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Simplifying fractions is a common method for making themathematical data easier to understand. Itis the process of reducing the denominator and the numerator to their smallest whole numbers possiblewith no common factors other than 1, resulting in the fraction to be in its simplest form. The value of the fractions will remain unchanged even if it is simplified. In mathematics, it is necessary to make sure that everything's beenreduced to its simplestform. Simplifying fractions willhelp the mathematicians to avoid the confusion when the numbers or the equations become very large and complex.
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Key Terms: Numerator, Denominator, Fractions, Number Line, Proper Fraction, Whole Number, Improper Fraction, Prime Numbers
What is Fraction?
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A fraction is a number that represents a portion of whole. Each part of a whole is said to be a fraction onlywhen it is divided into two or more parts. Usually, A fraction is written as 1/2 , 1/12 , 11/2 and so on.
A fraction is made up of two partsa numerator on thetop and a denominator on the bottom, separated by a horizontal bar. The total number of parts into which the whole is divided is represented by the denominator. And the numerator is represented bythe total number of parts removed. For anexample, in the fraction 5/8 , the numerator is 5 and the denominator is 8.
Fraction
Example for Fraction in Real Life
Your father has ordered pizza for you and your buddies to celebrate your sister's birthday. When your father receives the pizza, he discovers that it has been cut into 8 slices. Let's assume you have 7 friends. So, there will be 8 people eating the 8 slices of pizza.
Example for Fraction in Real Life
How much slice does each of your friends get? Well, if we divide the pizza into 8 equal parts, then each friend will get 1/8 or one-eighth of the pizza. The pizza can be sliced into a different number of slices which will create different fractions.
Read Also: Revision of Arithmetic Properties
Types of Fractions
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Fractions are mainlycategorized into 3 types:
- Proper Fraction
- Improper Fraction
- Mixed Fraction
There are also some other types of fractions, such as unlike fractions, like fractions, unit fractions, equivalent fractions, and so on.
1. Proper Fraction
It is said to beProper Fraction onlywhen numerator is less than the denominator. These fractions are all less than 1, and none of them lieson the number line beyond one.
Proper Fraction
2. Improper Fraction
It is said to be anImproper Fraction when the numerator is greater than the denominator. These fractions are greater than 1and lie on the number line beyond 1. Any natural number can be represented as an improper fraction as the denominator is always 1.
Improper Fraction
3. Mixed Fraction
A mixed fraction is a fraction that contains both a natural number and a fraction. A mixed fraction can be converted to an improper fraction and vice - versa and itis alwaysgreaterthan 1.
Mixed Fraction
How to Simplify Fractions?
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Divide boththe numerator as well asdenominator by the same number to simplify a fraction. On simplifying the fractions, it can easily convert fractions to decimals and percentages, add and substratefractions, and compare fractions. We divide boththe numerator as well asdenominator by the same number to simplify a fraction.
A number that divides boththe numerator and denominator evenly with no remainders is called as common factor. On dividing the fractions until both the numerator and denominator have 1 as a common factor. Ifthe numerator and denominator of a fraction contain prime numbers, it is already in its simplest form, as prime numbers are special numbers that cannot be divided further. Only whole numbers are used to divide the fraction. When decimals are used to simplify fractions, it becomes harder to see the proportions and the fractions lose their simplicity.
Fractions can be simplified by 2 methods
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Method 1:-
Follow the step given below to learn how to simplify fractions using method 1.
Step 1: Find the factors of both the numbers which are mentioned in numerator and
denominator of the fraction.
Step 2: In this step, find the common factors of both the numerator and denominator.
Step 3: Divide both the numerator and denominator by the common factors that have find out
in the above step until there is no common factor other than 1.
Below example is to understand the simplification of fraction using method 1
Example: Find the simplified form of the fraction 6/24 .
Solution. The fraction given is 6/24
Step 1: Find the factors of 6 and 24
Factors of 6 are: 1, 2, 3, 6.
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
Step 2: In this step find the common factors of 6 and 24
Common factors are: 1, 2, 3, 6.
Step 3: Keep on dividing both the numerator and denominator by the common factors.
