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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) β924 (ii) β14β35 (iii) 27β72 (iv) β55β99
Solution:
(i) β924
The denominator of the rational number β924 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 β924 by 3, we get
β924 = (β9)Γ·324Γ·3 = β38
Thus,
the standard form of β924 is β38.
(ii) β14β35
The denominator of the rational number β14β35 is negative. So, we first make it positive.
Multiplying the numerator and denominator of β14β35 by -1 we get
β14β35 = (β14)Γ(β1)(β35)Γ(β1) = 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 β14β35
is 25.
(iii) 27β72
The
denominator of 27β72 is negative. So, we first make it positive.
Multiplying the numerator and denominator of 27β72 by -1, we have
27β72 = 27Γ(β1)(β72)Γ(β1) = β2772
The greatest common divisor of 27 and 72 is 9.
Dividing the numerator and denominator
of β2772 by 9, we get
β2772 = (β27)Γ·972Γ·9 = β38
Hence, the standard form of 27β72 is β38.
(iv) β55β99
The denominator of β55β99 is negative. So, we first
make it positive.
Multiplying
the numerator and denominator of β55β99 by -1, we have
β55β99 = (β55)Γ(β1)(β99)Γ(β1)= 5599
The greatest common divisor of 55 and 99 is 11.
Dividing the numerator and denominator of by 5599 by 11, we get
5599 = 55Γ·1199Γ·11 = 59
Hence, the standard form of β55β99 is 59.
More examples on standard form of a rational number:
2. Express the rational number β247β228 in the standard form:
Solution:
The denominator of β247β228 is negative. So, we first make it positive.
Multiplying the numerator and denominator of β247β228 by -1, we get
β247β228 = (β247)Γ(β1)(β228)Γ(β1) = 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 β247β228 is 1312.
3. Express the rational number 299β161 in the standard form:
Solution:
The denominator of 299β161 is negative. So we first make it positive.
Multiplying the numerator and denominator of 299β161 by -1, we get
299β161 = 299Γ(β1)(β161)Γ(β1) = β299161
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 β299161
by 23 we get
β299161 = (β299)Γ·23161Γ·23 = β137
Hence, the standard form of a rational number 299β161 is β137.
β Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
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
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
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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β Rational Numbers - Worksheets
Worksheet on Equivalent Rational Numbers
Worksheet on Lowest form of a Rational Number
Worksheet on Standard form of a Rational Number
Worksheet on Equality of Rational Numbers
Worksheet on Comparison of Rational Numbers
Worksheet on Representation of Rational Number on a Number Line
Worksheet on Adding Rational Numbers
Worksheet on Properties of Addition of Rational Numbers
Worksheet on Subtracting Rational Numbers
Worksheet on Addition and
Subtraction of Rational Number
Worksheet on Rational Expressions Involving Sum and Difference
Worksheet on Multiplication of Rational Number
Worksheet on Properties of Multiplication of Rational Numbers
Worksheet on Division of Rational Numbers
Worksheet on Properties of Division of Rational Numbers
Worksheet on Finding Rational Numbers between Two Rational Numbers
Worksheet on Word Problems on Rational Numbers
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