Questions 1-10 : Reduce the given rational number to its standard form.
Question 1 :
²⁄₄
Question 2 :
⁶⁄₁₅
Question 3 :
²⁵⁄₃₅
Question 4 :
⁻⁹⁄₂₄
Question 5 :
⁶⁄₆.₃
Question 6 :
⁶⁰⁄₋₉₆
Question 7 :
⁻³⁴⁄₋₅₁
Question 8 :
⁰^{.}⁶⁄₀.₈
Question 9 :
⁰^{.}⁰⁰²⁄₀.₀₄
Question 10 :
⁰^{.}¹⁸⁄₁.₅₆
Examples 11-14 : Write the given decimal number as a rational number in standard form.
Question 11 :
0.5
Question 12 :
0.02
Question 13 :
0.0025
Question 14 :
0.104
Question 15 :
In the standard form of a rational number, the common factor of numerator and denominator is always :
(A) 0
(B) 1
(C) -2
(D) 2
Question 16 :
Which of the following determines the rational number 5/7 in standard form?
(A) Both the numerator 5 and denominator 7 are integers.
(B) The denominator 7 is a positive integer.
(C) The greatest common divsior of numerator 5 and denominator 7 is 1.
(D) All the above
1. Answer :
In the rational number ²⁄₄, both 2 and 4 are even numbers. So they are evenly divisible by 2.
²⁄₄ = ⁽² ÷ ²⁾⁄₍₄ ÷ ₂₎
= ½
The standard form of the rational number ²⁄₄ is ½.
2. Answer :
In the rational number ⁶⁄₁₅, both 6 and 15 are multiples of 3.. So they are evenly divisible by 3.
⁶⁄₁₅ = ⁽⁶ ÷ ³⁾⁄₍₁₅ ÷ ₃₎
= ⅖
The standard form of the rational number ⁶⁄₁₅ is ²⁄₅.
3. Answer :
In the rational number ²⁵⁄₃₅, both 15 and 35 are multiples of 5. So, they are evenly divisible by 5.
²⁵⁄₃₅ = ⁽²⁵ ÷ ⁵⁾⁄₍₃₅ ÷ ₅₎
= ⁵⁄₇
The standard form of the rational number ²⁵⁄₃₅ is ⁵⁄₇.
4. Answer :
In the rational number ⁻⁹⁄₂₄, both 9 and 21 are multiples of 3. So both are evenly divisible by 3.
⁻⁹⁄₂₄ = ⁻⁽⁹ ÷ ³⁾⁄₍₂₄ ÷ ₃₎
= ⁻³⁄₈
The standard form of the rational number ⁻⁹⁄₂₄ is ⁻³⁄₈.
5. Answer :
In the rational number ⁶⁄₆.₃, the denominator 6.3 is not an integer, it is a decimal number. Since there is one digit after the decimal point in 6.3, we have to multiply both numerator and denominator by 10 to get rid of the decimal point in 6.3.
⁶⁄₆.₃ = ⁽⁶ ˣ ¹⁰⁾⁄₍₆.₃ ₓ ₁₀₎
= ⁶⁰⁄₆₃
= ⁽⁶⁰ ÷ ³⁾⁄₍₆₃ ÷ ₃₎
= ²⁰⁄₂₁
The standard form of the rational number ⁶⁄₆.₃ is ²⁰⁄₂₁.
6. Answer :
Method 1 :
⁶⁰⁄₋₉₆ = ⁻⁶⁰⁄₉₆
In the rational number -60/96, both the numerator and denominator are two digit numbers. To make the process easier, we can do successive division by smaller numbers.
= ⁻⁽⁶⁰ ÷ ²⁾⁄₍₉₆ ÷ ₂₎
= ⁻³⁰⁄₄₈
= ⁻⁽³⁰ ÷ ²⁾⁄₍₄₈ ÷ ₂₎
= ⁻¹⁵⁄₂₄
= ⁻⁽¹⁵ ÷ ³⁾⁄₍₂₄ ÷ ₃₎
= ⁻⁵⁄₈
Method 2 :
⁶⁰⁄₋₉₆ = ⁻⁶⁰⁄₉₆
The greatest common divisor of 60 and 96 is 12. We can get the standard form -60/96 by dividing both numerator and denominator by 12.
= ⁻⁽⁶⁰ ÷ ¹²⁾⁄₍₉₆ ÷ ₁₂₎
= ⁻⁵⁄₈
The standard form of the rational number ⁶⁰⁄₋₉₆ is ⁻⁵⁄₈.