First divide it by 2,
(6 ÷ 2) / (24 ÷ 2) = 3/12
Now divide by 3,
(3 ÷ 3) / (12 ÷ 3) = 1/4
The simplified fraction of 3/12 is 1/4 .
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Method 2:-
Follow the step given below to learn how to simplify fractions using method 2.
Step 1: Find the prime factors of both numerator and denominator of the fraction.
Step 2: Find the HCF (highest common factor) of both numerator and denominator.
Step 3: In this stepdivide both the numerator and the denominator by the HCF (highest common factor) obtained.
Below example is to understand the simplification of fraction using method 2.
Example: Find the simplified form of the fraction 1628 .
Solution. The fraction given is 16/28
Step 1: Write 16 and 28 in terms of product of prime factor
16 = 2 × 2 × 2 × 2
28 = 2 × 2 × 7
Step 2: Find the HCF (highest common factor) of 16 and 28
HCF of 16 and 28 = 2 × 2 = 4
Step 3:In this step divide both numerator and denominator by the HCF obtained.
(16 ÷ 4)/ (28 ÷ 4) = 47
The simplified fraction of 16/18 is 4/7 .
Read Also: Quadratic Equations Formula
How to Simplify Fraction with Variables
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Rational expressions are fractions that have variables in the numerator and denominator. To simplify the fraction with variables, use the extended form of each term in boththe numerator and thedenominator.
Let's consider the fraction with variable (2x2z)/(xy).
Express the fraction in terms of the product of variables and cancel out the common variables.
(2x2y) / (xy) = (2 × x × x × y) / (x × y) = 2x
The simplified fraction of (2x2y) / (xy) is 2x.
How to Simplify Fraction with Exponents and Power
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Simplifying the fraction containing exponents and power in the numerator and denominator is similar to simplifying the fraction with variables. Use the expanded form of exponents in the numerator and denominator to make it easy for you to simplify the fraction with exponents.
Let’s consider the fraction with exponents and power 35/32.
Express the fraction in terms of product of numbers and cancel out the common numbers.
35/32= (3 × 3 × 3 × 3 × 3) / (3 × 3) = 3 × 3 × 3 = 27
The simplified fraction of 35/32 is 27.
Read Also: Polynomial Formula
Things to Remember
- A fraction is a number that represents a portion of something larger. A single or a group of objects might make up the whole.
- A number line can be used to represent fractions. Every fraction has a number line point associated with it.
- If the numerator of a fraction is less than the denominator then such fraction is called proper fraction.
- Improper fractions are those in which thenumerator is greater than the denominator.
- When an improper fraction is written as a mixture of a whole and a part, it is known as a mixed fraction.
- If the numerator and denominator of a fraction have no common factor other than 1, it is said to be in the simplestform.
Read Also: Arithmetic Progression Formula
Sample Questions
Ques. What fraction of a minute is 23 seconds? (2 marks)
Ans. There are 60 seconds in a minute
Then, required fraction is = 23/60
Ques. 35. Decimal 16.25 is equal to the fraction ______. (2 marks)
Ans. Decimal 16.25 is equal to the fraction 16 1/4 or 65/4.
Depending on the number of decimal places in the decimals, they can be converted to fractions by eliminating the decimal points and writing 10,100, etc. in the denominators.
16.25 = 1625/100
Divide both numerator and denominator by 25.
= 65/4
= 16 1/4
Ques. Write all the natural numbers between 1 to 11. What fraction of them are odd numbers? (2 marks)
Ans. The natural numbers between 1 to 11 are: 2, 3, 4, 5, 6, 7, 8, 9, 10
Among these odd numbers are: 3, 5, 7, 9
So, no. of odd numbers between 1 to 11 = 4
The fraction of odd numbers between 1 to 11 = no. of odd numbers between 1 to 11 / no. of natural numbers between 1 to 11
= 4/8
= 1/4
Ques. In the below diagram write the fractions showing the shaded portions (2 marks)
Ans.
