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Inequalities

Coded and direct inequalities.

45 solved questions · 15 easy · 20 medium · 10 hard

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Shortcuts & formulas

  • Combine statements into one chain only where the signs point the same way (all > / >= or all < / <=).
  • A > B >= C gives A > C; > with >= always gives > (the stronger sign wins).
  • A >= B >= C gives only A >= C; A = B = C gives A = C; mixing = with > keeps >.
  • Opposite signs in the middle (A > B < C) give no relation between A and C: the answer is 'cannot be determined'.
  • Either-or case: the two conclusions are complementary, such as P > Q and P <= Q, and neither is definite alone.
  • Coded questions: write the meaning of each symbol first, decode the whole statement, then solve like a direct inequality.
  • A conclusion follows only if it is true in every case; one counter-case makes it fail.

Solved examples

Try each one first, then open the answer and the step-by-step solution.

Easy questions

Q1.

Read the statement and decide which conclusion(s) definitely follow. Statement: B ≤ H < T = E Conclusions: I. B < T II. T > E

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. B < T: from B ≤ H < T, the relation between B and T is fixed, so this is true. II. T > E: the chain gives T ≥ E, so this is false. Answer: Only conclusion I follows

Q2.

Read the statement and decide which conclusion(s) definitely follow. Statement: L < Q < U = M Conclusions: I. Q ≤ L II. U ≥ M

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. Q ≤ L: the chain gives Q > L, so this is false. II. U ≥ M: from U = M, the relation between U and M is fixed, so this is true. Answer: Only conclusion II follows

Q3.

Read the statement and decide which conclusion(s) definitely follow. Statement: R = Y = X ≥ C Conclusions: I. C ≥ Y II. C > X

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. C ≥ Y: the relation between C and Y is not fixed by the statement (opposite signs break the chain), so this is not definite. II. C > X: the chain gives C ≤ X, so this is false. Answer: Neither conclusion I nor II follows

Q4.

Read the statement and decide which conclusion(s) definitely follow. Statement: Y ≥ J > S = M Conclusions: I. S < M II. J < S

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. S < M: the chain gives S ≥ M, so this is false. II. J < S: the chain gives J > S, so this is false. Answer: Neither conclusion I nor II follows

Q5.

Read the statement and decide which conclusion(s) definitely follow. Statement: R = S < H ≤ Y Conclusions: I. S ≤ Y II. R ≥ S

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

I. S ≤ Y: from S < H ≤ Y, the relation between S and Y is fixed, so this is true. II. R ≥ S: from R = S, the relation between R and S is fixed, so this is true. Answer: Both conclusions I and II follow

Q6.

Read the statement and decide which conclusion(s) definitely follow. Statement: B = F < G ≤ K Conclusions: I. K ≥ G II. F > K

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. K ≥ G: from K ≥ G, the relation between K and G is fixed, so this is true. II. F > K: the chain gives F < K, so this is false. Answer: Only conclusion I follows

Q7.

Read the statement and decide which conclusion(s) definitely follow. Statement: B < Q ≤ F ≤ J Conclusions: I. F > J II. F ≥ B

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. F > J: the chain gives F ≤ J, so this is false. II. F ≥ B: from F ≥ Q > B, the relation between F and B is fixed, so this is true. Answer: Only conclusion II follows

Q8.

Read the statement and decide which conclusion(s) definitely follow. Statement: G ≤ M < F ≤ A Conclusions: I. F = A II. M > G

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. F = A: the relation between F and A is not fixed by the statement (opposite signs break the chain), so this is not definite. II. M > G: the relation between M and G is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q9.

Read the statement and decide which conclusion(s) definitely follow. Statement: C ≥ E > N ≥ U Conclusions: I. N = U II. E = C

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. N = U: the relation between N and U is not fixed by the statement (opposite signs break the chain), so this is not definite. II. E = C: the relation between E and C is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q10.

Read the statement and decide which conclusion(s) definitely follow. Statement: L = A > N = D Conclusions: I. N ≤ A II. N ≤ L

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

I. N ≤ A: from N < A, the relation between N and A is fixed, so this is true. II. N ≤ L: from N < A = L, the relation between N and L is fixed, so this is true. Answer: Both conclusions I and II follow

Q11.

Read the statement and decide which conclusion(s) definitely follow. Statement: L ≤ T = C < H Conclusions: I. L ≤ C II. H = T

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. L ≤ C: from L ≤ T = C, the relation between L and C is fixed, so this is true. II. H = T: the chain gives H > T, so this is false. Answer: Only conclusion I follows

Q12.

