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CBSE Notes Class 6 Maths Chapter 7 Fractions
Fractions are an essential concept in CBSE Class 6 Maths that appear in many areas of life, from measuring ingredients to dividing a pizza! In CBSE Class 6, students are introduced to fractions in a deeper way, exploring various types and operations involving fractions. In this blog, we’ll take a closer look at key topics related to fractions, including fractional units, unit fractions, mixed fractions, equivalent fractions, and more. Let’s dive into the world of fractions!
1.0What is a Fraction?
A fraction represents a part of a whole or a part of a group. It is written as a numerator (the top number) and a denominator (the bottom number), separated by a slash or a horizontal line (Eg:). Understanding fractions involves recognising how a whole can be divided into equal parts.
2.0Fractional Units as Parts of a Whole
In the world of fractions, the fractional unit refers to one of the equal parts into which a whole is divided. For example, if you divide a pizza into 8 equal slices, each slice is a fractional unit. In this case, each slice represents of the whole pizza. When you divide something into fractional units, you're essentially splitting it into parts that represent a fraction of the entire object.
Unit Fractions
A unit fraction is a fraction where the numerator is always 1, and the denominator represents the number of equal parts the whole is divided into. For example:
- is a unit fraction representing one half of a whole.
- represents one third of a whole.
Unit fractions are the building blocks of all other fractions. They help us understand how a whole can be divided into smaller parts.
3.0Measuring Using Fractional Units
Measuring with fractional units allows us to quantify portions of objects or quantities. For instance, when you measure of a meter, it means that the meter has been divided into four equal parts, and three of those parts are being used. This is especially helpful in real-world applications like cooking, construction, or tailoring, where portions of a whole are often required.
4.0Marking Fraction Lengths on the Number Line
A number line is a powerful tool for visualising fractions. To mark fractional lengths on a number line, first, divide the segment between two whole numbers into equal parts. These parts represent the fractional units. For example, if you're working with , divide the segment between 0 and 1 into four equal parts, and mark the point that represents .
5.0Mixed Fractions
A mixed fraction is a combination of a whole number and a fraction.
For example, is a mixed fraction, where 2 is the whole number and is the fractional part. Mixed fractions are often used in everyday situations, such as when measuring distances or quantities that aren't whole numbers.
Converting Between Mixed Fractions and Improper Fractions
Sometimes it’s useful to convert a mixed fraction into an improper fraction (where the numerator is larger than the denominator), or vice versa. For example:
- can be written as the improper fraction .
- can be rewritten as the mixed fraction
6.0Equivalent Fractions
Two fractions are equivalent if they represent the same portion of a whole, even though they may have different numerators and denominators. For example:
- is equivalent to , and so on.
To find equivalent fractions, you can multiply or divide both the numerator and the denominator by the same number. For instance, if you multiply both the numerator and denominator of by 2, you get , which is an equivalent fraction.
7.0Comparing Fractions
To compare fractions, it’s helpful to:
- Find a common denominator: This makes the fractions easier to compare. For example, to compare and , you could find the least common denominator (LCD) and rewrite both fractions with that denominator.
- Use the number line: Marking fractions on a number line allows for a clear visual comparison. The fraction closer to 0 is the smaller fraction.
Which Fraction is Bigger?
To compare fractions with the same denominator, simply look at the numerators. The fraction with the larger numerator is the larger fraction. For example, is greater than because 3 is greater than 2.
8.0Addition and Subtraction of Fractions
Adding and subtracting fractions can be tricky, but once you understand the process, it becomes much easier.
- Addition of Fractions
If the denominators are the same, just add the numerators. For example:
If the denominators are different, first find the LCM (Least Common Multiple) of the denominators, rewrite the fractions with a common denominator, and then add the numerators. For example:
: The LCM of 3 and 6 is 6, so rewrite as
- Subtraction of Fractions
Subtraction is similar to addition. Ensure the fractions have the same denominator, then subtract the numerators. For example:
If the denominators are different, find a common denominator, just like in addition, and then subtract the numerators.
9.0A Pinch of History: The Origins of Fractions
Fractions have a long history that dates back thousands of years. The ancient Egyptians were among the first to use fractions, especially in their mathematical texts. They primarily used unit fractions, as they believed fractions could only be represented as a sum of unit fractions.
The concept of fractions evolved over time, especially during the Islamic Golden Age, when mathematicians like Al-Khwarizmi contributed to understanding fractions as parts of a whole. Today, fractions are a foundational concept in mathematics, essential for everything from algebra to calculus.
Key Concepts to Remember
- Fractions and equal shares: A fraction is about equal parts of a whole. For example, two children share one roti equally, each getting of the roti.
- Fractional units: When a whole is divided into equal parts, each part is a fractional unit. For example, if a roti is divided into four equal pieces, each piece represents
- Numerator and denominator: In the fraction 3 is the numerator and 5 is the denominator. The numerator tells us how many parts we are considering, while the denominator tells us how many equal parts make up the whole.
- Fractions as parts of a whole: A fraction can represent a part of an object, such as a chikki divided into six equal pieces. Each piece represents , and four pieces represent of the whole.
- Measuring with fractions: Fractions can help us measure quantities smaller than one whole. For example, three pieces of size make , while four such pieces make one whole.
- Fractions on a number line: To show a fraction such as , divide the distance from 0 to 1 into five equal parts. The third mark represents .
