Chapter Notes: Fractions
What is a Fraction?
A fraction names a part of a whole when that whole is divided into equal parts. A fraction has two parts: a numerator (the number above) and a denominator (the number below). The denominator tells into how many equal parts the whole is divided and the numerator tells how many of those parts are taken.
Fractions can be used to
- name a part of a single thing;
- Here, 1 part out of 4 equal parts is red, we write it as 1 / 4.
- name a part of a group of things.
In this aquarium, you have 7 fish, out of which 2 are starfish. So, we write 2 / 7 are starfish.
Representation of a Fraction on a Number Line
To place a proper fraction (less than 1) on a number line, draw the portion of the number line from 0 to 1 and divide it into as many equal parts as the denominator. Then count the numerator number of parts from 0.
For example, to show 3 / 8, divide the interval from 0 to 1 into 8 equal parts and mark the third division. Since 3 / 8 < 1, it lies between 0 and 1 on the number line.
Types of Fractions
Proper Fractions
- A fraction in which the numerator is less than the denominator is called a proper fraction.
- Every proper fraction is less than 1.
- Examples: 0 / 2, 1 / 4, 2 / 5, 3 / 7, 81 / 100.
Improper Fractions
- A fraction in which the numerator is greater than or equal to the denominator is called an improper fraction.
- Every improper fraction is greater than or equal to 1.
- Examples: 6 / 6, 7 / 4, 21 / 20, 121 / 100.
Mixed Numbers
A mixed number has a whole number part and a proper fraction part together. The whole number part is not zero.
For example,
are all mixed numbers.
Unit Fractions
- A unit fraction has a numerator 1 and a denominator greater than 1.
- Examples: 1 / 8, 1 / 3, 1 / 5.
Like and Unlike Fractions
- Like fractions have the same denominator. Examples: 8 / 12, 7 / 12, 17 / 12.
- Unlike fractions have different denominators. Examples: 4 / 6, 3 / 8, 12 / 15.
EduRev Tips: We can write a mixed number as an improper fraction and an improper fraction as a mixed number.
Example: Write 7 / 5 as a mixed number.
Method 1: Pictorial and number line methods
On the number line:
Jump five-fifths to reach 1, then two more fifths to reach 7 / 5. So 7 / 5 = 1 2/5.
Method 2: By division
EduRev Tips:
Example: Write 
as an improper fraction.
Pictorial method:
Multiply the whole number part by the denominator and add the numerator to the product to get the numerator of the improper fraction.
Alternatively,
Equivalent Fractions
Fractions that name the same part of a whole are called equivalent fractions. We find equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.
Thus, 1 / 4 = 2 / 8 = 3 / 12 because they all represent the same part of the whole.
EduRev's Tips:
To find equivalent fractions, multiply or divide numerator and denominator by the same number (except 0).
Example: Fill in the missing numbers.
An Important Property of Equivalent Fractions
- If two fractions are equivalent, then the product of the numerator of the first and the denominator of the second equals the product of the denominator of the first and the numerator of the second.
- This gives a simple test for equivalence.
Example: Check whether these fractions are equivalent:
Sol:
6 × 65 = 390 and 13 × 30 = 390
Both products are equal. Therefore, the fractions are equivalent.
Writing a Fraction in Its Simplest Form
A fraction is in its simplest form when its numerator and denominator have no common factor other than 1.
Examples of simplest form: 1 / 2, 3 / 5, 5 / 8. The fraction 10 / 15 is not in its simplest form because 5 is a common factor. 3 / 21 is not the simplest because 3 is a common factor.
Example: Express the following in simplest form
(a) 6 / 8
Divide the numerator and denominator by 2:
(b) 15 / 30
Hence 15 / 30 = 1 / 2.
Alternative method: Repeatedly divide by common factors until only 1 remains as a common factor.
(c)210 / 330
Comparing Fractions
1. Fractions with the same denominator (like denominators)
Rule: When denominators are equal, the fraction with the larger numerator is the larger fraction.
Example: For a singing competition, 6 / 15 of the children are girls and 9 / 15 are boys. Whose number is greater?
Compare 6 / 15 and 9 / 15.Method 1: Pictorially
From the shaded parts we see 9 / 15 > 6 / 15, so boys are more.
Method 2: Number line
9 / 15 lies to the right of 6 / 15, so 9 / 15 > 6 / 15.
Rule: If two or more fractions have the same denominator, the fraction with the greater numerator is the greater number, e.g., 6 / 7 > 5 / 7 or 5 / 7 < 6 / 7.
2. Fractions with different denominators (unlike denominators)
Example: Which is greater: 1 / 3 or 2 / 5?
Method 1: Pictorial comparison
From the shaded diagrams, 2 / 5 > 1 / 3.
Method 2: Number line
2 / 5 is to the right of 1 / 3, so 2 / 5 > 1 / 3.
3. By finding equivalent fractions using LCM
Convert both fractions to equivalent fractions having the same denominator. Choose the least common denominator as the LCM of the denominators.
For 1 / 3 and 2 / 5, LCM(3, 5) = 15.
EduRev Tip: Finding equivalent fractions using the LCM is the most commonly used method for comparing and ordering unlike fractions.
Ordering Fractions
Once you can compare fractions, you can arrange them in increasing or decreasing order.
Example: Arrange 7 / 8, 5 / 12, 15 / 16 in increasing order.
Step 1: Find LCM of denominators 8, 12 and 16. LCM = 48.
Step 2: Convert each fraction to an equivalent fraction with denominator 48.
Step 3: Compare the converted numerators and order the fractions.