Read the statement and decide which conclusion(s) definitely follow. Statement: W = H > R > S Conclusions: I. S > R II. R < W

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. S > R: the chain gives S < R, so this is false. II. R < W: from R < H = W, the relation between R and W is fixed, so this is true. Answer: Only conclusion II follows

Q13.

Read the statement and decide which conclusion(s) definitely follow. Statement: H = Q ≥ J = K Conclusions: I. Q > H II. K = Q

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. Q > H: the chain gives Q ≥ H, so this is false. II. K = Q: the relation between K and Q is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q14.

Read the statement and decide which conclusion(s) definitely follow. Statement: L > Q ≥ F > D Conclusions: I. F < Q II. L ≤ F

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. F < Q: the relation between F and Q is not fixed by the statement (opposite signs break the chain), so this is not definite. II. L ≤ F: the chain gives L > F, so this is false. Answer: Neither conclusion I nor II follows

Q15.

Read the statement and decide which conclusion(s) definitely follow. Statement: H > W > D ≥ Q Conclusions: I. W ≥ Q II. D < H

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

I. W ≥ Q: from W > D ≥ Q, the relation between W and Q is fixed, so this is true. II. D < H: from D < W < H, the relation between D and H is fixed, so this is true. Answer: Both conclusions I and II follow

Medium questions

Q16.

Read the statement and decide which conclusion(s) definitely follow. Statement: D < B > G ≥ V = U Conclusions: I. B > V II. U > G

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. B > V: from B > G ≥ V, the relation between B and V is fixed, so this is true. II. U > G: the chain gives U ≤ G, so this is false. Answer: Only conclusion I follows

Q17.

Read the statement and decide which conclusion(s) definitely follow. Statement: W > V ≥ Y ≥ M < G Conclusions: I. W = V II. Y < W

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. W = V: the chain gives W > V, so this is false. II. Y < W: from Y ≤ V < W, the relation between Y and W is fixed, so this is true. Answer: Only conclusion II follows

Q18.

Read the statement and decide which conclusion(s) definitely follow. Statement: N ≤ G > A > P ≤ F Conclusions: I. F ≤ A II. F > A

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

I. F ≤ A: the relation between F and A is not fixed by the statement (opposite signs break the chain), so this is not definite. II. F > A: the relation between F and A is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q19.

Read the statement and decide which conclusion(s) definitely follow. Statement: S ≤ F < C ≥ D < N Conclusions: I. S = D II. C ≤ S

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. S = D: the relation between S and D is not fixed by the statement (opposite signs break the chain), so this is not definite. II. C ≤ S: the chain gives C > S, so this is false. Answer: Neither conclusion I nor II follows

Q20.

Read the statement and decide which conclusion(s) definitely follow. Statement: R ≥ N = G > Z ≤ H Conclusions: I. Z ≤ N II. G ≤ R

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

I. Z ≤ N: from Z < G = N, the relation between Z and N is fixed, so this is true. II. G ≤ R: from G = N ≤ R, the relation between G and R is fixed, so this is true. Answer: Both conclusions I and II follow

Q21.

Read the statement and decide which conclusion(s) definitely follow. Statement: G ≥ C = F = Y ≥ K Conclusions: I. Y ≤ C II. C ≥ G

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. Y ≤ C: from Y = F = C, the relation between Y and C is fixed, so this is true. II. C ≥ G: the relation between C and G is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Only conclusion I follows

Q22.

Read the statement and decide which conclusion(s) definitely follow. Statement: Z = B < F ≤ Y < Q Conclusions: I. F < Y II. Y > B

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. F < Y: the relation between F and Y is not fixed by the statement (opposite signs break the chain), so this is not definite. II. Y > B: from Y ≥ F > B, the relation between Y and B is fixed, so this is true. Answer: Only conclusion II follows

Q23.

Read the statement and decide which conclusion(s) definitely follow. Statement: J ≥ U ≤ K > L ≥ F Conclusions: I. L ≤ F II. L > F

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

I. L ≤ F: the relation between L and F is not fixed by the statement (opposite signs break the chain), so this is not definite. II. L > F: the relation between L and F is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q24.

