Reasoning

Inequality: Practice Questions with Answers

16 practice questions on Inequality, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.

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1. Q.1

   In the statements, '@' means '>', '#' means '=', and '$' means '<'. Statements: A @ B, B # C, C $ D. Conclusions: I. A @ D. II. D @ A. Which of the conclusions follow (stated in plain terms)?

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   Answer: Neither conclusion follows

   Solution:

   1. Translating: A > B, B = C, C < D. Combining: A > B = C, and separately C < D — so both A and D are simply "greater than C," with no direct link established between A and D themselves.
   2. Try two valid assignments: (B=C=1, A=5, D=10) satisfies everything and gives D > A. But (B=C=1, A=100, D=2) also satisfies everything and gives A > D. Since both outcomes are possible, neither "A > D" nor "D > A" is guaranteed.

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2. Q.2

   Statements: P ≥ Q, Q = R, R > S. Conclusions: I. P > S. II. P ≥ S. Which of the conclusions follow?

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   Answer: Both conclusions follow

   Solution:

   1. Since Q = R, the chain becomes P ≥ R > S. Even in the boundary case where P equals R exactly, R > S still makes P > S strictly — so P > S is forced in every valid case.
   2. Since P > S is proven, the weaker P ≥ S is also always true (> always implies ≥). Both conclusions follow, with I being the exact relationship.

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3. Q.3

   Statements: M ≤ N, N < O, O = P. Conclusions: I. M < P. II. M < O. Which of the conclusions follow?

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   Answer: Both conclusions follow

   Solution:

   1. N < O and O = P combine to N < P. Since M ≤ N, even in the boundary case M = N, the strict N < P still forces M < P. So Conclusion I is forced.
   2. Similarly, M ≤ N < O forces M < O by the same logic. Conclusion II is forced too.

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4. Q.4

   Statements: A > B, C > B, C > D. Conclusions: I. A > D. II. D > A. Which of the conclusions follow?

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   Answer: Neither conclusion follows

   Solution:

   1. A and C both relate to B, but neither relates directly to D through a single consistent chain — A connects to B, and D connects to C, with no link bridging A and D together.
   2. Try a valid number assignment: B = 1, C = 10, A = 3, D = 3 satisfies every statement (3 > 1, 10 > 1, 10 > 3) but gives A = D exactly — ruling out both "A > D" and "D > A" as safe conclusions.

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5. Q.5

   Statements: A ≥ B, B ≥ C. Conclusions: I. A > C. II. A = C. Which of the conclusions follow?

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   Answer: Either Conclusion I or Conclusion II follows

   Solution:

   1. A ≥ B ≥ C combines to A ≥ C — but this is only guaranteed as a non-strict relationship, since both links could be equalities (A = B = C, giving A = C) or could involve a strict step somewhere (e.g. A=3, B=2, C=1, giving A > C).
   2. Since neither "A > C" nor "A = C" is individually guaranteed, but A ≥ C IS guaranteed (and ≥ means one of those two must hold), the answer is "either I or II follows."

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6. Q.6

   Statements: A > B, B > C, C > D. Conclusions: I. A > D. II. B > D. Which of the conclusions logically follow?

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   Answer: Both conclusions follow

   Solution:

   1. The whole chain points the same direction: A > B > C > D.
   2. Combining consistently: A > D (skipping through B and C), and B > D (skipping C). Both are forced by the chain.

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7. Q.7

   Statements: X < Y, Y < Z, Z = W. Conclusions: I. X < W. II. X < Z. Which of the conclusions follow?

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   Answer: Both conclusions follow

   Solution:

   1. X < Y < Z combines directly to X < Z — Conclusion II is forced.
   2. Since Z = W, substituting gives X < Z = W, so X < W too — Conclusion I is forced as well.

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8. Q.8

   Statements: P = Q, Q < R, R = S. Conclusions: I. P < S. II. P < R. Which of the conclusions follow?

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   Answer: Both conclusions follow

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9. Q.9

   Statements: X > Y, Z > Y, Z > W. Conclusions: I. X > W. II. W > X. Which of the conclusions follow?

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   Answer: Neither conclusion follows (Y is the shared reference point — there is no direct chain between X and W)

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10. Q.10

    Statements: P ≥ Q, Q ≥ R. Conclusions: I. P > R. II. P = R. Which of the conclusions follow?

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    Answer: Either Conclusion I or II follows (P ≥ R is guaranteed, but not which of the two it specifically is).

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11. Q.11

    Statements: W ≥ X, X > Y, Y ≥ Z. Conclusions: I. W > Z. II. W ≥ Z. Which of the conclusions follow?

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    Answer: Both conclusions follow (the strict X > Y link in the middle forces W > Z even with ≥ at both ends).

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12. Q.12

    Statements: J ≤ K, K < L, L = M. Conclusions: I. J < M. II. J < L. Which of the conclusions follow?

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    Answer: Both conclusions follow (the strict K < L link forces strict results even though J ≤ K is non-strict).

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13. Q.13

    In the statements, '*' means '>', '~' means '=', and '%' means '<'. Statements: A * B, B ~ C, C % D. Conclusions: I. A * D. II. D * A. Which of the conclusions follow (stated in plain terms)?

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    Answer: Neither conclusion follows — same structure as the "shared reference point" trap, just with coded symbols.

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14. Q.14

    Statements: M > N, N ≥ O, O > P. Conclusions: I. M > P. II. N > P. Which of the conclusions follow?

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    Answer: Both conclusions follow

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15. Q.15

    In these statements, "P @ Q" means P is greater than Q, and "P % Q" means P is equal to Q. Statements: A @ B, B % C, C @ D. Conclusions: I. A @ D. II. B @ D. Which of the conclusions follow?

    1. A Both conclusions follow
    2. B Only conclusion I follows
    3. C Only conclusion II follows
    4. D Neither conclusion follows
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    Correct answer: (A) Both conclusions follow

    Decoding the symbols: A > B, B = C, C > D. This combines to A > B = C > D, so A > D (Conclusion I, decoded as A @ D) and B > D (Conclusion II, decoded as B @ D) are both forced.

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16. Q.16

    In these statements, "P δ Q" means P is greater than or equal to Q, and "P θ Q" means P is less than Q. Statements: A δ B, B δ C, D θ C. Conclusions: I. A δ C. II. D δ A. Which of the conclusions follow?

    1. A Both conclusions follow
    2. B Only conclusion I follows
    3. C Only conclusion II follows
    4. D Neither conclusion follows
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    Correct answer: (B) Only conclusion I follows

    Decoding: A ≥ B, B ≥ C, D < C. This combines to A ≥ B ≥ C, so A ≥ C (Conclusion I, decoded as A δ C) is forced. For Conclusion II: D < C ≤ A means D < A, which is the OPPOSITE of D ≥ A (D δ A) — so Conclusion II is not just unproven, it is actually always false.

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