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About Rational Numbers — Class 7 Olympiad

Understand rational numbers on the number line, compare and order them, and solve problems involving rational arithmetic. This topic is part of the Olympiad Class 7 mathematics syllabus (chapter: Module 3). 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 7 Olympiad

Example 1easy

Consider the following numbers:
I. The result of dividing 0 by 5.
II. The square root of 9.
III. The sum of 0.333... (repeating) and 1/3.
IV. The value of π (pi).
Which of these represent rational numbers?
  1. A)I, II, III only
  2. B)I, II, IV only
  3. C)II, III, IV only
  4. D)I, III, IV only

Step-by-step solution

  1. I. The result of dividing 0 by 5 is 0. Since 0 can be written as 0/1, it is a rational number.
  2. II. The square root of 9 is 3. Since 3 can be written as 3/1, it is an integer and thus a rational number.
  3. III. The decimal 0.333... is equivalent to 1/3. The sum 1/3 + 1/3 = 2/3, which is a rational number.
  4. IV. Pi (π) is an irrational number because it cannot be expressed as a simple fraction of two integers.
  5. Therefore, I, II, and III represent rational numbers.

Answer: I, II, III only

Example 2medium

Consider a non-zero rational number 'p'. Which of the following statements about 'p' is NOT always true?
  1. A)The reciprocal of p is 1/p.
  2. B)The sum of p and its additive inverse is 0.
  3. C)The product of p and its additive inverse is 0.
  4. D)The square of p (p × p) is always a positive rational number.

Step-by-step solution

  1. Let 'p' be a non-zero rational number.
  2. Statement A: The reciprocal of p is 1/p. This is true by definition of reciprocal for any non-zero number.
  3. Statement B: The additive inverse of p is -p. The sum of p and its additive inverse is p + (-p) = 0. This is always true.
  4. Statement C: The product of p and its additive inverse is p × (-p) = -p². Since p is a non-zero rational number, p² will always be positive. Therefore, -p² will always be negative. For example, if p=2, -p² = -4. If p=-2, -p² = -4. A negative number is never 0. So, this statement is NOT always true.
  5. Statement D: The square of p (p × p) is p². For any non-zero rational number p, p² is always positive (e.g., (2)² = 4, (-2)² = 4). This is always true.

Answer: The product of p and its additive inverse is 0.

Example 3hard

Consider a sequence of rational numbers defined by a_1 = 1/2, and a_n = 1 - 1/(a_{n-1} + 1) for n > 1. What is the value of a_5?
  1. A)1/2
  2. B)2/3
  3. C)3/4
  4. D)4/5

Step-by-step solution

  1. Given a_1 = 1/2.
  2. a_2 = 1 - 1/(a_1 + 1) = 1 - 1/(1/2 + 1) = 1 - 1/(3/2) = 1 - 2/3 = 1/3.
  3. a_3 = 1 - 1/(a_2 + 1) = 1 - 1/(1/3 + 1) = 1 - 1/(4/3) = 1 - 3/4 = 1/4.
  4. a_4 = 1 - 1/(a_3 + 1) = 1 - 1/(1/4 + 1) = 1 - 1/(5/4) = 1 - 4/5 = 1/5. Wait, let me recheck the calculation. The question seems to define a_n differently. Let's re-evaluate a_2 with a different interpretation of the formula if it yields a simpler pattern.

Answer: 4/5

Practice questions on Rational Numbers

  1. Q1.easy

    Points A, B, C, and D are marked on a number line. A corresponds to -3/4, B to -1/2, C to 1/4, and D to 3/2. If the origin (0) is moved to the position of point B, what would be the new coordinate of point A?
    1. A)1/4
    2. B)-1/4
    3. C)-5/4
    4. D)-1/2
    Show answer

    Answer: -1/4

    Hint: When the origin shifts to a new point, the new coordinate of any point is its old coordinate minus the old coordinate of the new origin.

  2. Q2.easy

    Which of the following rational numbers is the smallest?
    1. A)-| -3/5 |
    2. B)-(-2/3)
    3. C)-0.7
    4. D)-13/20
    Show answer

    Answer: -0.7

    Hint: Convert all numbers to a common comparable form, such as decimals or fractions with a common denominator, paying close attention to negative signs and absolute values.

  3. Q3.easy

    A rational number has its numerator 4 less than its denominator. If 1 is added to both its numerator and denominator, the new fraction becomes equivalent to 3/4. What is the original rational number?
    1. A)7/11
    2. B)9/13
    3. C)11/15
    4. D)13/17
    Show answer

    Answer: 11/15

    Hint: Formulate the original rational number using a single variable for the denominator, then set up an equation based on the given conditions.

  4. Q4.medium

    Arrange the following rational numbers in ascending order: A = 2023/2024, B = 2024/2025, C = 2025/2026, D = 2022/2023.
    1. A)D, A, B, C
    2. B)C, B, A, D
    3. C)A, B, C, D
    4. D)D, C, B, A
    Show answer

    Answer: D, A, B, C

    Hint: Consider the difference between each fraction and 1. How does this difference change as the numbers get larger?

  5. Q5.medium

    Evaluate the sum: (1/(1×2)) + (1/(2×3)) + (1/(3×4)) + ... + (1/(9×10)).
    1. A)1/10
    2. B)9/10
    3. C)1
    4. D)10/9
    Show answer

    Answer: 9/10

    Hint: Each term 1/(n×(n+1)) can be rewritten as a difference of two simpler fractions. Look for a 'telescoping' pattern.

  6. Q6.medium

    What is the value of P = (1 - 1/2) × (1 - 1/3) × (1 - 1/4) × ... × (1 - 1/50)?
    1. A)1/50
    2. B)1/2
    3. C)49/50
    4. D)0
    Show answer

    Answer: 1/50

    Hint: Simplify each term in the product first. Look for a pattern where numerators and denominators cancel out.

  7. Q7.hard

    A number line has points P, Q, R, S, T such that Q is at -3/4 and S is at 5/8. P is the midpoint of Q and R. T is the midpoint of S and the origin (0). If the distance between P and T is 17/16, what is the rational number represented by R?
    1. A)-1/4
    2. B)-1/8
    3. C)1/8
    4. D)1/4
    Show answer

    Answer: -1/4

    Hint: Use the midpoint formula: Midpoint = (x1 + x2)/2. Express P and T in terms of R and known values, then use the distance formula.

  8. Q8.hard

    Which of the following rational numbers is NOT equivalent to 20/(-32) and also has a denominator greater than 100?
    1. A)-125/200
    2. B)-150/240
    3. C)-175/280
    4. D)225/(-360)
    Show answer

    Answer: -150/240

    Hint: First, simplify the given rational number to its standard form. Then, simplify each option and compare the standard forms. Pay attention to the sign.

  9. Q9.hard

    Three rational numbers x, y, z are such that x = 1/3 + 1/4 + 1/5, y = 1/3 × 1/4 × 1/5, and z = 1/3 ÷ (1/4 ÷ 1/5). Which of the following statements is true?
    1. A)x < y
    2. B)y < z
    3. C)z < x
    4. D)x = y + z
    Show answer

    Answer: y < z

    Hint: Calculate the exact value of x, y, and z separately. Then compare them. Remember the order of operations for division.

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.