Read the statement and decide which conclusion(s) definitely follow. Statement: H < D > N ≥ K < P Conclusions: I. H = K II. P < N

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. H = K: the relation between H and K is not fixed by the statement (opposite signs break the chain), so this is not definite. II. P < N: the relation between P and N is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q25.

Read the statement and decide which conclusion(s) definitely follow. Statement: P > X = W < A > M Conclusions: I. W < A II. X < P

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

I. W < A: from W < A, the relation between W and A is fixed, so this is true. II. X < P: from X < P, the relation between X and P is fixed, so this is true. Answer: Both conclusions I and II follow

Q26.

Read the statement and decide which conclusion(s) definitely follow. Statement: J < D > S ≥ A > P Conclusions: I. S > P II. D = J

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

I. S > P: from S ≥ A > P, the relation between S and P is fixed, so this is true. II. D = J: the chain gives D > J, so this is false. Answer: Only conclusion I follows

Q27.

Read the statement and decide which conclusion(s) definitely follow. Statement: D ≤ R ≥ N = U = E Conclusions: I. D ≥ U II. U ≥ E

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

I. D ≥ U: the relation between D and U is not fixed by the statement (opposite signs break the chain), so this is not definite. II. U ≥ E: from U = E, the relation between U and E is fixed, so this is true. Answer: Only conclusion II follows

Q28.

Read the statement and decide which conclusion(s) definitely follow. Statement: T ≤ K = E < P > U Conclusions: I. E < U II. E ≥ U

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

I. E < U: the relation between E and U is not fixed by the statement (opposite signs break the chain), so this is not definite. II. E ≥ U: the relation between E and U is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q29.

Read the statement and decide which conclusion(s) definitely follow. Statement: C ≥ K ≥ S = F ≤ D Conclusions: I. K = S II. D < K

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

I. K = S: the relation between K and S is not fixed by the statement (opposite signs break the chain), so this is not definite. II. D < K: the relation between D and K is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q30.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols &, $, @, #, * are used with the meanings given: A $ B means A is smaller than B, A & B means A is greater than B, A * B means A is equal to B, A # B means A is smaller than or equal to B, A @ B means A is greater than or equal to B. Statement: Y & R $ A $ H Conclusions: I. A @ R II. R # Y

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

Decode the statement using the legend: Y & R $ A $ H becomes Y > R < A < H Conclusion I decodes to A ≥ R; conclusion II decodes to R ≤ Y. I. A ≥ R: from A > R, the relation between A and R is fixed, so this is true. II. R ≤ Y: from R < Y, the relation between R and Y is fixed, so this is true. Answer: Both conclusions I and II follow

Q31.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, *, #, @, & are used with the meanings given: A % B means A is greater than B, A @ B means A is smaller than or equal to B, A & B means A is equal to B, A * B means A is smaller than B, A # B means A is greater than or equal to B. Statement: J * X * A * L Conclusions: I. J * L II. A & L

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

Decode the statement using the legend: J * X * A * L becomes J < X < A < L Conclusion I decodes to J < L; conclusion II decodes to A = L. I. J < L: from J < X < A < L, the relation between J and L is fixed, so this is true. II. A = L: the chain gives A < L, so this is false. Answer: Only conclusion I follows

Q32.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols &, %, $, *, @ are used with the meanings given: A @ B means A is equal to B, A $ B means A is greater than or equal to B, A & B means A is greater than B, A % B means A is smaller than B, A * B means A is smaller than or equal to B. Statement: N $ W * U @ G Conclusions: I. W % N II. G $ U

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

Decode the statement using the legend: N $ W * U @ G becomes N ≥ W ≤ U = G Conclusion I decodes to W < N; conclusion II decodes to G ≥ U. I. W < N: the relation between W and N is not fixed by the statement (opposite signs break the chain), so this is not definite. II. G ≥ U: from G = U, the relation between G and U is fixed, so this is true. Answer: Only conclusion II follows

Q33.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols @, #, &, %, * are used with the meanings given: A % B means A is smaller than or equal to B, A # B means A is smaller than B, A & B means A is greater than or equal to B, A * B means A is equal to B, A @ B means A is greater than B. Statement: X & S % C @ T Conclusions: I. S % T II. S @ T

