Denominator

Introduction

A denominator is the number that appears below the horizontal line. In mathematics, a fraction is represented by a number that identifies a portion of a whole. A fraction is a component or section taken from a whole, which can be any number, a certain amount, or an object.

The denominator is the fraction's divisor. In a fraction, the denominator is the number or integer that lies below the horizontal line. A fraction's numerator is above the line. If a denominator is equal to 0, the result is an indefinite value.

In this tutorial, we will discuss the denominator, fraction, and its types.

Fractions

Fractions are expressed in mathematics as a numerical value that defines a part of the whole. This whole can be a number, region, or collection. The word fraction comes from the Latin word "fraction" which means "break".

Fractions are also known as parts or sections of any quantity.

For example − $\mathrm{\frac{1}{2}\:,\:\frac{2}{5}\:,\:\frac{1}{7}}$

Denominator

A fraction's denominator is the portion of the fraction that falls below the fractional bar. The denominator represents the total number of parts that comprise a whole.

A fraction is distinguished by a horizontal bar between two numbers and, on occasion, by the symbol "/."

The "fractional bar" is the name given to this bar or symbol. The number on top is referred to as the "numerator," and the number below the fractional bar is referred to as the "denominator."

Types of fractions

Suppose a fraction is written in the form of 𝑥, where x and y are part of the given fraction, where x denotes the numerator, and y denotes the denominator. Let see an example for better understanding.

For example − 1 is a fraction where 2 is a denominator and 1 is a numerator.

There are three types of fractions. That is the improper fraction, the proper fraction, and the mixed fraction.

Below is a brief description of each type.

• Proper Fractions − A proper fraction is one whose numerator is smaller than the denominator.

For example, $\mathrm{\frac{1}{8}}$ is a proper fraction because the numerator is less than the denominator.

• Improper Fractions − Improper fractions are those fractions whose numerator is larger than the denominator.

For example, $\mathrm{\frac{6}{5}}$ is an improper fraction because the numerator is greator than denominator.

• Mixed Fractions −

For example, $\mathrm{2\frac{1}{2}}$ which can be written as,

$$\mathrm{=\:\frac{(2\times\:2)\:+\:1}{2}\:=\:\frac{5}{2}}$$

Algebra of Fractions

Addition of fractions

There is a simple rule for adding fractions −

$$\mathrm{\frac{a}{b}\:+\:\frac{c}{d}\:=\:\frac{ad\:+\:bc}{bd}}$$

For example − $\mathrm{\frac{x}{3}\:+\:\frac{y}{4}\:=\:\frac{4x\:+\:3y}{12}}$

Subtraction of fractions

It is similar to the addition rule except for the signs.

$$\mathrm{\frac{a}{b}\:-\:\frac{c}{d}\:=\:\frac{ad\:-\:bc}{bd}}$$

For example − $\mathrm{\frac{x}{2}\:-\:\frac{y}{3}\:=\:\frac{3x\:-\:2y}{6}}$

Division of fractions

First, flip the fraction to be divided, and then use the same method as multiplication.

$$\mathrm{\frac{a}{b}\:\div\:\frac{d}{c}\:=\:\frac{a}{b}\times\:\frac{c}{d}\:=\:\frac{a\:.\:c}{b\:.\:d}}$$

Multiplication of fractions

Fractional multiplication is defined as the product of a particular fraction and a fraction, an integer, or a variable. To multiply a fraction, follow the following steps −

• Step 1 − Multiply the numerator by the numerator.

• Step 2 − Multiply the denominator by the denominator

• Step 3 − Simplify the fraction as needed.

$$\mathrm{\frac{a}{b}\:\times\:\frac{c}{d}\:=\:\frac{a\:.\:c}{b\:.\:d}}$$

$$\mathrm{Product\:of\:fraction\:=\:\frac{Product\:of\:numerator}{Product\:of\:the\:denominator}}$$

Decimal Expansion on the basis of the denominator

Terminating decimals, non-terminating repeating decimals, and non-terminating nonrepeating decimals are the three different categories of decimal expansion.

• Terminating decimals: − After a certain number of steps, the decimal expansion concludes. Terminating decimals is the name given to several sorts of decimal expansion. It implies that the numbers reach their maximum value following the decimal point.

