Observe the following rational numbers.
⅔, ⅘, ⁻³⁄₇, ¼, ⁻⁶⁄₁₁, ⁻⁴⁷⁄₅₁
(i) Both numerator and denominator of all the rational numbers above are integers.
(i) The denominators of the rational numbers are all positive integers
(ii) 1 is the only common factor between the numerator and the denominator of each of them.
(iii) The negative sign occurs only in the numerator.
Such rational numbers are said to be in standard form.
A rational number is said to be in standard form, if its denominator is a positive integer and both the numerator and denominator have no common factor other than 1.
If a rational number is not in the standard form, then it can be simplified to arrive at the standard form.
Note :
In a rational number, if there is a negative sign in denominator, it can be moved to the numerator.
For example,
ᵃ⁄₋b = ⁻ᵃ⁄b
Examples 1-6 : Reduce the given rational number to its standard form.
Example 1 :
⁶⁄₈
Solution :
In the rational number 6/8, both 6 and 8 are even numbers, so they re evenly divisible by 2.
⁶⁄₈ = ⁽⁶ ÷ ²⁾⁄₍₈ ÷ ₂₎
= ¾
The standard form of ⁶⁄₈ is ¾.
Example 2 :
³⁄₄.₅
Solution :
In the rational number ³⁄₄.₅, the denominator is not an integer, it is a decimal number. To make the denominator 4.5 as an integer, multiply both numerator and denominator by 10. Since there is one digit after the decimal point in 4.5, we have to multiply by 10.
³⁄₄.₅ = ⁽³ ˣ ¹⁰⁾⁄₍₄.₅ ₓ ₁₀₎
= ³⁰⁄₄₅
= ⁽³⁰ ÷ ⁵⁾⁄₍₄₅ ÷ ₅₎
= ⁶⁄₉
= ⁽⁶ ÷ ³⁾⁄₍₉ ÷ ₃₎
= ⅔
The standard form of ³⁄₄.₅ is ⅔.
Example 3 :
⁴⁸⁄₋₈₄
Solution :
Method 1 :
In the rational number ⁴⁸⁄₋₈₄, since both the numerator and denominator are two digit numbers, we can do successive division by smaller numbers to make the process easier.
⁴⁸⁄₋₈₄ = ⁻⁴⁸⁄₈₄
= ⁻⁽⁴⁸ ÷ ²⁾⁄₍₈₄ ÷ ₂₎
= ⁻²⁴⁄₄₂
= ⁻⁽²⁴ ÷ ²⁾⁄₍₄₂ ÷ ₂₎
= ⁻¹²⁄₂₁
= ⁻⁽¹² ÷ ³⁾⁄₍₂₁ ÷ ₃₎
= ⁻⁴⁄₇
Method 2 :
⁴⁸⁄₋₈₄ = ⁻⁴⁸⁄₈₄
The highest common factor of 48 and 84 is 12. Thus, we can get its standard form by dividing it by 12.
= ⁻⁽⁴⁸ ÷ ¹²⁾⁄₍₈₄ ÷ ₁₂₎
= ⁻⁴⁄₇
The standard form of ⁴⁸⁄₋₈₄ is ⁻⁴⁄₇.
Example 4 :
⁻¹⁸⁄₋₄₂
Solution :
In the rational number -18/(-42), since both the numerator and denominator are negative even numbers, we can start dividing both -18 and -42 by -2.
⁻¹⁸⁄₋₄₂ = ⁽⁻¹⁸ ÷ ⁻²⁾⁄₍₋₄₂ ÷ ₋₂₎
= ⁹⁄₂₁
= ⁽⁹ ÷ ³⁾⁄₍₂₁ ÷ ₃₎
= ³⁄₇
The standard form of ⁻¹⁸⁄₋₄₂ is ³⁄₇.
Example 5 :
⁰^{.}⁴⁄₀.₆
Solution :
In the rational number 0.4/0.6, both the numerator and denominator are not integers, they are decimal numbers. To make both the numerator and denominator as integers, multiply both numerator and denominator by 10. Since there is one digit after the decimal point in both the numerator and denominator, we have to multiply by 10.
⁰^{.}⁴⁄₀.₆ = ⁽⁰^{.}⁴ ˣ ¹⁰⁾⁄₍₀.₆ ₓ ₁₀₎
= ⁴⁄₆
= ⁽⁴ ÷ ²⁾⁄₍₆ ÷ ₂₎
= ⅔
The standard form of ⁰^{.}⁴⁄₀.₆ is ⅔.
Example 6 :
⁰^{.}⁰³⁄₀.₉
Solution :
In the rational number 0.03/0.9, both the numerator and denominator are not integers, they are decimal numbers. Comparing the numerator 0.03 and denominator 0.9, there are more number of digits (two digits) after the decimal point in the numerator 0.03. Since there are two digits after the decimal point in numerator 0.03, we have to multiply both numerator and denominator by 100 to get rid of the decimal points in both.
⁰^{.}⁰³⁄₀.₉ = ⁽⁰^{.}⁰³ ˣ ¹⁰⁰⁾⁄₍₀.₉ ₓ ₁₀₀₎
= ³⁄₉₀
= ⁽³ ÷ ³⁾⁄₍₉₀ ÷ ₃₎
= ¹⁄₃₀
The standard form of ⁰^{.}⁰³⁄₀.₉ is ¹⁄₃₀.
Examples 7-10 : Write the given decimal number as a rational number in standard form.
Example 7 :
0.2
Solution :
Since there is one digit after the decimal point in 0.2, it can be written as a fraction with the denominator 10.
0.2 = ²⁄₁₀
= ⁽² ÷ ²⁾⁄₍₁₀ ÷ ₂₎
= ⅕
Example 8 :
0.05
Solution :
Since there are two digits after the decimal point in 0.05, it can be written as a fraction with the denominator 100.
0.05 = ⁵⁄₁₀₀
= ⁽⁵ ÷ ⁵⁾⁄₍₁₀₀ ÷ ₅₎
= ¹⁄₂₀
Example 9 :
0.001
Solution :
Since there are three digits after the decimal point in 0.001, it can be written as a fraction with the denominator 1000.
0.001 = ¹⁄₁₀₀₀
Example 10 :
0.502
Solution :
Since there are three digits after the decimal point in 0.502, it can be written as a fraction with the denominator 1000.
0.502 = ⁵⁰²⁄₁₀₀₀
= ⁽⁵⁰² ÷ ²⁾⁄₍₁₀₀₀ ÷ ₂₎
= ²⁵¹⁄₅₀₀
Example 11 :
In the standard form of a rational number, what is the common factor of numerator and denominator?
(A) 0
(B) 1
(C) -2
(D) 2
Solution :
In the standard form of any rational number, the common factor of numerator and denominator is always 1.
The correct answer is (B).
Example 12 :
The rational number 2/3 is in standard form, because
(A) both numerator and denominator are integers.
(B) the denominator is a positive integer
(C) the common factor of numerator and denominator is always 1
(D) all the above
Solution :
The correct answer is (D).
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