CBSE Class 8 Mathematics Rational Numbers MCQ Assignment Set 03

Choose the correct answer from the multiple answers.

Question. To write a number in the form of a rational number \( \frac{p}{q} \) , the restriction on \( q \) is
(a) \( q = 0 \)
(b) \( q \neq 0 \)
(c) \( q > 0 \)
(d) \( q < 0 \)
Answer: (b) \( q \neq 0 \)
In simple words: A rational number is like a fraction. The bottom number of a fraction can never be zero because division by zero is not possible.

Exam Tip: Remember that the denominator of a rational number cannot be zero, though the numerator can be any integer, including zero.

Question. The reciprocal of \( \frac{-8}{17} \) is
(a) \( \frac{17}{-8} \)
(b) \( +\frac{8}{17} \)
(c) \( \frac{-17}{-8} \)
(d) \( \frac{-8}{-17} \)
Answer: (a) \( \frac{17}{-8} \)
In simple words: To find the reciprocal of a fraction, just turn it upside down. The top number goes to the bottom, and the bottom number goes to the top.

Exam Tip: Keep the sign of the number the same when finding its reciprocal - only flip the numerator and denominator.

Question. The additive inverse of \( -\frac{a}{b} \) is
(a) \( \frac{b}{a} \)
(b) \( \frac{a}{-b} \)
(c) \( \frac{a}{b} \)
(d) \( \frac{-b}{a} \)
Answer: (c) \( \frac{a}{b} \)
In simple words: The additive inverse is the number you add to get zero. To find it, just change the sign from negative to positive.

Exam Tip: The sum of a number and its additive inverse is always zero. Use this rule to check your answer.

Question. If \( \frac{7}{8} \) and \( \frac{4}{7} \) are two rational numbers then \( \frac{7}{8} \times \frac{4}{7} \) is also a rational number by which property?
(a) closure
(b) Commutative
(c) Associative
(d) D.L
Answer: (a) closure
In simple words: When we multiply two fractions, the answer is always another fraction. This rule is called the closure property.

Exam Tip: The closure property means that the result of an operation on numbers in a set stays in that same set.

Question. Between any two rational numbers, how many numbers you can find?
(a) 10
(b) 20
(c) some finite numbers
(d) Unlimited numbers
Answer: (d) Unlimited numbers
In simple words: You can find infinitely many fractions between any two given fractions. There is no limit to them.

Exam Tip: Remember that rational numbers are dense, meaning there is always another rational number between any two of them.

Question. Which property is used in \( \frac{3}{8} \times 1 = \frac{3}{8} \) ?
(a) closure
(b) Commutative
(c) Property of one
(d) Associative
Answer: (c) Property of one
In simple words: Multiplying any number by 1 does not change its value. This is why 1 is called the multiplicative identity.

Exam Tip: The property of one is also known as the identity property of multiplication because the number keeps its identity.

Question. The rational number which lies between \( \frac{1}{2} \) and \( \frac{1}{4} \) is
(a) \( \frac{3}{8} \)
(b) \( \frac{5}{8} \)
(c) \( \frac{7}{8} \)
(d) \( \frac{9}{8} \)
Answer: (a) \( \frac{3}{8} \)
In simple words: To find a number exactly between two fractions, add them and divide by 2. Here, the middle is \( 3/8 \).

Exam Tip: Converting the given fractions to a common denominator makes it easy to find and verify the number in between.

Question. The numbers \( \frac{1}{2} + ( \frac{1}{4} + \frac{1}{8} ) \) is equivalent to
(a) \( \frac{6}{8} \)
(b) \( \frac{5}{8} \)
(c) \( \frac{4}{8} \)
(d) \( \frac{7}{8} \)
Answer: (d) \( \frac{7}{8} \)
In simple words: To add these fractions, make their bottom numbers the same. Use 8 as the common bottom number, then add the top numbers.

Exam Tip: Always find the least common multiple of the denominators before adding or subtracting fractions.

