addition, and multiplication facts:

• Division Fact: 1 ÷ 4 = 1/4
• Addition Fact: 1 = 1/4 + 1/4 + 1/4 + 1/4
• Multiplication Fact: 1 = 4 x 1/4

Expressing a Fraction in Lowest Terms (or in its Simplest Form)

• A fraction is in lowest terms or simplest form when its numerator and denominator have no common factors other than 1. This means the numbers are as small as possible.
• To express a fraction in lowest terms, you can find an equivalent fraction with the smallest possible numerator and denominator.

Step-by-Step Example
Now, let’s simplify 36/60:

1. Step 1: Notice that both 36 and 60 are even numbers, so divide both by 2:
   

2. Step 2: Again, 18 and 30 are even, so divide both by 2:
   

3. Step 3: Now, 9 and 15 are both multiples of 3, so divide both by 3:
   

4. Now, 3/5 is in its simplest form because 3 and 5 have no common factors other than 1.

Comparing Fractions

Comparing Fractions means determining which of two or more fractions is larger or smaller. To compare fractions easily, we can convert them to equivalent fractions with the same denominator. Once the fractions have the same denominator, we can simply compare the numerators (the top numbers) to see which fraction is greater.

Example

Let's compare the fractions 4/9 and 5/7:

1. Step 1: Find a common denominator. A common denominator is a number that both denominators (9 and 7) can divide into. In this case, the common denominator is 63.
2. Step 2: Convert each fraction to have the same denominator:
   • 4/9 becomes 28/63 because 4 × 7 = 28 and 9 × 7 = 63.
   • 5/7 becomes 45/63 because 5 × 9 = 45 and 7 × 9 = 63.
3. Step 3: Now compare the numerators (28 and 45):
   • Since 45 > 28, 5/7 is greater than 4/9.

Another Example

Let's compare 7/9 and 5/21:

1. Step 1: Find a common denominator. For 9 and 21, the common denominator is 63.
2. Step 2: Convert each fraction to have the same denominator:
   • 7/9 becomes 49/63 because $7\times 7=49$ and $9\times 7=639\; \backslash times\; 7\; =\; 63$.
   • 5/21 becomes 15/63 because $5\times 3=155\; \backslash times\; 3\; =\; 15$ and $21\times 3=63$.
3. Step 3: Compare the numerators (49 and 15):
   • Since 49 > 15, 7/9 is greater than 5/21.

Addition and Subtraction of Fractions

Addition and Subtraction of Fractions involve combining or taking away parts of a whole. To add or subtract fractions, they must have the same denominator (the bottom number of the fraction). Once the fractions have the same denominator, you can simply add or subtract the numerators (the top numbers) and keep the denominator the same.

Understanding with an Example

Example: Addition of Fractions

Let's solve the example from Meena and her brother:

1. Step 1: Meena ate 1/2 of the chikki, and her brother ate 1/4 of the chikki.
2. Step 2: To add these fractions, they need the same denominator. The common denominator for 1/2 and 1/4 is 4.
   • 1/2 becomes 2/4 because $1\times 2=21\; \backslash times\; 2\; =\; 2$1×2=2 and $2\times 2=42\; \backslash times\; 2\; =\; 4$2×2=4.
   • 1/4 stays as 1/4.
3. Step 3: Now, add the numerators:
   • 2/4 + 1/4 = 3/4.

So, together, Meena and her brother ate 3/4 of the chikki.

Example: Subtraction of Fractions

Let's imagine Meena wanted to subtract 1/4 of the chikki from what she ate:

1. Step 1: Meena ate 1/2 of the chikki, and we want to subtract 1/4.
2. Step 2: Convert 1/2 to 2/4 (as done above).
3. Step 3: Now, subtract the numerators:
   • 2/4 - 1/4 = 1/4.

So, after subtracting 1/4, Meena has 1/4 of the chikki left.

Key Point
To add or subtract fractions:

• Make sure the fractions have the same denominator.
• Add or subtract the numerators.
• Keep the denominator the same.

Adding Fractions with the Same Fractional Unit or Denominator

Adding Fractions with the Same Denominator means combining fractions that have the same bottom number (denominator), which represents the same size or part of the whole. When fractions have the same denominator, you can add them directly by adding their numerators (top numbers), while keeping the denominator the same.

Understanding with Examples

Example 1: Find the sum of 2/5 and 1/5.

1. Step 1: Represent both fractions using rectangular strips. Each strip is divided into 5 equal parts because the denominator is 5.
   • 2/5 is represented by 2 shaded parts out of 5.
     
   • 1/5 is represented by 1 shaded part out of 5.
     
2. Step 2: To add the fractions, count the total number of shaded parts:
   • 2 shaded parts + 1 shaded part = 3 shaded parts.
3. Step 3: Since each shaded part represents 1/5, the total is:

So, the sum of 2/5 and 1/5 is 3/5.

Example 2: Find the sum of 4/7 and 6/7.

1. Step 1: Represent both fractions using rectangular strips. Each strip is divided into 7 equal parts because the denominator is 7.
   • 4/7 is represented by 4 shaded parts out of 7.
     
   • 6/7 is represented by 6 shaded parts out of 7.
     
