Operations on Rational Numbers

Introduction

Operations on rational numbers such as basic operations like addition, subtraction, multiplication and division. Rational numbers are fractions containing a numerator and a denominator (the denominator should be a non-zero integer). In algebra, rational numbers are used in various science, mathematics, and economics sectors. Moreover, various operations such as addition, subtraction, division, and multiplication are necessary to obtain a third rational number. All these operations are quite different from the algebraic operations of integers. In this tutorial, we will discuss number systems, rational numbers, comparison between rational numbers, and their various operations with solved examples.

Number System

A number is a mathematical term with some value used to count or measure objects. Various numbers are included in mathematics, such as natural numbers, integers, rational numbers, and irrational numbers. However, the number system is defined as the representation of numbers in various forms that computers can understand. The number system is also known as the numeral system.

There are four types of number systems in algebra such as

• Decimal (base-10) − In this system, a total of 10 digits, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

• Binary (base-2) − It uses only two digits, such as 0 and 1, to represent any number

• Octal (base-8) − It uses 8 digits, such as 0, 1, 2, 3, 4, 5, 6, and 7, to represent any number.

• Hexadecimal (base-16) − It uses 16 digits or alphabets, including 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and A, B, C, D, E, F.

Rational Numbers

In mathematics, the rational number is one type of number known as a fractional number containing two terms, such as one numerator and the other one denominator.

The rational numbers can be expressed as "𝑝", where p and q refer to the numerator and denominator, respectively, and both are integers. The necessary condition of the rational number is that the denominator should be a non-zero integer.

In real life, rational numbers are used in various ways, as below.

• Dividing pizza slices equally

• Evaluating the discounted price of a product or material

• Splitting a bill at a restaurant

• Expressing the marks obtained in an examination

• Calculate the interest amount on investments

In addition, there are four types of rational numbers used in algebra.

• Integers − Example, -10, 0, 6, etc.

• Fractional numbers − For example $\mathrm{\frac{3}{4}\:,\:\frac{9}{14}\:etc}$

• Terminating decimals − Example, 0.1, 0.2568, 0.47, etc.

• Non-terminating decimals − Example, 0.333…, 0.141414…, etc.

Conversion of Rational Numbers into Fractions

Each rational number can be represented may or may not be in the form of $\mathrm{\frac{p}{q}}$ Hence, those rational numbers can be converted to fractions.

• Non-terminating decimals − The non-terminating decimals can be converted into fractions using the following formula

$$\mathrm{0.\overline{pqrs}\:=\:\frac{Repeated\:term}{Number\:of\:9's\:for\:the\:repeated\:term}}$$

For example, $\mathrm{\overline{0.3}\:=\:\frac{3}{9}\:,\:\overline{0.125}\:=\:\frac{125}{999}}$

• Terminating decimals − In this case, write all the integers in the numerator place, and the denominator will be a power of 10. The exponent or power of base 10 depends on the integers present left of the decimal.

For example, $\mathrm{0.4567\:=\:\frac{4567}{10^{4}}\:=\:\frac{4567}{10000}}$.

Comparing Rational Numbers

The comparison between two rational numbers can be carried out as per the following instructions.

• If the denominators are the same, then compare numerators. The maximum value of the numerator will be the greater rational number among them.

For example, compare $\mathrm{\frac{2}{15}\:and\:\frac{11}{15}\:Since\:11\:>\:2\:,\:therefore\:\frac{11}{15}\:>\:\frac{2}{15}}$.

• If the denominators are not the same, then find the LCM or lowest common multiple of the denominators of two rational numbers. Now, multiply the numerator and denominator of each fraction in such a way that the denominator will be the LCM value. Now, we can easily compare those numbers since the denominators of two fractions become equal.

For example, compare $\mathrm{\frac{2}{5}\:and\:\frac{7}{3}}$ The LCM of 5 and 3 is 15. Now, $\mathrm{\frac{2\times\:3}{5\times\:3}\:=\:\frac{6}{15}\:and\:\frac{7\:\times\:5}{3\times\:5}\:,\:Among\:\frac{6}{15}\:and\:\frac{35}{15}\:,\:\frac{35}{15}\:is\:greater\:then\:\frac{6}{15}\:.\:Hence\:,\:\frac{7}{3}\:>\:\frac{2}{5}}$

Operations on Rational Numbers

The four basic algebraic operations, such as addition, subtraction, multiplication, and divisions of integers, are explained below.

Addition of rational numbers:

There are two cases in addition to rational numbers.

• Rational numbers having like or the same denominator − In this case, we will add all the numerators and write the common denominator.

For example, $\mathrm{\frac{3}{8}\:+\:\frac{7}{8}\:=\:\frac{3\:+\:7}{8}\:=\:\frac{10}{8}}$

• Rational numbers with unequal or different denominators − Find the LCM or lowest common multiple of the denominators of two rational numbers. Now, multiply the numerator and denominator of each fraction in such a way that the denominator will be the LCM value. Now, add all the numerators and write the common denominator.

