Class 6 — Mathematics (NCERT)
Fractions: Understanding and Operations
Class 6
- ✓By the end of this lesson students will be able to define and identify proper, improper, and mixed fractions.
- ✓By the end of this lesson students will be able to convert improper fractions to mixed fractions and vice-versa.
- ✓By the end of this lesson students will be able to understand and find equivalent fractions.
- ✓By the end of this lesson students will be able to add and subtract fractions with the same denominator.
- ✓By the end of this lesson students will be able to add and subtract fractions with different denominators.
Key concepts
A fraction represents a part of a whole. It is written as a numerator over a denominator. The numerator tells us how many parts are being considered, and the denominator tells us the total number of equal parts the whole is divided into.
A proper fraction is a fraction where the numerator is less than the denominator. Its value is always less than 1. For example, 3/5, 1/2.
An improper fraction is a fraction where the numerator is greater than or equal to the denominator. Its value is always greater than or equal to 1. For example, 7/4, 5/5.
A mixed fraction (or mixed numeral) is a combination of a whole number and a proper fraction. Improper fractions can be converted into mixed fractions, and vice-versa. For example, 2 1/3, 1 3/4.
Equivalent fractions are fractions that represent the same value, even though they have different numerators and denominators. We can obtain equivalent fractions by multiplying or dividing both the numerator and the denominator by the same non-zero number. For example, 1/2, 2/4, 3/6 are equivalent fractions.
To add fractions with the same denominator, we simply add their numerators and keep the denominator common.
To subtract fractions with the same denominator, we simply subtract their numerators and keep the denominator common.
To add fractions with different denominators, first find the Least Common Multiple (LCM) of the denominators. Then, convert each fraction into an equivalent fraction with the LCM as the new denominator. Finally, add the numerators of these equivalent fractions and keep the common denominator.
To subtract fractions with different denominators, first find the Least Common Multiple (LCM) of the denominators. Then, convert each fraction into an equivalent fraction with the LCM as the new denominator. Finally, subtract the numerators of these equivalent fractions and keep the common denominator.
Key facts to remember
- 1A fraction represents a part of a whole, with the numerator indicating parts taken and the denominator indicating total equal parts.
- 2In a proper fraction, the numerator is always less than the denominator (e.g., 2/5).
- 3In an improper fraction, the numerator is greater than or equal to the denominator (e.g., 7/3).
- 4A mixed fraction combines a whole number and a proper fraction (e.g., 1 2/3).
- 5To convert an improper fraction to a mixed fraction, divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator remains the same.
- 6Equivalent fractions represent the same value and are obtained by multiplying or dividing both the numerator and denominator by the same non-zero number.
- 7To add or subtract fractions with the same denominator, simply add or subtract their numerators and keep the common denominator.
- 8To add or subtract fractions with different denominators, first find the LCM of the denominators, convert the fractions to equivalent fractions with the LCM as the denominator, and then perform the operation.
Worked examples
Example 1
Identify the type of fraction for (a) 3/5, (b) 7/4, (c) 2 1/3. Also, convert 7/4 into a mixed fraction.
Answer
(a) Proper fraction, (b) Improper fraction, (c) Mixed fraction. 7/4 = 1 3/4.
Example 2
Find two equivalent fractions for 2/3.
Answer
Two equivalent fractions for 2/3 are 4/6 and 6/9.
You can choose any non-zero integer to multiply or divide by.
Example 3
Simplify: 5/6 - 1/4
Answer
7/12
Always find the LCM when denominators are different.
Example 4
Simplify: 2 1/5 + 1 3/10
Answer
3 1/2
Always convert mixed fractions to improper fractions before performing addition or subtraction.
Common mistakes
- ✗Adding or subtracting the denominators along with the numerators when performing operations.
- ✗Incorrectly finding the Least Common Multiple (LCM) of denominators, leading to wrong equivalent fractions.
- ✗Forgetting to convert mixed fractions to improper fractions (or vice-versa) before performing addition or subtraction, or doing it incorrectly.
- ✗Not simplifying the final answer to its lowest terms or converting an improper fraction answer back to a mixed fraction when required.
- ✗Confusing proper, improper, and mixed fractions.
Exam tips
- ★Always read the question carefully to understand what type of fraction is expected in the answer (e.g., improper or mixed).
- ★Show all steps clearly, especially when finding the LCM and converting fractions to equivalent forms. This helps in identifying errors and earns partial marks.
- ★Practice converting between improper and mixed fractions regularly to become proficient.
- ★Double-check your calculations, especially when dealing with multiple steps or larger numbers.
- ★Simplify your final answer to its lowest terms unless otherwise specified.
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