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Standard form of a Rational Number

What is the standard form of a rational number?

A rational number ab is said to be in the standard form if b is positive, and the integers a and b have no common divisor other than 1.

How to convert a rational number into standard form?

In order to express a given rational number in the standard form, we follow the following steps:

Step I: Obtain the rational number.

Step II: See whether the denominator of the rational number is positive or not. If it is negative, multiply or divide numerator and denominator both by -1 so that denominator becomes positive.

Step III: Find the greatest common divisor (GCD) of the absolute values of the numerator and the denominator.

Step IV: Divide the numerator and denominator of the given rational number by the GCD (HCF) obtained in step III. The rational number so obtained is the standard form of the given rational number.

The following examples will illustrate the above procedure to convert a rational number into standard form.


1. Express each of the following rational numbers in the standard form:

(i) βˆ’924         (ii) βˆ’14βˆ’35         (iii) 27βˆ’72         (iv) βˆ’55βˆ’99

Solution: 

(i) βˆ’924

The denominator of the rational number βˆ’924 is positive. In order to express it in standard form, we divide its numerator and denominator by the greatest common divisor of 9 and 24 is 3.

Dividing the numerator and denominator of βˆ’924 by 3, we get

βˆ’924 = (βˆ’9)Γ·324Γ·3 = βˆ’38

Thus, the standard form of βˆ’924 is βˆ’38.


(ii) βˆ’14βˆ’35

The denominator of the rational number βˆ’14βˆ’35 is negative. So, we first make it positive.

Multiplying the numerator and denominator of βˆ’14βˆ’35 by -1 we get

βˆ’14βˆ’35 = (βˆ’14)Γ—(βˆ’1)(βˆ’35)Γ—(βˆ’1) = 1435

The greatest common divisor of 14 and 35 is 7.

Dividing the numerator and denominator of 1435 by 7, we get

1435 = 14Γ·735Γ·7 = 25

Hence, the standard form of a rational number βˆ’14βˆ’35  is 25.

(iii)  27βˆ’72

The denominator of 27βˆ’72 is negative. So, we first make it positive.

Multiplying the numerator and denominator of 27βˆ’72 by -1, we have

27βˆ’7227Γ—(βˆ’1)(βˆ’72)Γ—(βˆ’1) = βˆ’2772

The greatest common divisor of 27 and 72 is 9.

Dividing the numerator and denominator of βˆ’2772 by 9, we get

βˆ’2772(βˆ’27)Γ·972Γ·9 = βˆ’38

Hence, the standard form of  27βˆ’72 is βˆ’38.

(iv) βˆ’55βˆ’99

The denominator of βˆ’55βˆ’99 is negative. So, we first make it positive.

Multiplying the numerator and denominator of βˆ’55βˆ’99 by -1, we have

βˆ’55βˆ’99  = (βˆ’55)Γ—(βˆ’1)(βˆ’99)Γ—(βˆ’1)= 5599

The greatest common divisor of 55 and 99 is 11.

Dividing the numerator and denominator of by 5599 by 11, we get

559955Γ·1199Γ·11 = 59

Hence, the standard form of βˆ’55βˆ’99 is 59.

More examples on standard form of a rational number:

2. Express the rational number βˆ’247βˆ’228 in the standard form:

Solution:  

The denominator of βˆ’247βˆ’228 is negative. So, we first make it positive.

Multiplying the numerator and denominator of βˆ’247βˆ’228 by -1, we get

βˆ’247βˆ’228 = (βˆ’247)Γ—(βˆ’1)(βˆ’228)Γ—(βˆ’1) = 247228

Now, we find the greatest common divisor of 247 and 228.

247 = 13 Γ— 19 and 228 = 2 Γ— 2 Γ— 3 Γ— 19

Clearly, the greatest common divisor of 228 and 247 is equal to 19.

Dividing the numerator and denominator of 247228 by 19, we get

247228 = 247Γ·19228Γ·19 = 13/12

Hence, the standard form of βˆ’247βˆ’228 is 1312.


3. Express the rational number 299βˆ’161 in the standard form:

Solution:  

The denominator of 299βˆ’161 is negative. So we first make it positive.

Multiplying the numerator and denominator of 299βˆ’161 by -1, we get

299βˆ’161 = 299Γ—(βˆ’1)(βˆ’161)Γ—(βˆ’1) = βˆ’299161

Now, we find the greatest common divisor of 299 and 161:

299 = 13 Γ— 23 and 161 = 7 Γ— 23

Clearly, the greatest common divisor of 299 and 161 is equal to 23.

Dividing the numerator and denominator of βˆ’299161
by 23 we get

βˆ’299161 =  (βˆ’299)Γ·23161Γ·23 = βˆ’137

Hence, the standard form of a rational number 299βˆ’161 is βˆ’137.

● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers






8th Grade Math Practice 

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● Rational Numbers - Worksheets

Worksheet on Rational Numbers

Worksheet on Equivalent Rational Numbers

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Worksheet on Standard form of a Rational Number

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Objective Questions on Rational Numbers