2. Step 2: To add the fractions, count the total number of shaded parts:
   • 4 shaded parts + 6 shaded parts = 10 shaded parts.
3. Step 3: Since each shaded part represents 1/7, the total is:

Or as a mixed fraction:
So, the sum of 4/7 and 6/7 is 10/7, which can also be written as .

Key Point: When adding fractions with the same denominator, simply add the numerators and keep the denominator the same. If the result is an improper fraction (where the numerator is larger than the denominator), you can also express it as a mixed number.

Adding Fractions with Different Fractional Units or Denominators

Adding Fractions with Different Denominators means combining fractions that have different bottom numbers (denominators). To do this, you first need to convert the fractions to have the same denominator, which makes them easier to add. This involves finding a common denominator, then adding the fractions as usual by summing the numerators and keeping the denominator the same.

Example: Find the sum of 2/3 and 1/5.

1. Step 1: Find a common denominator. The denominators are 3 and 5. The lowest common denominator is 15.
2. Step 2: Convert each fraction to an equivalent fraction with the common denominator:
   • For 2/3, multiply the numerator and denominator by 5:
     
   • For 1/5, multiply the numerator and denominator by 3:
     
3. Step 3: Add the fractions with the same denominator:
   

So, the sum of 2/3 and 1/5 is 13/15.

Brahmagupta's method for adding fractions

• Find equivalent fractions so that the fractional unit is common for all fractions. This can be done by finding a common multiple of the denominators (e.g., the product of the denominators, or the smallest common multiple of the denominators).
• Add these equivalent fractions with the same fractional units by adding the numerators and keeping the same denominator.
• If needed, express the result in lowest terms.

Subtraction of Fractions with Different Fractional Units or Denominators

Subtraction of Fractions with the Same Denominator means taking away one fraction from another when both fractions have the same bottom number (denominator). Since the denominators are the same, you can subtract the numerators (the top numbers) directly and keep the denominator the same.

Example: Let's subtract 4/7 from 6/7.

1. Step 1: Both fractions have the same denominator, which is 7. This means the fractional units are the same.
2. Step 2: Represent the larger fraction, 6/7, using a rectangular strip model where each part represents 1/7. Shade 6 out of the 7 parts to represent 6/7.
3. Step 3: To subtract 4/7, remove 4 of the shaded parts from the strip:
   • Start with 6 shaded parts representing 6/7.
   • Remove 4 shaded parts, which represents 4/7.
4. Step 4: After removing 4 parts, 2 shaded parts are left, representing 2/7.

So, the result of subtracting 4/7 from 6/7 is 2/7.

Brahmagupta's Method for Subtracting Two Fractions

• Convert the given fractions into equivalent fractions with the same denominator.
• Subtract the numerators while keeping the denominator the same.
• Simplify the result if necessary.

A Pinch of History

• In ancient India, a fraction was referred to as bhinna in Sanskrit, which means "broken." It was also called bhaga or ansha, meaning "part" or "piece."
• The modern global notation for writing fractions originated in ancient India. In texts like the Bakshali manuscript(around 300 CE), fractions were written in a way similar to today’s method.
• This method was used by famous mathematicians such as Aryabhata(499 CE), Brahmagupta(628 CE), Sridharacharya(c. 750 CE), and Mahaviracharya (c. 850 CE).
• The line segment between the numerator and denominator in fractions was introduced by A1-Hassar, a Moroccan mathematician, in the 12th century.
• Fractions were also used in ancient cultures like Egypt and Babylon, but they mainly dealt with fractional units, where the numerator was 1. More complex fractions were expressed as sums of these units, known as Egyptian fractions.
• India was the first to introduce general fractions (where the numerator is not necessarily 1) along with rules for arithmetic operations like addition, subtraction, multiplication, and division of fractions.
• The Sulba-sutras from Vedic times show that Indians had discovered rules for operating with fractions long ago.
• Brahmagupta formally codified these rules and methods for fractions in a modern form. His methods for adding and subtracting fractions are still in use today.
• For example, Brahmagupta explained that to add or subtract fractions, you should make their denominators the same by multiplying the numerator and denominator of each fraction by the other denominators. Then, you add or subtract the numerators.
• Indian concepts and methods related to fractions were transmitted to Europe via the Arabs and became common in Europe around the 17th century, eventually spreading worldwide.

Key Points

• Fraction as equal share: When a whole number of units is divided into equal parts and shared equally, a fraction results.
• Fractional Units: When one whole basic unit is divided into equal parts, then each part is called a fractional unit.
• Reading Fractions: In a fraction such as -, 5 is called the numerator and 6 is called the denominator.
• Mixed fractions contain a whole number part and a fractional part.
• Number line: Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.
• Equivalent Fractions: When two or more fractions represent the same share/number, they are called equivalent fractions.
• Lowest terms: A fraction whose numerator and denominator have no common factor other than 1 is said to be in lowest terms or in its simplest form.
• Brahmagupta's method for adding fractions: When adding fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then add the number of fractional units in each fraction to obtain the sum. This is accomplished by adding the numerators while keeping the same denominator.
• Brahmagupta's method for subtracting fractions: When subtracting fractions, convert them into equivalent fractions with the same fractional unit (i.e., the same denominator), and then subtract the number of fractional units. This is accomplished by subtracting the numerators while keeping the same denominator.