SUMMARY: The chapter on Rational Numbers introduces students to the concept of rational numbers and their properties, operations, and representation on the number line. KEY TOPICS: definition of rational numbers, properties of rational numbers, representation on the number line, addition and subtraction of rational numbers, multiplication and division of rational numbers, standard form of rational numbers, comparison of rational numbers, rational numbers between two rational numbers.
Which of the following represents the rational number 5/8 on the number line?
ABetween 0 and 1
BBetween 1 and 2
CBetween 2 and 3
DBetween 3 and 4
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Correct answer: Option 1 — Between 0 and 1
Q41 Mark
If a rational number is in the form of 12/16, what is its standard form?
A3/4
B4/3
C2/3
D1/2
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Correct answer: Option 1 — 3/4
Q51 Mark
Which of the following pairs of rational numbers has a rational number between them?
A1/2 and 1/3
B2/5 and 3/5
C4/7 and 4/8
D1 and 2
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Correct answer: Option 2 — 2/5 and 3/5
Short Answer Questions5 questions
Q63 Marks
Define a rational number and provide two examples.
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A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, where p is the numerator, q is the denominator, and q is not zero. Examples include 1/2 and -3/4.
Q73 Marks
What are the properties of rational numbers? List at least two properties.
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Rational numbers have several properties, including closure under addition and multiplication, meaning the sum or product of two rational numbers is also a rational number. Additionally, they can be compared using the concept of a common denominator.
Q83 Marks
How do you represent the rational number 3/4 on a number line?
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To represent 3/4 on a number line, first identify the segment between 0 and 1. Divide this segment into 4 equal parts, then count 3 parts from 0. The point where you reach after counting 3 parts represents 3/4 on the number line.
Q93 Marks
Perform the addition of -2/3 and 1/6. Provide the answer in standard form.
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To add -2/3 and 1/6, first find a common denominator, which is 6. Convert -2/3 to -4/6. Now add: -4/6 + 1/6 = -3/6. Simplifying gives -1/2, which is the answer in standard form.
Q103 Marks
Explain how to compare two rational numbers, for example, 2/5 and 3/10.
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To compare 2/5 and 3/10, convert them to a common denominator. The common denominator for 5 and 10 is 10. Convert 2/5 to 4/10. Now compare 4/10 and 3/10; since 4/10 is greater than 3/10, we conclude that 2/5 is greater than 3/10.
Long Answer Questions5 questions
Q116 Marks
Define rational numbers and provide three examples. Explain how they can be represented on a number line.
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Rational numbers are numbers that can be expressed in the form of a fraction a/b, where 'a' and 'b' are integers and 'b' is not equal to zero. Examples of rational numbers include 1/2, -3/4, and 5 (which can be written as 5/1). To represent rational numbers on a number line, we first identify the position of whole numbers and then divide the segments between them into equal parts to place the fractions accurately. For instance, 1/2 would be located halfway between 0 and 1.
Q126 Marks
Discuss the properties of rational numbers, focusing on closure, commutativity, and associativity for addition and multiplication.
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Rational numbers exhibit several key properties. The closure property states that the sum or product of any two rational numbers is also a rational number. For example, adding 1/2 and 3/4 yields 5/4, which is rational. The commutative property indicates that the order of addition or multiplication does not affect the result; thus, a/b + c/d = c/d + a/b and a/b × c/d = c/d × a/b. Lastly, the associative property shows that when adding or multiplying three rational numbers, the grouping does not change the result; for instance, (a/b + c/d) + e/f = a/b + (c/d + e/f).
Q136 Marks
Explain how to add and subtract rational numbers with an example. What steps must be followed to ensure the correct result?
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To add or subtract rational numbers, the first step is to ensure that they have a common denominator. For example, to add 1/3 and 1/4, we find the least common multiple of 3 and 4, which is 12. We convert the fractions to have this common denominator: 1/3 becomes 4/12 and 1/4 becomes 3/12. Now, we can add: 4/12 + 3/12 = 7/12. For subtraction, the process is similar; if we were to subtract 1/4 from 1/3, we would again convert to a common denominator and then subtract the numerators. Thus, the steps include finding a common denominator, converting the fractions, and then performing the addition or subtraction.
Q146 Marks
Describe the process of multiplying and dividing rational numbers. Provide an example for each operation.
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Multiplying rational numbers involves multiplying the numerators together and the denominators together. For instance, to multiply 2/3 by 4/5, we calculate (2 × 4) / (3 × 5) = 8/15. On the other hand, dividing rational numbers requires multiplying by the reciprocal of the divisor. For example, to divide 3/4 by 2/5, we multiply 3/4 by the reciprocal of 2/5, which is 5/2. Thus, (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8. This method simplifies the division of fractions into a multiplication problem.
Q156 Marks
How do you compare two rational numbers? Illustrate your answer with an example of finding a rational number between two given rational numbers.
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To compare two rational numbers, we can convert them to have a common denominator or convert them to decimal form. For instance, to compare 1/2 and 2/3, we can find a common denominator, which is 6. Thus, 1/2 becomes 3/6 and 2/3 becomes 4/6, allowing us to see that 3/6 < 4/6, hence 1/2 < 2/3. To find a rational number between 1/2 and 2/3, we can take the average of the two fractions. The average is (1/2 + 2/3) / 2. Converting to a common denominator gives us (3/6 + 4/6) / 2 = 7/12 / 2 = 7/24, which lies between 1/2 and 2/3.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): Every integer is a rational number.
Reason (R): A rational number can be expressed as the quotient of two integers.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The sum of two rational numbers is always a rational number.
Reason (R): Rational numbers are closed under addition.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): 0 is a rational number.
Reason (R): 0 can be expressed as 0/1, which is a fraction.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The product of a rational number and an irrational number is always irrational.
Reason (R): Rational numbers cannot be expressed as fractions involving irrational numbers.
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Correct answer: Option 4 —
A is false, but R is true.
Q201 Mark
Assertion (A): Rational numbers can be represented on a number line.
Reason (R): Every rational number corresponds to a unique point on the number line.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: A rational number can be expressed as a fraction where both the numerator and denominator are integers and the denominator is not zero.
Statement 2: Every integer is a rational number.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The sum of two rational numbers is always a rational number.
Statement 2: The product of two rational numbers is always an integer.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: Rational numbers can be represented on the number line.
Statement 2: The number 0 is not a rational number.
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Correct answer: Option 4 —
Both statements are false.
Q241 Mark
Statement 1: To compare two rational numbers, we can convert them to a common denominator.
Statement 2: The rational number 1/2 is greater than 3/4.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: The standard form of a rational number is when the numerator and denominator have no common factors other than 1.
Statement 2: The rational number -3/4 is in standard form.
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Correct answer: Option 1 —
Both statements are true.
