Fraction - Definition, How to Learn & Examples

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The term fraction is derived from the Latin word ‘fractus’, which means, ‘broken’. It is one of the oldest concepts in mathematics (i.e. ever since the Egyptian civilisation). It represents a part of a whole. A fraction can be described as the number of equal parts of a certain size, forming the whole collection.

For example, consider a pizza that is cut into 8 slices of equal size. Then 3 slices of that pizza can be written as 3/8 or three-eights. Similarly, 6 slices can be written as 6/8 which when reduced to the lowest form becomes 3/4.

The number that forms the top part or is displayed before the slash is called numerator and a number written below the line or displayed to the right side of the slash is called denominator. The numerator tells how many equal parts of the whole or collection are taken. The denominator shows the total number of divisible equal parts which are there in a collection.

For example, there are 5 children.

(a) 3 out of 5 are girls. So, the fraction of girls is 3/5 or three-fifths.

(b) 2 out of 5 are boys. So, the fraction of boys is 2/5 or two-fifths.

Few things to note—

• The object must only be divided into equal parts for the concept of fractions to be stated true.
• There are different types of fractions which are explained in the second part of this article.
• Every number is a fraction. If it doesn't have a denominator that means, the denominator is 1 and it need not be mentioned. For example, 42/1 is commonly written as 42.
• Any fraction with a denominator of 0 is not valid. This will result in infinite.
• Fractions are a mode of comparison of numbers. It is used to represent ratios or division. Therefore 3/4 can also be written as 3:4 (three is to four) or 3÷4 (three divided by four).
• Both the numerator and the denominator must be integers, other than zero.
• Fractions must be reduced to the lowest format by dividing both the numerator and the denominator by a common integer, to the extent possible. For example—The numerator and the denominator of 15/35 can be divided by 5 to reduce it to 3/7.

In mathematics, we have different types of fractions. Let us understand what these types are and how to understand and solve these with examples.

Types of Fractions

1. Simple fractions— These fractions are also called common or vulgar fractions. It is a rational number written as a/b, where both a and b are integers and the denominator cannot be zero. Examples include 1/2, 5/7, 9/10 etc. Simple fractions can be positive or negative, and they can be proper or improper.

2. Proper fractions— When the numerator and the denominator are positive, the fraction is called proper if the numerator is less than the denominator. The absolute value of such fractions is strictly less than one. Examples of proper fractions are 2/3, 3/4, and 4/9.

3. Improper fractions— When the numerator and the denominator are positive, the fraction is called proper if the numerator is more than the denominator. It is also called top-heavy fractions. The absolute value of such fractions is greater than or equal to one. Examples of improper fractions are 9/4, 4/3, and 3/3.

4. Mixed fractions— Also known as compound fractions, it is a combination of a non-zero integer and a proper fraction (having the same sign). It is a way of representing an improper fraction.

For example, 21 (4/5) has a whole number part i.e. 21 and a proper fraction i.e. 4/5. These fractions can be converted into simple fractions. The numerator is [(denominator * whole number part)+ existing numerator] and the denominator remains the same. In our case 21 (4/5) can be written as [(5*21)+4]/5 which on solving becomes 109/5.

Note:-

Improper fractions can be converted into mixed fractions by dividing the numerator by the denominator. For example, 11/4 can be converted into a mixed fraction by keeping 4 as the divisor and 11 as the dividend.

4)11(2

8

——

3

——

The quotient becomes the whole number part i.e. 2, the divisor becomes the denominator i.e. 4, and the remainder is the numerator i.e. 3. So the mixed fraction looks like 2(3/4).

1. Equivalent fractions— When two or more fractions, when reduced to the simplest form, results in the same figure, the fractions are said to be equal to each other or equivalent fractions. For example:-

(a) 1/2 is equivalent to 5/10 or 25/50.

(b) 1/3 and 3/9 are equivalent fractions.

1. Unit fractions— A unit fraction is a fraction where the numerator is 1. For example, 1/7 or 1/10.

2. Like fractions— The fractions which have the same denominator figure are called like-fractions. For example 7/2, 3/2, 5/2 are like fractions.

3. Unlike fractions— The fractions which have different denominators are called, unlike-fractions. For example 1/11, 1/3, 1/2, 1/9 are unlike fractions.