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

Decode the statement using the legend: X & S % C @ T becomes X ≥ S ≤ C > T Conclusion I decodes to S ≤ T; conclusion II decodes to S > T. I. S ≤ T: the relation between S and T is not fixed by the statement (opposite signs break the chain), so this is not definite. II. S > T: the relation between S and T is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q34.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, #, $, @, * are used with the meanings given: A @ B means A is smaller than or equal to B, A % B means A is greater than B, A # B means A is smaller than B, A * B means A is equal to B, A $ B means A is greater than or equal to B. Statement: S $ K % G # C Conclusions: I. G $ S II. C @ K

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

Decode the statement using the legend: S $ K % G # C becomes S ≥ K > G < C Conclusion I decodes to G ≥ S; conclusion II decodes to C ≤ K. I. G ≥ S: the chain gives G < S, so this is false. II. C ≤ K: the relation between C and K is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Neither conclusion I nor II follows

Q35.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols @, *, $, &, % are used with the meanings given: A $ B means A is greater than or equal to B, A % B means A is equal to B, A & B means A is smaller than or equal to B, A @ B means A is greater than B, A * B means A is smaller than B. Statement: T % K * W % F Conclusions: I. F % W II. W $ K

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

Decode the statement using the legend: T % K * W % F becomes T = K < W = F Conclusion I decodes to F = W; conclusion II decodes to W ≥ K. I. F = W: from F = W, the relation between F and W is fixed, so this is true. II. W ≥ K: from W > K, the relation between W and K is fixed, so this is true. Answer: Both conclusions I and II follow

Hard questions

Q36.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, &, *, @, $ are used with the meanings given: A $ B means A is equal to B, A @ B means A is smaller than or equal to B, A * B means A is greater than or equal to B, A & B means A is smaller than B, A % B means A is greater than B. Statement: E & T $ K * X $ N Conclusions: I. K * X II. E & N

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

Decode the statement using the legend: E & T $ K * X $ N becomes E < T = K ≥ X = N Conclusion I decodes to K ≥ X; conclusion II decodes to E < N. I. K ≥ X: from K ≥ X, the relation between K and X is fixed, so this is true. II. E < N: the relation between E and N is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Only conclusion I follows

Q37.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols &, %, $, @, * are used with the meanings given: A * B means A is equal to B, A & B means A is greater than B, A $ B means A is greater than or equal to B, A @ B means A is smaller than or equal to B, A % B means A is smaller than B. Statement: U $ V @ B $ G * F Conclusions: I. G & F II. B $ F

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

Decode the statement using the legend: U $ V @ B $ G * F becomes U ≥ V ≤ B ≥ G = F Conclusion I decodes to G > F; conclusion II decodes to B ≥ F. I. G > F: the chain gives G ≥ F, so this is false. II. B ≥ F: from B ≥ G = F, the relation between B and F is fixed, so this is true. Answer: Only conclusion II follows

Q38.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, &, $, @, * are used with the meanings given: A @ B means A is smaller than or equal to B, A * B means A is equal to B, A % B means A is greater than B, A & B means A is smaller than B, A $ B means A is greater than or equal to B. Statement: L & Z $ M * S @ W Conclusions: I. M $ L II. M & L

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

Decode the statement using the legend: L & Z $ M * S @ W becomes L < Z ≥ M = S ≤ W Conclusion I decodes to M ≥ L; conclusion II decodes to M < L. I. M ≥ L: the relation between M and L is not fixed by the statement (opposite signs break the chain), so this is not definite. II. M < L: the relation between M and L is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q39.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols *, $, @, %, & are used with the meanings given: A * B means A is greater than B, A @ B means A is greater than or equal to B, A % B means A is smaller than or equal to B, A $ B means A is smaller than B, A & B means A is equal to B. Statement: A @ X $ Q & N * U Conclusions: I. X & Q II. Q & U

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

Decode the statement using the legend: A @ X $ Q & N * U becomes A ≥ X < Q = N > U Conclusion I decodes to X = Q; conclusion II decodes to Q = U. I. X = Q: the chain gives X < Q, so this is false. II. Q = U: the chain gives Q > U, so this is false. Answer: Neither conclusion I nor II follows

Q40.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols #, $, %, *, @ are used with the meanings given: A $ B means A is smaller than B, A @ B means A is equal to B, A * B means A is smaller than or equal to B, A # B means A is greater than B, A % B means A is greater than or equal to B. Statement: T @ S % G $ F $ J Conclusions: I. S * T II. G * S

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

Decode the statement using the legend: T @ S % G $ F $ J becomes T = S ≥ G < F < J Conclusion I decodes to S ≤ T; conclusion II decodes to G ≤ S. I. S ≤ T: from S = T, the relation between S and T is fixed, so this is true. II. G ≤ S: from G ≤ S, the relation between G and S is fixed, so this is true. Answer: Both conclusions I and II follow