Question. The simplified form of \( \frac{7}{8} \times \frac{3}{2} \times \frac{4}{7} \times \frac{4}{3} \) is
(a) 0
(b) 1
(c) 2
(d) \( \frac{3}{7} \)
Answer: (b) 1
In simple words: We can cancel matching numbers on the top and bottom. This simplifies the math and leaves us with 1.

Exam Tip: Simplify by cross-cancelling common factors across numerators and denominators to save time during calculations.

Question. What should be subtracted from \( \frac{5}{8} \) to get \( ( \frac{-1}{24} ) \) ?
(a) \( \frac{2}{3} \)
(b) \( \frac{3}{2} \)
(c) \( \frac{5}{6} \)
(d) \( \frac{1}{4} \)
Answer: (a) \( \frac{2}{3} \)
In simple words: To find the missing number, add the two given fractions after changing the sign of the target value.

Exam Tip: Form a simple equation like \( \frac{5}{8} - x = -\frac{1}{24} \) and solve for \( x \) to avoid sign errors.

Question. Is Subtraction Commutative in the set of different rational numbers ?
(a) yes
(b) No
(c) Sometimes
(d) None of the options
Answer: (b) No
In simple words: Changing the order of numbers in subtraction changes the sign of the answer, so subtraction is not commutative.

Exam Tip: For any two different rational numbers \( a \) and \( b \), \( a - b \) is not equal to \( b - a \).

Question. Is \( \frac{2}{3} + (\frac{5}{6} + \frac{1}{3}) = (\frac{2}{3} + \frac{5}{6}) + \frac{1}{3} \) true by which property ?
(a) closure
(b) Commutative
(c) Associative
(d) None of the options
Answer: (c) Associative
In simple words: This is the associative property. It says we can group numbers differently when adding, and the answer remains the same.

Exam Tip: The associative property deals with how numbers are grouped using parentheses, whereas commutative deals with their order.

Question. Is division closed in the set of rational numbers.
(a) yes
(b) No
(c) Sometimes
(d) None of the options
Answer: (b) No
In simple words: Dividing by zero is not allowed. Since zero is a rational number, division is not closed for this set.

Exam Tip: The closure property fails for division of rational numbers solely because division by zero is undefined.

Question. The product of two numbers is 80 . If one of them is \( 13\frac{1}{3} \) , the other number is
(a) 6
(b) 7
(c) 8
(d) \( \frac{4}{5} \)
Answer: (a) 6
In simple words: Convert the mixed number into a fraction, then divide the total product by this fraction to get the answer.

Exam Tip: Change mixed numbers to improper fractions before solving. Here, \( 13\frac{1}{3} \) becomes \( \frac{40}{3} \).

Question. \( 12\frac{1}{4} \) metre cloth cost Rs \( 212\frac{1}{3} \) . The Cost of 1 metre cloth is
(a) \( 17\frac{1}{3} \)
(b) \( 27\frac{1}{3} \)
(c) \( 17\frac{2}{3} \)
(d) \( 27\frac{2}{3} \)
Answer: (a) \( 17\frac{1}{3} \)
In simple words: To find the cost of one metre, divide the total cost by the total length of the cloth.

Exam Tip: Convert mixed numbers to improper fractions. Divide the total cost by the total length and simplify by cross-cancelling common factors.

Question. The product of the additive inverse and the multiplicative inverse of -5 is
(a) 1
(b) 0
(c) -1
(d) -5
Answer: (c) -1
In simple words: The additive inverse of -5 is 5, and its reciprocal is -1/5. Multiplying these together gives -1.

Exam Tip: Remember that the additive inverse changes the sign of a number, while the multiplicative inverse flips it.

Question. \( \frac{3}{-4} \times \frac{-16}{7} \) is
(a) \( \frac{-7}{12} \)
(b) \( \frac{12}{7} \)
(c) \( \frac{-12}{7} \)
(d) \( \frac{7}{12} \)
Answer: (b) \( \frac{12}{7} \)
In simple words: Multiplying two negative numbers gives a positive result. We can simplify by dividing 16 by 4.

Exam Tip: Determine the sign of your answer first. Multiplying two negative fractions always results in a positive fraction.