Note:-

Some fraction types which are meant for higher-level discussions are as follows—

1. Complex fractions— A complex fraction is a fraction where either the numerator, or the denominator, or both contain a fraction. For example—(3/4)/5, 6/(8/9), or (8/5)/(1/3).

2. Partial fractions— It is a concept for higher-level algebraic operations where the numerator, or denominator, or both have polynomials. For example—(5x-4)/(x^2-x-2).

Coming to the third and the last part of the article, we will discuss some simple operations that could be performed on fractions. These operations are explained in a step by step approach for easy understanding. We encourage each and every reader to follow these steps properly.

How to Learn Fractions

Cuemath explains how to learn fractions using learning aids:

(A) With Same Denominators —

Step 1 — Add/Subtract the numerator and keep the denominator the same.

Step 2 — Then reduce the fraction to the lowest form.

Example 1 — Add 2/9 and 5/9.

Solution — (2+5)/9 = 7/9.

Example 2 — Subtract 8/9 from 11/9

Solution — (11-8)/9 = 3/9 = ⅓

(B) With Different denominators —

Method 1

Step 1 — Find the LCM of the numbers in the denominators of different fractions. This will be the denominator of the final answer.

Step 2 — Find how many times the denominator of the fractions forms the LCM.

Step 3 — Note down the multiple

Step 4 — Multiply numerators with multiples you got in step 3

Step 5 — Perform the desired operation on the numbers we got in step 4. This will form our numerator of the final answer.

Step 6 — Reduce to the lowest form

Method 2

Step 1 — Convert the fractions into decimals.

Step 2 — Perform the operation.

Step 3 — Convert the decimal back to fraction.

Method 3

Step 1 —  Make the denominator of the fractions the same

Step 2 — Proceed like Part (A)

Example 1 — Add 3/5 and 7/10.

Solution

Step 1 — The LCM of 5 and 10 is 10.

Step 2 — 5*2 forms 10 and 10*1 forms 10

Step 3 — Note down 2 and 1

Step 4 — (2*3) = 6 and (1*7) = 7

Step 5 — (6+7)/10 = 13/10 or 1 (3/10) or 1.3

Example 2 — Subtract 11/20 from 6/10

Solution

Step 1 — 6/10 can be written as 0.6 and 11/20 can be written as 0.55

Step 2 — 0.6 - 0.55 = 0.05

Step 3 — 0.05 can be written as 5/100 or 1/20.

Example 3 — Subtract 6/10 from 25/30

Solution

Step 1 — 6/10 can be written as 18/30

Step 2 — Perform (25-18)/30 = 7/30

Multiplication of Fractions--

Method 1--

Step 1 -- Multiply the numerators and multiply the denominators

Step 2 -- Reduce the result in the lowest form

Example 1 — Multiply 11/20 from 2/10

Solution

Step 1 — 11*2 = 22 and 20*10 = 200

Step 2 — 22/200 = 11/100 or 0.11

Method 2

Step 1 — Convert the fractions into decimals.

Step 2 — Perform the operation.

Step 3 — Convert the decimal back to fraction.

Example 2 — Multiply 2/4 from 5/7

Solution

Step 1 — 2/4 = 0.5 and 5/7 = 0.714

Step 2 — 0.5 * 0.714 = 0.3571

Step 3 -- 0.3571 can be written as 5/14

Division of Fractions--

Step 1 -- Keep the first fraction to be the same and reciprocate/inverse the second fraction.

Step 3 -- Perform multiplication.

Example 1 — Divide 3/4 from 2/7

Solution

Step 1 — 2/7 becomes 7/2

Step 2 — 3*7 = 21 and 4*2 = 8

Step 3 -- 21/8 is the final answer

This brings us to the end of this article. We hope this article was useful for our readers. Cuemath regularly comes up with articles like these to help its students. A step by step approach of understanding makes it easier for the students to grasp the concepts and remember them for their lifetime. Not only this, but Cuemath also provides its students with enough practice problems and assignments so that they can undergo a regular practice. We also give allocated time to the doubts of the students. We provide descriptive solutions to the students to understand each and every concept. Our maths gym app ensures that the students gain mental skills by solving various puzzles and quizzes which are available for Android and iOS users. We encourage our students to learn and develop.

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