As an illustration, the decimal expansion of the rational integer $\mathrm{\frac{1}{2}\:is\:0.5.}$

• Non-terminating repeating decimals − Non-terminating decimals have an infinite number of digits and an unbounded decimal expansion. The decimals that evenly repeat a particular number of digits after the decimal point are known as repeating decimals.

1.454545 is an illustration of a non-terminating and repeating decimal. After the decimal point, the number 45 appears repeatedly.

• Non-terminating non-repeating decimals − One sort of decimal expansion involves non-terminating and non-repeating decimals, in which neither the number following the decimal point, nor the decimal numbers repeat.

3.34765………. is an illustration of a non-terminating and non-terminating decimal. The numbers following the decimal point, in this case, are infinite and do not repeat.

Solved Examples

1) Simplify the following equation $\mathrm{\frac{2}{4}\:\times\:\frac{2}{3}}$

Answer − The given equation is $\mathrm{\frac{2}{4}\:\times\:\frac{2}{3}}$

$$\mathrm{=\:\frac{4}{12}}$$

$$\mathrm{=\:\frac{1}{3}}$$

2) Multiply $\mathrm{\frac{8}{3}\:with\:\frac{3}{8}}$

Answer − According to the question we have to find

$$\mathrm{\frac{8}{3}\:\times\:\frac{3}{8}}$$

$$\mathrm{=\:\frac{24}{24}}$$

$$\mathrm{=\:1}$$

3) Simplify the following equation $\mathrm{\frac{4}{4}\:\times\:\frac{8}{2}\:+\:\frac{8}{4}\:\times\:\frac{4}{8}}$

Answer − The given equation is $\mathrm{\frac{4}{4}\:\times\:\frac{8}{2}\:+\:\frac{8}{4}\:\times\:\frac{4}{8}}$

$$\mathrm{=\:\frac{32}{8}\:+\:\frac{32}{32}}$$

$$\mathrm{=\:4\:+\:1}$$

$$\mathrm{}=\:5$$

4) $\mathrm{2\:-\:\frac{1}{4}\:+\:\frac{1}{2}\:\times\:\frac{1}{4}}$

Answer − $\mathrm{\:\:\:2\:-\:\frac{1}{4}\:+\:\frac{1}{4}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:2\:-\:\frac{1}{8}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{15}{8}}$

5) Riya went to a restaurant and ordered a pizza. Each piece of pizza represents a component of a larger whole. The pizza is cut into six equal slices. If she ate one slice, express this fact as a fraction and write the fraction's denominator.

Answer − Total number of pizza slices = 6

Riya consumed a total of one piece.

She consumed one-sixth of a pizza i.e $\mathrm{\frac{1}{6}}$

In this case, 6 represents the total number of pizza slices. As a result, the denominator is 6.

6) A grandmother purchased an apple for her grandson. She divided it into four pieces. If she gives the child one piece of an apple, express this fact as a fraction and write the fraction's denominator

Answer − The total number of apple pieces is four. If the child is given one piece of an apple, the fraction for this fact will be expressed as $\mathrm{\frac{1}{4}}$

As a result, the denominator is 4, which represents the total number of apple pieces.

Conclusion

• Fractions are expressed in mathematics as a numerical value that defines a part of the whole. This whole can be a number, region, or collection. The word fraction comes from the Latin word "fraction" which means "break".

• Fractions are also known as parts or sections of any quantity.

• A fraction's denominator is the number that appears below the horizontal line. The bottom number of a fraction represents the total number of equal parts into which an object is divided.

FAQs

1. What do you mean by fractions?

Fractions are expressed in mathematics as a numerical value that defines a part of the whole. Fractions are also known as parts or sections of any quantity.

2. What is a proper fraction?

A proper fraction is one whose numerator is smaller than the denominator

For example $\mathrm{\frac{3}{8}}$

3. What is an improper fraction?

Improper fractions are those fractions whose numerator is larger than the denominator

For example $\mathrm{\frac{7}{5}}$

4. What do you mean by mixed fractions?

A mixed fraction is a combination of an integer part and a proper fraction

For example $\mathrm{2\frac{1}{2}}$

5. How many types of fractions are in mathematics?

There are three types of fractions improper, proper, and mixed fractions