- Fractions less than or greater than one: A fraction with a smaller numerator than denominator is less than one, such as . A fraction with a larger numerator than denominator is greater than one, such as.
- Mixed numbers: A mixed number has a whole number and a fraction together. For example, is equal to 2 because it contains two whole units and one-third of another unit.
- Equivalent fractions: Fractions can look different but represent the same amount. For example, , and are equivalent fractions.
- Fractions in simplest form: A fraction is said to be in lowest terms if the only factor that the numerator and denominator have in common is 1. For instance,
can be reduced to
by dividing both numbers by 4 .
- Comparing fractions: To compare two fractions with the same denominator , you compare the numerators . For example, is larger than because five eighths is larger than three eighths.
- Adding fractions: Fractions with the same denominator can be added by adding their numerators and keeping the denominator unchanged. For example,
+ = . If the denominators differ, first convert the fractions to equivalent fractions with a common denominator.
- Subtracting fractions: When the denominators are the same, subtract the numerators and keep the denominator unchanged. For example, − = . If the denominators differ, first find a common denominator.
10.0Important Formulas Fractions
These rules will help you solve fraction questions more easily.
The denominators must not be zero. When simplifying a fraction, divide its numerator and denominator by the same common factor.
Converting a Fraction Greater Than One into a Mixed Number
Divide the numerator by the denominator. The quotient gives the whole number, the remainder becomes the numerator of the fractional part, and the denominator stays the same.
Example:
Since 8 = 3 × 2 + 2, the fraction contains two whole units and two-thirds of another unit.
Converting a Mixed Number into a Fraction
Multiply the whole number by the denominator, add the numerator, and keep the original denominator.
Example:
Adding Fractions with Different Denominators
First, find a common denominator. Then convert the fractions into equivalent fractions and add their numerators.
Example: Find +
+
&=+
&=
Subtracting Fractions with Different Denominators
Convert both fractions to equivalent fractions with a common denominator. Then subtract their numerators.
Example: Find
−
-
&=-
&=
11.0Learning Guide for Class 6 Maths Fractions - Chapter 7
- Sharing things equally: Imagine three rotis being shared equally among four children. Each child receives 3/4 of a roti. This is one way to understand how fractions represent equal shares.
- Understanding fractions with chikki: Suppose a chikki is divided into six equal pieces. Each piece is 1/6 of the whole. If you take four pieces, you have 4/6 of the chikki.
- Using paper strips: Take a paper strip and fold it into two equal parts to show halves. Divide it into four equal parts to show quarters. Three of the four parts represent 3/4 of the strip.
- Showing fractions on a number line: Draw a line from 0 to 1 and divide it into five equal intervals. Mark the points 0, 1/5, 2/5, 3/5, 4/5 and 1. This makes it easier to see where 3/5 lies between 0 and 1.
- Understanding fractions greater than one: Take 7/3. Three one-third units make one whole, so six one-third units make two wholes. One-third remains, giving 7/3 = 2 1/3.
- Converting mixed numbers: Consider 2 3/4. Each whole contains four quarters, so two wholes contain eight quarters. Add the remaining three quarters to get eleven quarters. Therefore, 2 3/4 = 11/4.
- Comparing equivalent fractions: Draw equal-length strips and divide them into two, four and eight equal parts. Half of each strip covers the same length, showing that 1/2 = 2/4 = 4/8.
- Finding equivalent fractions: Start with 2/3 and multiply its numerator and denominator by 2. You get 4/6. Both fractions represent the same amount, even though they are written differently.
- Simplifying fractions: Take 36/60. Divide both numbers by 2 to get 18/30, then divide by 3 to get 6/10. Divide by 2 once more to get 3/5, which is the simplest form.
- Comparing fractions with the same denominator: Compare 4/7 and 6/7. Both fractions are made up of sevenths. Since six sevenths are more than four sevenths, 6/7 > 4/7.
- Comparing fractions with different denominators: To compare 3/4 and 7/10, convert them to fractions with denominator 20. You get 15/20 and 14/20. Therefore, 3/4 > 7/10.
- Adding fractions: To find 2/3 + 1/5, use 15 as the common denominator. Convert the fractions to 10/15 and 3/15. Their sum is 13/15.
- Subtracting fractions: To find 5/6 − 1/3, convert 1/3 into 2/6. Then subtract: 5/6 − 2/6 = 3/6 = 1/2.
- Checking your answer: After solving a question, see whether the fraction can be simplified further. You can also check whether the answer makes sense. For example, subtracting a smaller positive fraction from a larger fraction should give a result smaller than the original fraction.
12.0Explore Chapter-wise Class 6 Maths Notes
13.0Additional Study Materials for Class 6 Maths
14.0Find the Right Study Resources for Class 6
15.0Benefits of CBSE Notes for Class 6 Maths Chapter 7 - Fractions
- Visual Aids: Some CBSE Notes for Class 6 incorporate diagrams and number lines to help visualise fractions, enhancing comprehension.
- Real-World Connections: Good notes might include real-life examples of how fractions are used, making the topic more relatable.
- Practice Opportunities: Some CBSE Notes include practice questions or refer to important exercises, reinforcing learning.
- Foundation for Future Concepts: A strong understanding of fractions is crucial for future mathematical topics like decimals, percentages, and ratios.
- Improved Problem-Solving Skills: Working with fractions helps develop logical reasoning and problem-solving abilities.