Q41.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols #, $, &, @, * are used with the meanings given: A $ B means A is smaller than B, A & B means A is greater than or equal to B, A * B means A is equal to B, A @ B means A is smaller than or equal to B, A # B means A is greater than B. Statement: J $ Y & P @ H & M Conclusions: I. P @ H II. Y & M

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: A. Only conclusion I follows

Solution

Decode the statement using the legend: J $ Y & P @ H & M becomes J < Y ≥ P ≤ H ≥ M Conclusion I decodes to P ≤ H; conclusion II decodes to Y ≥ M. I. P ≤ H: from P ≤ H, the relation between P and H is fixed, so this is true. II. Y ≥ M: the relation between Y and M is not fixed by the statement (opposite signs break the chain), so this is not definite. Answer: Only conclusion I follows

Q42.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols $, #, *, &, @ are used with the meanings given: A # B means A is smaller than B, A * B means A is greater than or equal to B, A $ B means A is greater than B, A & B means A is smaller than or equal to B, A @ B means A is equal to B. Statement: S # L $ N @ J * A $ T Conclusions: I. A * L II. N * A

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: B. Only conclusion II follows

Solution

Decode the statement using the legend: S # L $ N @ J * A $ T becomes S < L > N = J ≥ A > T Conclusion I decodes to A ≥ L; conclusion II decodes to N ≥ A. I. A ≥ L: the chain gives A < L, so this is false. II. N ≥ A: from N = J ≥ A, the relation between N and A is fixed, so this is true. Answer: Only conclusion II follows

Q43.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, *, @, &, # are used with the meanings given: A % B means A is greater than B, A @ B means A is greater than or equal to B, A * B means A is smaller than B, A # B means A is equal to B, A & B means A is smaller than or equal to B. Statement: N @ U % E * V @ G % Y Conclusions: I. N % V II. N & V

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: C. Either conclusion I or II follows

Solution

Decode the statement using the legend: N @ U % E * V @ G % Y becomes N ≥ U > E < V ≥ G > Y Conclusion I decodes to N > V; conclusion II decodes to N ≤ V. I. N > V: the relation between N and V is not fixed by the statement (opposite signs break the chain), so this is not definite. II. N ≤ V: the relation between N and V is not fixed by the statement (opposite signs break the chain), so this is not definite. The two conclusions are complementary (one must be true), so either I or II follows. Answer: Either conclusion I or II follows

Q44.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols %, @, *, $, & are used with the meanings given: A @ B means A is smaller than B, A & B means A is equal to B, A * B means A is greater than or equal to B, A % B means A is greater than B, A $ B means A is smaller than or equal to B. Statement: N & Q % D % C $ H % W Conclusions: I. Q @ H II. C * N

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: D. Neither conclusion I nor II follows

Solution

Decode the statement using the legend: N & Q % D % C $ H % W becomes N = Q > D > C ≤ H > W Conclusion I decodes to Q < H; conclusion II decodes to C ≥ N. I. Q < H: the relation between Q and H is not fixed by the statement (opposite signs break the chain), so this is not definite. II. C ≥ N: the chain gives C < N, so this is false. Answer: Neither conclusion I nor II follows

Q45.

Read the statement and decide which conclusion(s) definitely follow. In the following question, the symbols #, @, *, %, & are used with the meanings given: A * B means A is greater than or equal to B, A % B means A is smaller than or equal to B, A & B means A is equal to B, A @ B means A is smaller than B, A # B means A is greater than B. Statement: C @ P % U * L @ B @ J Conclusions: I. B % J II. L @ J

  1. AOnly conclusion I follows
  2. BOnly conclusion II follows
  3. CEither conclusion I or II follows
  4. DNeither conclusion I nor II follows
  5. EBoth conclusions I and II follow
Show answer & solution

Answer: E. Both conclusions I and II follow

Solution

Decode the statement using the legend: C @ P % U * L @ B @ J becomes C < P ≤ U ≥ L < B < J Conclusion I decodes to B ≤ J; conclusion II decodes to L < J. I. B ≤ J: from B < J, the relation between B and J is fixed, so this is true. II. L < J: from L < B < J, the relation between L and J is fixed, so this is true. Answer: Both conclusions I and II follow

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