For example, $\mathrm{\frac{4}{7}\:+\:\frac{9}{2}}$ .The LCM of 7 and 2 is 14.

Hence,$\mathrm{\frac{4}{7}\:+\:\frac{9}{2}\:=\:\frac{4\times\:2}{7\times\:2}\:+\:\frac{9\times\:7}{2\times\:7}\:=\:\frac{8}{14}\:\:+\:\frac{63}{14}\:=\:\frac{8\:+\:63}{14}\:=\:\frac{71}{14}}$

Subtraction of rational numbers:

The procedure of subtraction of rational numbers is the same as that of addition.

For example,$\mathrm{\frac{3}{5}\:-\:\frac{19}{5}\:=\:\frac{3\:-\:19}{5}\:=\:\frac{3\:-\:19}{5}\:=\:\frac{-16}{5}}$

Multiplication of rational numbers:

Let’s consider the multiplication of $\mathrm{\frac{p}{q}\:and\:\frac{r}{s}}$

• Multiply the numerator of one fraction by the numerator of another. In this case, $\mathrm{p\times\:q}$

• Multiply the denominator of one fraction by the denominator of another. In this case,$\mathrm{q\times\:s}$

• Write the multiplication results of numerators and denominators in their corresponding places and simplify the fraction (if any). In this case,$\mathrm{\frac{p\times\:r}{q\times\:s}}$

Division of rational numbers:

During the division of rational numbers, the second fraction gets inversed. Then, the multiplication operation is performed.

For example , $\mathrm{\frac{3}{2}\:\div\:\frac{9}{4}\:=\:\frac{3}{2}\:\times\:\frac{4}{9}\:=\:\frac{3\times\:4}{2\times\:9}\:=\:\frac{12}{18}}$

Solved Examples

1)Evaluate the difference between $\mathrm{-\:\frac{9}{10}\:and\:\frac{2}{3}}$

Answer − The denominators of both rational numbers are not the same.

Hence, the LCM of 10 and 3 is 30.

$$\mathrm{(-\frac{9}{10})\:-\:(\frac{2}{3})\:=\:(-\frac{9\times\:3}{10\times\:3})\:-\:(\frac{2\times\:10}{3\times\:10})}$$

$$\mathrm{=\:(-\frac{27}{30})\:-\:(\frac{20}{30})}$$

$$\mathrm{=\:\frac{-27\:-\:20}{30}\:=\:\frac{-47}{30}}$$

∴ The difference between $\mathrm{-\frac{9}{10}\:and\:\frac{2}{3}\:is\:\frac{-47}{30}}$

2)Geeta uses $\mathrm{\frac{4}{7}}$ of the flour if she has to make a pancake. How much flour will she use to bake $\mathrm{\frac{1}{3}}$ portion of the pancake?

Answer − It is given that,

Total flour consumed to make a pancake $\mathrm{=\:\frac{4}{7}}$

Now, the amount of flour used to make $\mathrm{\frac{1}{3}}$ portion of the pancake is $\mathrm{=\:\frac{4}{7}\:\times\:\frac{1}{3}}$

$$\mathrm{=\:\frac{4\times\:1}{7\times\:3}\:=\:\frac{4}{21}}$$

∴ Geeta has to use $\mathrm{\frac{4}{21}}$ portion of the flour.

3)Evaluate $\mathrm{\frac{10}{17}\:\div\:\frac{1}{2}}$

Answer − $\mathrm{\frac{10}{17}\:\div\:\frac{1}{2}\:=\:\frac{10}{17}\:\times\:\frac{2}{1}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{10\times\:2}{17\times\:1}\:=\:\frac{20}{17}}$

Conclusion

The present tutorial gives a brief introduction to the basic concept of rational numbers and their various operations. The basic definition of the number system, and rational numbers, comparison of rational numbers, and operations on rational numbers have been described in this tutorial. Moreover, some solved examples have been provided for better clarity of this concept. In conclusion, the present tutorial may be useful for understanding the basic concept of operations on rational numbers.

FAQs

1. What are the additive and multiplicative identities?

Zero and one are known as an additive and multiplicative identities. That means the addition of any rational number with 0 results in itself. Similarly, the multiplication of any rational number with 1 is the same number.

2. Can a rational number be represented on a number line?

Yes, rational numbers can also be represented on a number line.

3. Which algebraic operations satisfy commutative and associative properties for rational numbers?

The addition and multiplication of rational numbers obey the commutative and associative laws.

4. What are the properties of the multiplication of rational numbers?

There are several properties of the multiplication of rational numbers, which are described below.

• If the order of multiplication is reversed, then the result remains unaffected.

• The multiplication of any rational numbers with 1 gives the same rational numbers.

• The multiplication of any rational numbers with 0 gives 0.

5. The division of rational numbers obeys the inverse property. Is the statement true?

No. The division of the rational numbers does not satisfy the inverse property.