Fractions

A fraction shows part of a whole. This whole can be a region or a collection. The word fraction is derived from the Latin word 'fractio' which means 'to break'. The Egyptians, being the earliest civilization to study fractions, used fractions to resolve their mathematical problems, which included the division of food, supplies, and the absence of a bullion currency.

In Ancient Rome, fractions were only written using words to describe a part of the whole. In India, the fractions were first written with one number above another (numerator and denominator), but without a line known as the fraction bar. It was the Arabs, who added the line which is used to separate the numerator and the denominator. Let us learn more about fractions, and fraction examples, along with a few fraction practice problems.

Fractions, in Mathematics, are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing. Let us understand this concept using an example. The following figure shows a pizza that is divided into 8 equal parts. Now, if we want to express one selected part of the pizza, we can express it as 1/8 which shows that out of 8 equal parts, we are referring to 1 part.

It means one in eight equal parts. It can also be read as:

• One-eighth, or
• 1 by 8

If we select 2 parts of the pizza, it will be expressed as 2/8. Similarly, if we are referring to 6 parts of this pizza, we would write it as 6/8 as a fraction.

Fraction Definition

Fraction is defined as a part of something, and a quantity that is not a whole number. It is expressed as the number of equal parts being counted over the total number of parts in the whole.

Fraction Bar

Fraction bar is the line that is drawn to separate the numerator and the denominator. Let us learn more about the parts of a fraction in the following section.

All fractions consist of a numerator and a denominator and they are separated by a horizontal bar known as the fraction bar.

• The denominator indicates the number of parts in which the whole has been divided into. It is placed in the lower part of the fraction below the fractional bar.
• The numerator indicates how many sections of the fraction are represented or selected. It is placed in the upper part of the fraction above the fractional bar.

Based on the numerator and denominator, which are parts of a fraction, there are different types of fractions as listed below:

Proper Fraction

Proper fractions are the fractions in which the numerator is less than its denominator. For example, 5/7, 3/8, 2/5, and so on are proper fractions.

Improper Fraction

An improper fraction is the type of fraction in which the numerator is more than or equal to its denominator. It is always the same or greater than the whole. For example, 4/3, 5/2, 8/5, and so on.

Unit Fraction

Fractions in which the numerator is 1 are known as unit fractions. For example, 1/4, 1/7, 1/9, and so on.

Mixed Fraction

A mixed fraction is a mixture of a whole number and a proper fraction. For example, \(5\frac{1}{3}\), where 5 is the whole number and 1/3 is the proper fraction, or, \(2\frac{2}{5}\), \(7\frac{9}{11}\), and so on.

Equivalent Fraction

Equivalent fractions are the fractions that represent the same value after they are simplified. To get equivalent fractions of any given fraction:

• We can multiply both the numerator and the denominator of the given fraction by the same number.
• We can divide both the numerator and the denominator of the given fraction by the same number.

Example: Find the two fractions that are equivalent to 5/7.

Solution:

Equivalent Fraction 1: Let us multiply the numerator and the denominator with the same number 2. This means, 5/7= (5 × 2)/(7 × 2) = 10/14

Equivalent Fraction 2: Let us multiply the numerator and the denominator with the same number 3. This means, 5/7 = (5 × 3)/(7 × 3) = 15/21

Therefore, 10/14, 15/21, and 5/7 are equivalent fractions.

Like and Unlike Fractions

Like fractions are the fractions that have the same denominators. For example, 5/15, 3/15, 17/15, and 31/15 are like fractions.

Unlike fractions are the fractions which have different denominators. For example, 2/7, 9/11, 3/13, and 39/46 are unlike fractions.

The representation of fractions on a number line demonstrates the intervals between two integers, which also shows us the fundamental principle of fractional number creation. The fractions on a number line can be represented by making equal parts of a whole, i.e., from 0 to 1. The denominator of the fraction would represent the number of equal parts in which the number line will be divided and marked. For example, if we need to represent 1/8 on the number line, we need to mark 0 and 1 on the two ends and divide the number line into 8 equal parts. Then, the first interval can be marked as 1/8. Similarly, the next interval can be marked as 2/8, the next one can be marked as 3/8, and so on. It should be noted that the last interval represents 8/8 which means 1. Observe the following number line that represents these fractions on a number line.

Fraction Examples in Real Life

Let us know about a few fraction examples in real life.

• When we divide a cake into 3 equal parts, then each part is 1/3rd of the whole.
• We express the time as 'half an hour' which is a common way of expressing 30 minutes. Half is a fraction which is represented as 1/2.
• We represent the scores of tests as fractions, like 15/20, or, 7/20
• We use fractions while we use different recipes. For example, when we say 1/2 teaspoon of sugar or 3/4 tablespoon of salt.

A few fraction practice problems are given on this page so that the students can get an idea about the concept of fractions.

☛Related Articles

• Multiplication of Fractions
• Division of Fractions
• Addition and Subtraction of Fractions

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What are Fractions in Math?

Fractions in Math, represent a numerical value that expresses a part of a whole. The whole can be any number, a specific value, or a thing. Fractions are represented in the form of p/q. For example, ¼, ½, ¾, and so on.

What are the Different Types of Fractions?

Fractions are classified on the following basis:

• On the basis of the numerator and the denominator, they are categorized as proper fractions, improper fractions, mixed fractions.
• On the basis of groups they are categorized as like fractions, unlike fractions and equivalent fractions.

What are the Three Parts of a Fraction?

A fraction has two parts, the numerator and the denominator.