Thus in increasing order:
And in decreasing order:
Ordering Unit Fractions
A unit fraction has 1 as its numerator, but the denominator can vary.
For example, 1 / 2, 1 / 3, 1 / 4, 1 / 5 are unit fractions. What conclusion do you draw from the following diagrams?
It can be clearly seen that
Thus, if the numerator is the same, the fraction with the least denominator is the greatest.
Addition and Subtraction of Fractions
1. Adding or subtracting like fractions (same denominator)
Add or subtract the numerators and keep the denominator the same.
2. Adding or subtracting unlike fractions (different denominators)
Example: Ravi and Nisha painted a wall. Ravi painted 1 / 2 of the wall and Nisha painted 1 / 4. What part of the wall did they paint altogether?
Wall painted by Ravi = 1 / 2
Wall painted by Nisha = 1 / 4
Total painted = 1 / 2 + 1 / 4
Make denominators the same. Convert 1 / 2 to 2 / 4, then add:
Total painted = 3 / 4.
Rule: To add or subtract fractions with different denominators, first change them to equivalent fractions with the least common denominator (LCD), which is the LCM of the denominators, then add or subtract the numerators.
What is the least common denominator?
Common denominators are numbers that both denominators divide into. The smallest such number is the least common denominator (LCD), which equals the LCM of the denominators.
For example, 12 is an LCD of 1/6 and 3/4 because 12 is an LCM of 6 and 4.
Example: Add
Method 1: LCM(4, 5) = 20. Convert and add.
Method 2: Work in the column style
Steps: LCM = 20. Divide 20 by each denominator and multiply the result by the corresponding numerator to get the converted numerators, then add.
Another example: LCM of 6 and 10 is 30. Convert 5 / 6 and 1 / 10 to denominator 30 and then add:
Example: Subtract
Method 1: LCM(8, 4) = 8. Convert and subtract.
Method 2:
LCM of 7 and 11 is 77:
Addition of Mixed Fractions
Type I: Mixed numbers with the same denominators
Example: When released, a gas balloon first rose by 
and then again by
What is the total height to which the gas balloon rose?
Pictorially:
Add the whole number parts and then add the fractional parts. If the fractional parts add to an improper fraction, convert to a whole number and remainder.
Example: A cow gave 
litres of milk in the morning and 
litres in the evening. Find the total quantity of the milk given by the cow.
Method 1:
Method 2: Convert each mixed number to an improper fraction, add, and convert back if needed.
So the total milk is:
Type II: Mixed numbers with different denominators
Example: Add
Method 1:
LCM(2, 4) = 4. Convert and add.
Method 2: Convert each mixed number to an improper fraction, then find a common denominator and add.
Example: Add
Method 1:
LCM of 4, 6 and 12 is 12. Convert each fraction to denominator 12 and add.
Method 2:
EduRev Tip: Method 1 (adding whole parts first) is helpful when whole-number parts are large.
Subtraction of Mixed Fractions
Example: Subtract 
Example: Subtract
Method 1:
Step 1: Subtract whole numbers.
Step 2: Change fractions to equivalent fractions with a common denominator.
Step 3: Subtract the fractional parts.Method 2: Convert both mixed numbers into improper fractions, then subtract.
Subtracting from Whole Numbers
Subtracting from 1
Example: Mira is preparing a sweet dish. She used 5 / 8 litre of milk from a jug containing 1 litre milk. How much milk is left in the jug?
Quantity remaining = 1 - 5 / 8.
Or by writing 1 as 8 / 8 and subtracting:
Mira has 3 / 8 litre left.
Example: David lives 3 km away from a town. Paul lives 5/6 of a kilometre away from the town. How much closer does Paul live than David?
Method 1:
Paul livesDifference = 3 - 5/6 = 2 1/6 km closer (computed after converting).
Method 2:
Paul LivesCompute the difference by converting to improper fractions or common denominators.
Thus, Paul liveskm closer to the town as compared to David.
EduRev's Tip: In subtraction sums like
where whole number parts are large numbers, it may be easier and more convenient to apply method 1.
Thus,
Example: Subtract
Method 1:
Method 2:
Method 2 (converting to improper fractions) is usually preferred for clarity.
Properties of Addition of Fractions
- Commutative property: Changing the order of two addends does not change the sum.
Pictorially, it can be shown as: 
- Associative (grouping) property: Changing the way in which we group addends does not change the sum.
Thus,
- Additive identity (0):The sum of a fraction and 0 is the fraction itself.
Problems Based on Real Life Situations
Example: Deepa used 5 / 9 L of milk to prepare kheer. She added 1 / 2 L more. How many litres did she use in all?
Milk used = 5 / 9 L
Milk added = 1 / 2 L
Total = 5 / 9 + 1 / 2
LCM of 9 and 2 is 18. Convert to denominator 18 and add.
∴ Deepa used
of milk for the kheer.
Example:Pinki had 
of a ribbon. She used 
of the ribbon to wrap some gifts. How much ribbon is left with her?
Ribbon with Pinki:
Ribbon used:
Ribbon left = initial - used.
LCM of 2 and 3 is 6. Convert and subtract to find the remaining ribbon.
FAQs on Chapter Notes: Fractions
| 1. How do I add and subtract fractions with different denominators? |  |
| 2. What's the difference between proper fractions, improper fractions, and mixed numbers? | |
| 3. How do I compare fractions to find which one is bigger or smaller? | |
| 4. Why do I need to simplify fractions and how do I do it correctly? | |
| 5. How do I multiply and divide fractions, and why is the method different from addition? | |