Question. Which property of multiplication is used here? \( -\frac{2}{3} \times ( \frac{5}{6} + \frac{-2}{3} ) = (-\frac{2}{3}) \times (\frac{5}{6}) + (-\frac{2}{3}) \times (\frac{-2}{3}) \)
(a) closure
(b) Commutative
(c) Associative
(d) D.L
Answer: (d) D.L
In simple words: This is the distributive law. It lets us multiply a number outside parentheses by each number inside them.

Exam Tip: Distributive law (D.L) shows how multiplication spreads over addition, written as \( a \times (b + c) = (a \times b) + (a \times c) \).

Question. If x and y are two rational numbers then \( |x+y| \) is
(a) \( |x+y| \le |x| + |y| \)
(b) \( |x+y| < |x| + |y| \)
(c) \( |x+y| \ge |x| + |y| \)
(d) \( |x+y| = |x| + |y| \)
Answer: (a) \( |x+y| \le |x| + |y| \)
In simple words: The absolute value of two numbers added together is always less than or equal to adding their absolute values separately.

Exam Tip: This is the triangle inequality. The two sides are equal if both numbers have the same sign.

Question. \( - | \frac{5}{6} - \frac{2}{3} | \) is equal to
(a) \( \frac{1}{6} \)
(b) \( -\frac{1}{6} \)
(c) \( \frac{9}{6} \)
(d) \( \frac{6}{9} \)
Answer: (b) \( -\frac{1}{6} \)
In simple words: Subtract inside the vertical bars first to get a positive value. Then, the negative sign outside makes the answer negative.

Exam Tip: Absolute value is always positive, but any negative sign placed outside the absolute value bars makes the result negative.

Question. How many rational numbers are there whose absolute value is \( \frac{5}{6} \)
(a) Infinite
(b) only 2
(c) only 3
(d) only one
Answer: (b) only 2
In simple words: Only two rational numbers, namely \( 5/6 \) and \( -5/6 \), have an absolute value of \( 5/6 \).

Exam Tip: Any positive rational number has exactly two numbers (one positive and one negative) that share its absolute value.

Question. If \( \frac{a}{b} \) and \( \frac{c}{d} \) are two different rational numbers then is \( ( \frac{ad+bc}{bd} ) \) always a rational number ?
(a) No
(b) yes
(c) Sometime
(d) None of the options
Answer: (b) yes
In simple words: This expression is the sum of two fractions. Adding any two fractions always gives another fraction, which is rational.

Exam Tip: The sum of any two rational numbers is always rational. This illustrates the closure property of addition.

Question. If \( x = \frac{-5}{12} \) , \( y = \frac{5}{12} \) then \( |x+y| \) is
(a) \( \frac{10}{12} \)
(b) \( \frac{12}{10} \)
(c) \( \frac{-10}{12} \)
(d) \( \frac{0}{12} \)
Answer: (d) \( \frac{0}{12} \)
In simple words: Adding these two numbers gives zero because they cancel each other out. The absolute value of zero is zero.

Exam Tip: Since \( x \) and \( y \) are additive inverses, their sum is zero, which is represented as \( 0/12 \).

Question. The product of \( (-2\frac{3}{4}) \) with \( (-1\frac{1}{3}) \) is
(a) \( \frac{-11}{3} \)
(b) \( \frac{3}{11} \)
(c) \( \frac{11}{3} \)
(d) None of the options
Answer: (c) \( \frac{11}{3} \)
In simple words: Turn the mixed numbers into improper fractions and multiply. The minus signs cancel out, and the 4s cancel out too.

Exam Tip: Convert mixed numbers to improper fractions first. Multiplying two negative fractions results in a positive fraction.

Question. Is zero a rational number ?
(a) yes
(b) No
(c) sometime
(d) never
Answer: (a) yes
In simple words: Yes, zero is a rational number because we can write it as a fraction, like 0 divided by 1.

Exam Tip: Zero can be written in \( p/q \) form where \( p = 0 \) and \( q \) is any non-zero integer, making it rational.