• Numerator: The numerator represents the number that is placed above the fractional bar. For example, in 6/7, 6 is the numerator.
• Denominator: The denominator indicates the number that is placed below the fractional bar. For example, in 6/7, 7 is the denominator.
• Fraction bar: Fraction bar is the line that is drawn to separate the numerator and the denominator.

What is 0.125 as a Fraction?

0.125 as a fraction can be written as, 1/8. We can convert the decimal to a fraction as follows. 0.125 = 125/1000 = 5/40 = 1/8

How are Fractions and Decimals Related?

Fractions and decimals are different ways to represent numbers. Fractions are written in the form of p/q, where q≠0, for example, 3/5; while in decimals, the whole number part and fractional part is connected with a decimal point, for example, 3.56. A fraction can be converted to a decimal if we divide the given numerator by the denominator. Similarly, to convert a decimal into a fraction, we write the given decimal as the numerator, we place a fractional bar below it. Then, we place 1 right below the decimal point followed by the number of zeros required accordingly. Then, this fraction can be simplified. For example, if we need to convert 0.5 to a fraction, we place 10 in the denominator and remove the decimal point which makes it 5/10. After reducing the fraction, we get (5 ÷ 5) / (10 ÷ 5) = 1/2.

How do you Simplify Fractions?

In order to simplify a fraction, we first write down the factors for the numerator and the denominator. Then, determine the largest factor that is common between the two and divide the numerator and denominator by the Greatest Common Factor (GCF). The reduced fraction that we get is the simplest form of the given fraction. For example, in order to simplify 36/45, we will find the GCF of 36 and 45. The GCF of 36 and 45 = 9. Now, we will divide the numerator and the denominator by 9, that is, (36 ÷ 9)/(45 ÷ 9) = 4/5

How to Multiply Fractions?

To multiply any two fractions, we first multiply the numerators, then multiply the denominators. Then, simplify the resultant fraction. For example, 3/5 × 15/18 = 45/90 = 1/2.

How to Divide Fractions?

To divide one fraction by another, we first write the reciprocal of the second fraction and then multiply the fractions. In other words, we multiply the first fraction with the reciprocal of the second fraction. After writing the reciprocal of the second fraction we multiply the fractions in the usual way. We multiply the numerators, and then multiply the denominators. Then, simplify the resultant fraction, if required. For example, 5/6 ÷ 1/5 = 5/6 × 5/1 = 25/6 = \(4\frac{1}{6}\)

What do you Call Fractions with the Same Denominator?

Fractions with the same denominator are known as like fractions. For example, 4/7, 3/7, 5/7 are like fractions because they have the same denominator.

How do you Determine Which Fraction is Greater?

To determine the greater fraction, we first need to check if the given fractions are like fractions. For this, we need to compare the denominators.

• In case of the same denominators, the fraction with the greater numerator is the greater fraction. For example, to compare 3/4 and 2/4 we can easily check the numerators and say that 3/4 > 2/4
• In the case of different denominators, we convert the given fractions to like fractions by writing a common denominator for them and then compare the numerators. For example, to compare 2/3 and 4/5, we will find the Least Common Multiple (LCM) of the denominators. Once the denominators are made the same, we can compare the fractions easily. The LCM of 3 and 5 is 15. Now, let us convert them in such a way that the denominators become the same. Let us multiply the first fraction 2/3 with 5/5, that is, 2/3 × 5/5 = 10/15. Now, let us multiply the second fraction 4/5 with 3/3 that is, 4/5 × 3/3 = 12/15. Compare the fractions: 10/15 and 12/15. Since the denominators are the same, we will compare the numerators, and we can see that, 12 > 10. The fraction with the larger numerator is the larger fraction, that is, 10/15 < 12/15. Therefore, 2/3 < 4/5.

Are All Fractions Less Than 1?

No, all fractions are not less than 1.

• Proper fractions are greater than 0 but less than 1. (The numerator is less than the denominator).
• Improper fractions are always 1 or greater than 1. (The numerator is greater than or equal to the denominator)

How to Add Fractions with Different Denominators?

In order to add fractions with different denominators, we need to use the following steps. Let us add the fractions 4/5 + 6/7

• Step 1: Since the denominators in the given fractions are different, we will find the LCM of 5 and 7 to make them the same. LCM of 5 and 7 = 35.
• Step 2: After this step, we will multiply 4/5 with 7/7, that is, (4/5) × (7/7) = 28/35, and 6/7 with 5/5, (6/7) × (5/5) = 30/35. This step converts them to like fractions that have the same denominators.
• Step 3: Now, the denominators are the same, so we can add the numerators and keep the common denominator. The new fractions with common denominators are 28/35 and 30/35. So, 28/35 + 30/35 = (28 + 30)/35 = 58/35 = \(1\frac{23}{35}\).

How to Multiply Fractions with Whole Numbers?

In order to multiply fractions with whole numbers, we write the whole number in the fraction form by placing 1 in the denominator and then we follow the usual procedure of multiplication of fractions. For example, let us multiply 5/8 × 3. Here 3 is a whole number and we will write it as 3/1. Now, let us multiply 5/8 × 3/1 = 15/8 = \(1\frac{7}{8}\)

What is Comparing Fractions?

Comparing fractions means finding the larger and the smaller fraction between any two or more fractions. For example, let us compare 3/16 and 7/16. We first observe the denominators of the given fractions: 3/16 and 7/16. Since the denominators are the same, we can compare the numerators. We can see that 3 < 7. The fraction with the larger numerator is the larger fraction. Therefore, 3/16 < 7/16. In case, if the fractions have different denominators, we will convert them to like fractions by finding the LCM of the denominators and writing the respective equivalent fractions. Once the denominators become the same, we can easily compare the numerators and spot the greater fraction.