FRACTIONS: ITS TYPES & USES IN MATHEMATICS

What are the Fractions?

In Maths, a portion is characterized as a piece of the entire thing. When an apple is cut into four equal pieces, for instance, each piece is represented by ¼. Ratios and divisions are also represented by fractions. 

The term "fractional part" refers to the numerator and denominator of a fraction. On the other hand, a fraction may occasionally include both a whole number and a fractional part. Similarly, the numerator and denominator of positive common fractions are natural numbers. 

Additionally, the denominator tells how many equal parts make up a unit or a whole, while the numerator represents the number of equal parts. Since there can never be zero parts in a whole, the denominator cannot be zero. 

For instance, in the fraction  ¼, the denominator is 4 and the numerator is 1, respectively, indicating that the fraction consists of four parts that are equal to one another. 

Which are the Main Types of Fractions?

The common fraction and the decimal fraction are the two main forms of fractions. Further subdivided into distinct forms are these two forms. 

1. Common Fraction: 

A number that stands for a rational number is called a common fraction. The same number can also be expressed as a negative exponent, a percent, or a decimal. 

For instance, 0.05, 5%, and 10⁻³ are equivalent to the portion 1/1000. Similarly, Proper Fractions, Improper Fractions, and Mixed Fractions are the three subclasses of the common fraction.

• Proper Fractions: A proper fraction is one in which the denominator is greater than the numerator (¼).
• Improper Fractions: An improper fraction is one in which the numerator is greater than the denominator (5/3).
• Mixed Fractions: We have a whole number and a fractional part in a mixed fraction (1¼).

2. Decimal Fraction: 

A decimal part is a division whose denominator isn't given expressly, however, is perceived to be a whole number power of ten. 

Decimal divisions are regularly communicated involving the decimal documentation in which the suggested not entirely settled by the number of digits to one side of a decimal separator.

In this way, for 0.75 the numerator is 75 and the suggested denominator is 10 to the subsequent power, in particular, 100 since there are two digits to one side of the decimal separator.

Note: Decimal Fractions are also changed/converted into mixed or common fractions and vice versa. 

Example:    

 3.75             (Decimal Fraction)          

 375/100       (Improper Fraction)   

.           (Mixed Fraction)

What are the Complex Fractions?

In a perplexing portion or a complex fraction, either the numerator, the denominator, or both, is a small portion of a blended number, compared to the division of parts. To decrease a mind-boggling portion to a basic part, treat the longest part line as addressing division. 

For instance:

${\displaystyle {\frac {12{\tfrac {3}{4}}}{26}}=12{\tfrac {3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {12\cdot 4+3}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{4}}\cdot {\tfrac {1}{26}}={\tfrac {51}{104}}}$

${\displaystyle {\frac {\tfrac {3}{2}}{5}}={\tfrac {3}{2}}\times {\tfrac {1}{5}}={\tfrac {3}{10}}}$

${\displaystyle {\frac {8}{\tfrac {1}{3}}}=8\times {\tfrac {3}{1}}=24.}$

What are the Compound Fractions?

A compound fraction is a fractional part drawn from another fraction (Complex Fraction). In other words, when we further divide a complex fraction into more parts, we call those parts compound fractions.  

For instance,

 ${\displaystyle {\tfrac {3}{4}}}$ of ${\displaystyle {\tfrac {5}{7}}}$ is a compound fraction, corresponding to ${\displaystyle {\tfrac {3}{4}}\times {\tfrac {5}{7}}={\tfrac {15}{28}}}$. 

Similarly, the compound fraction  is equivalent to the complex fraction.

Uses of Fractions in Mathematics 

Fractions are used in various mathematical concepts and applications, including:

1. Representation of Rational Numbers:  Fractions are a way to represent numbers that are not whole, such as 1/2, 3/4, or -2/5.

2. Arithmetic Operations: Fractions are involved in addition, subtraction, multiplication, and division of rational numbers. For example, adding 1/4 and 1/3 involves fractions: 1/4 + 1/3 = 7/12.

3. Comparing Quantities: Fractions are used to compare quantities or numbers. For instance, you can compare 3/4 to 2/3 to determine which is larger.

4. Proportions and Ratios: Fractions are essential in expressing proportions and ratios, which are used in various real-world scenarios, such as cooking, finance, and engineering.

5. Decimals to Fractions: Converting decimals to fractions helps in understanding the relationship between the two number forms. For example, 0.75 is equivalent to 3/4.

6. Probability: Probability involves fractions. For instance, the probability of rolling a 6 on a fair six-sided die is 1/6.

7. Geometry: Fractions are used in geometry to represent parts of a whole. For example, 1/2 represents half of a circle's circumference.

8. Measurement: Fractions are used when measuring, especially with non-standard units. For example, 1/4 inch represents a quarter of an inch.

9. Scaling: When you scale an object or quantity, fractions come into play. Doubling something means multiplying it by 2, which can be expressed as a fraction: 2/1.

10. Algebra: Fractions appear in algebraic equations, inequalities, and expressions. Solving equations often requires working with fractions.

11. Statistics: In statistics, fractions are used when calculating probabilities, percentages, and proportions in data analysis.

12. Engineering and Science: Fractions are crucial in various scientific and engineering calculations, such as fluid dynamics, electrical circuits, and chemical reactions.

These are just a few examples of how fractions are used extensively in mathematics and various real-world applications. They provide a flexible and precise way to represent and work with non-integer quantities.

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