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About Rational Numbers — Class 8 Olympiad

Apply properties of rational numbers, represent them on the number line, and solve challenging arithmetic problems. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 1). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Rational Numbers — solved examples for Class 8 Olympiad

Example 1easy

Evaluate: (7/11) × ((-3)/13) + (7/11) × ((-10)/13) + (7/11)
  1. A)7/11
  2. B)1
  3. C)0
  4. D)-7/11

Step-by-step solution

  1. The expression is (7/11) × ((-3)/13) + (7/11) × ((-10)/13) + (7/11).
  2. We can factor out (7/11) from each term, remembering that (7/11) can be written as (7/11) × 1.
  3. Applying the distributive property: (7/11) × [((-3)/13) + ((-10)/13) + 1].
  4. Simplify inside the brackets: (7/11) × [(-3 - 10)/13 + 1] = (7/11) × [(-13)/13 + 1] = (7/11) × [-1 + 1] = (7/11) × 0 = 0.

Answer: 0

Example 2medium

Evaluate the expression: (1/2 × 3/5 + 1/2 × 2/5) × (3/4 - 3/4 × 1/3) ÷ (1/2 + 1/3 - 1/6).
  1. A)1/4
  2. B)1/2
  3. C)3/8
  4. D)2/3

Step-by-step solution

  1. Simplify the first bracket using the distributive property: 1/2 × (3/5 + 2/5) = 1/2 × 1 = 1/2.
  2. Simplify the second bracket: 3/4 × (1 - 1/3) = 3/4 × 2/3 = 1/2.
  3. Simplify the third bracket by finding a common denominator: (3/6 + 2/6 - 1/6) = 4/6 = 2/3.
  4. Perform the remaining operations: (1/2) × (1/2) ÷ (2/3) = (1/4) ÷ (2/3) = 1/4 × 3/2 = 3/8.

Answer: 3/8

Example 3hard

Evaluate: (7/13) × (11/19) + (7/13) × (17/19) - (7/13) × (9/19)
  1. A)7/13
  2. B)7/19
  3. C)13/19
  4. D)19/13

Step-by-step solution

  1. Identify the common factor (7/13) in all terms.
  2. Apply the distributive property: (7/13) × (11/19 + 17/19 - 9/19).
  3. Simplify the expression inside the parenthesis: (11 + 17 - 9)/19 = (28 - 9)/19 = 19/19 = 1.
  4. Multiply the common factor by the simplified result: (7/13) × 1 = 7/13.

Answer: 7/13

Practice questions on Rational Numbers

  1. Q1.easy

    If 'a' is the additive inverse of ((-5)/7) and 'b' is the reciprocal of ((-2)/3) × (1/7), find the value of a × b.
    1. A)-15/2
    2. B)15/2
    3. C)-10/7
    4. D)10/7
    Show answer

    Answer: -15/2

    Hint: First, find 'a' and 'b' separately using their definitions. Remember, the reciprocal of a product is the product of reciprocals.

  2. Q2.easy

    A rational number 'x' is located on the number line such that it is exactly halfway between -3/5 and 1/3. Which of the following describes the location of 'x'?
    1. A)x lies to the right of 0 but left of 1/5.
    2. B)x lies to the left of 0 but right of -1/5.
    3. C)x lies exactly at -1/5.
    4. D)x lies to the left of -1/5.
    Show answer

    Answer: x lies to the left of 0 but right of -1/5.

    Hint: To find the number exactly halfway between two numbers, calculate their average. Then compare 'x' with the given values.

  3. Q3.easy

    How many rational numbers with denominator 24 are strictly between 1/6 and 1/3?
    1. A)2
    2. B)4
    3. C)3
    4. D)5
    Show answer

    Answer: 3

    Hint: Convert the boundary rational numbers to equivalent fractions with the given denominator, then count the integers between their numerators.

  4. Q4.medium

    How many rational numbers between 1/3 and 1/2 have a denominator of 36 when written in their simplest form?
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 2

    Hint: Convert the given fractions to equivalent fractions with denominator 36, then identify numerators that result in a fraction in simplest form.

  5. Q5.medium

    Which of the following statements is TRUE?
    1. A)1/3 + 1/4 < 1/2 + 1/5
    2. B)(1/2)^2 > 1/2
    3. C)The product of the additive inverse of (1/3 - 1/4) and its multiplicative inverse is 1.
    4. D)If x = 1/2 and y = 1/3, then (x - y) / (x + y) < 1/5.
    Show answer

    Answer: 1/3 + 1/4 < 1/2 + 1/5

    Hint: Carefully evaluate each part of the statements and recall the definitions of additive and multiplicative inverses.

  6. Q6.medium

    What is the value of (1 - 1/2)(1 - 1/3)(1 - 1/4)...(1 - 1/n) for n = 2024?
    1. A)1/2
    2. B)1/2024
    3. C)2023/2024
    4. D)n
    Show answer

    Answer: 1/2024

    Hint: Simplify each term in the product individually and look for a pattern that cancels out terms.

  7. Q7.hard

    On a number line, point P represents the rational number x. Point Q is at 1/2, and point R is at 3/4. If P is equidistant from Q and R, and P is to the left of Q, what is the value of x?
    1. A)5/8
    2. B)1/4
    3. C)3/8
    4. D)7/16
    Show answer

    Answer: 1/4

    Hint: The condition 'equidistant' implies P is the midpoint of Q and R or Q is the midpoint of P and R. The condition 'P is to the left of Q' helps determine the correct configuration.

  8. Q8.hard

    Simplify the expression: 1 / (1 + 1 / (1 + 1 / (1 + 1/2)))
    1. A)5/8
    2. B)3/5
    3. C)8/5
    4. D)2/3
    Show answer

    Answer: 3/5

    Hint: Start simplifying the innermost fraction and work your way outwards, step by step.

  9. Q9.hard

    A student mistakenly calculates the sum of a rational number 'x' and its multiplicative inverse as 13/6. If the student had correctly calculated the difference between 'x' and its multiplicative inverse, what would be the result? (Assume x > 1)
    1. A)5/6
    2. B)1
    3. C)7/6
    4. D)11/6
    Show answer

    Answer: 5/6

    Hint: Form an algebraic equation from the given incorrect sum. Solve for 'x' and then use the correct value to find the difference.

These are 9 of the 60 questions available for Rational Numbers. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.