7. Answer :
In the rational number ⁻³⁴⁄₋₅₁, both the numerator and denominator are negative integers and also multiples of 34. So, we can get its standard form by dividing both -34 and -51 by -17.
⁻³⁴⁄₋₅₁ = ⁽⁻³⁴ ÷ ⁻¹⁷⁾⁄₍₋₅₁ ÷ ₋₁₇₎
= ⅔
The standard form of the rational number ⁻³⁴⁄₋₅₁ is ⅔.
8. Answer :
In the rational number ⁰^{.}⁶⁄₀.₈, both the numerator and denominator are not integers, they are decimal numbers. Since there is one digit after the decimal point in both the numerator and denominator, multiply both by 10 to get rid of the decimal point in both numerator and denominator.
⁰^{.}⁶⁄₀.₈ = ⁽⁰^{.}⁶ ˣ ¹⁰⁾⁄₍₀.₈ ₓ ₁₀₎
= ⁶⁄₈
= ⁽⁶ ÷ ²⁾⁄₍₈ ÷ ₋₂₎
= ¾
The standard form of the rational number ⁰^{.}⁶⁄₀.₈ is ¾.
9. Answer :
In the rational number ⁰^{.}⁰⁰²⁄₀.₀₄, both the numerator and denominator are not integers, they are decimal numbers. If the numerator 0.002 and denominator 0.04 are compared, there are more number of digits (three digits) after the decimal point in the numerator 0.002. Since there are three digits after the decimal point in numerator 0.002, multiply both numerator and denominator by 1000 to get rid of the decimal points in both.
⁰^{.}⁰⁰²⁄₀.₀₄ = ⁽⁰^{.}⁰⁰² ˣ ¹⁰⁰⁰⁾⁄₍₀.₀₄ ₓ ₁₀₀₀₎
= ²⁄₄₀
= ⁽² ÷ ²⁾⁄₍₄₀ ÷ ₂₎
= ¹⁄₂₀
The standard form of the rational number ⁰^{.}⁰⁰²⁄₀.₀₄ is ¹⁄₂₀.
10. Answer :
In the rational number ⁰^{.}¹⁸⁄₁.₅₆, both the numerator and denominator are not integers, they are decimal numbers. Since there are two digits after the decimal point in both the numerator 0.18 and denominator 1.53, multiply both numerator and denominator by 100 to get rid of the decimal points in both.
⁰^{.}¹⁸⁄₁.₅₆ = ⁽⁰^{.}¹⁸ ˣ ¹⁰⁰⁾⁄₍₁.₅₆ ₓ ₁₀₀₎
= ¹⁸⁄₁₅₆
= ⁽¹⁸ ÷ ²⁾⁄₍₁₅₆ ÷ ₂₎
= ⁹⁄₇₈
= ⁽⁹ ÷ ³⁾⁄₍₇₈ ÷ ₃₎
= ³⁄₂₆
The standard form of the rational number ⁰^{.}¹⁸⁄₁.₅₆ is ³⁄₂₆.
11. Answer :
In 0.5, there is one digit after the decimal point. So, it can be written as a fraction with the denominator 10.
0.5 = ⁵⁄₁₀
= ⁽⁵ ÷ ⁵⁾⁄₍₁₀ ÷ ₅₎
= ½
12. Answer :
In 0.02, there are two digits after the decimal point. So, it can be written as a fraction with the denominator 100.
0.02 = ²⁄₁₀₀
= ⁽² ÷ ²⁾⁄₍₁₀₀ ÷ ₂₎
= ¹⁄₅₀
13. Answer :
In 0.0025, there are four digits after the decimal point. So, it can be written as a fraction with the denominator 10000.
0.0025 = ²⁵⁄₁₀₀₀₀
= ⁽²⁵ ÷ ²⁵⁾⁄₍₁₀₀₀₀ ÷ ₂₅₎
= ¹⁄₄₀₀
14. Answer :
In 0.104, there are three digits after the decimal point. So, it can be written as a fraction with the denominator 1000.
0.104 = ¹⁰⁴⁄₁₀₀₀
= ⁽¹⁰⁴ ÷ ⁸⁾⁄₍₁₀₀₀ ÷ ₈₎
= ¹³⁄₁₂₅
15. Answer :
The correct answer is (B).
16. Answer :
The correct answer is (D).
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