- Numbers of Parts = 4
Shaded portion = 1
Fraction = 1/4
- Numbers of Parts = 6
Shaded portion = 2
Fraction = 2/6
Ques. Represent the below mentioned fractions on the number line (2 marks)
(a) 1/5 (b) 3/5
Ans. 
- Point A represents 1/5
- Point C represents 3/5
Ques. The value of 50 coins of 50 paisa = Rs.______. (3 marks)
Ans. The value of 50 coins of 50 paisa = Rs.25.
We know that, Rs.1 = 100 paisa
So, 50 coins of 50 paisa = 50 × 50
= 2500 paisa.
Then,
= 2500/100
= Rs. 25
Ques. Write any 3 proper fractions with denominator 7, 3 improper fractions with denominator 5 and 3 mixed fractions with denominator 3. (3 marks)
Ans. Proper fractions with denominator 7 are: 3/7 , 1/7 , 5/7
Improper fractions with denominator 5 are: 7/5 , 9/5 , 11/5
Mixed fractions with denominator 3 are: 2 5/3 , 5 7/3 , 4 7/3
Ques. Find the Error, if any (3 marks)
Ans.
- Since, the parts are not equal the shaded portion of a triangle is not 1/2
- Since, the parts are not equal the shaded portion of a rectangle is not 1/4
- Since, the parts are not equal the shaded portion of a circle is not 3/4
Ques. Find the sum of 1 2/3 and 3 2/5 (3 marks)
Ans.
1 2/3 + 3 2/5 = 1 + 2/3 + 3 + 2/5 = 1 + 3 + 2/3 + 2/5
= 4 + ( 2 x 5/3 x 5 ) + ( 2 x 3/5 x 3 ) = 4 + ( 10/15 + 6/15 )
= 4 + ( 16/15 ) = 5 1/15
Hence, 1 2/3 + 3 2/5 = 5 1/15
Ques. Pratham, Abhi and Vishal shared their lunch. Pratham has brought 2 pizzas, one made of vegetable and one of sauce. The other 2 boys forgot to bring their lunch. Pratham agreed to share his pizza so that each of his friend will have an equal share of each pizza.
(a) How can Pratham divide his pizzas so that each person has an equal share?
(b) What part of a pizzas will each boy receive? (5 marks)
Ans.
(a) Pratham has divided the pizzas into 3 equal parts. Hence, each person will get one part.
(b) Each boy receives 1/3 part.
Ques. Express the following mixed fraction in improper fraction. (5 marks)
(a) 9 1/4 (b) 7 3/4 (c) 2 2/7 (d) 5 3/2 (e) 1 5/4
Ans.
(a) (9 x 4 + 1) / 4 = 37/4
The improper fraction is 37/4
(b) (7 x 4 + 3) / 4 = 31/4
The improper fraction is 31/4
(c) (2 x 7 + 2) / 7 = 16/7
The improper fraction is 16/7
(d) (5 x 2 + 3) / 2 = 13/2
The improper fraction is 13/2
(e) (1 x 4 + 5) / 4 = 9/4
The improper fraction is 9/4
Ques. Write each of the following divisions in simplified fraction form. (5 marks)
(a) 6 ÷ 3
(b) 50 ÷ 100
(c) 125 ÷ 500
(d) 55 ÷ 11
(e) 6 ÷ 7
Ans.
(a) The division 6 ÷ 3 can be written as 2/1
(b) The division 50 ÷ 100 can be written as 1/2
(c) The division 125 ÷ 500 can be written as 1/4
(d) The division 55 ÷ 11 can be written as 5/1
(e) The division 6 ÷ 7 can be written as 6/7
Ques. Find answer to the following. (5 marks)
(a) Is 6/7 equal to 12/14 ?
(b) Is 4/3 equal to 8/5 ?
(c) Is 1/2 equal to 14/26 ?
(d) Is 10/100 equal to 1/10 ?
(d) Is 25/5 equal to 5/1 ?
Ans.
(a) Yes, 6/7 is equal to 12/14
(b) No, 4/3 is not equal to 8/5
(c) No, 1/2 is not equal to 14/26
(d) Yes, 10/100 is equal to 1/10
(e) Yes, 25/5 is equal to 5/1
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