Q: What is the perimeter of a semicircle? I. The radius of the semicircle equals half the side of a square. II. The area of the square is 196 sq cm.

Introduction / Context:The goal is to compute a numeric perimeter for a semicircle. Conventionally, the perimeter of a semicircle includes the curved length plus the diameter: P = π*r + 2*r. We must see whether the given statements fix r uniquely.Given Data / Assumptions:I: r = (side of square)/2.II: Area of square = 196 sq cm ⇒ side = 14 cm.Concept / Approach:Use II to get the square’s side numerically; use I to link that side to r; then compute P = π*r + 2*r.Step-by-Step Solution:1) From II: side s = √196 = 14 cm.2) From I: r = s/2 = 14/2 = 7 cm.3) Perimeter of semicircle: P = π*r + 2*r = 7π + 14 cm.4) Sufficiency: I alone gives only a relation; II alone gives only s. Together they yield a unique numeric answer.Verification / Alternative check:Some texts ask for curved length only (π*r); here “perimeter” explicitly includes the straight edge (diameter), which is standard for perimeter.Why Other Options Are Wrong:A/B/C/D: Neither I nor II alone is enough; together they are.Common Pitfalls:Omitting the diameter; misreading II as radius instead of side.Final Answer:Both statements together are sufficient, but NEITHER alone is sufficient.

Q: What is the exact average of the six numbers: n, 35, 39, 42, p, and w? I. n is six more than w. II. w is four less than p.

Introduction / Context:We must determine the exact arithmetic mean of six values given only linear relations among three variables (n, p, w). The question is sufficient only if a unique numeric average (not an expression in a variable) results.Given Data / Assumptions:I: n = w + 6.II: w = p − 4 (equivalently, p = w + 4).Constants: 35, 39, 42.Concept / Approach:Combine relations and attempt to express the average purely as a number. If a free variable remains, the data are not sufficient.Step-by-Step Solution:1) Average A = (n + 35 + 39 + 42 + p + w)/6.2) Substitute I and II: n = w + 6 and p = w + 4 ⇒ numerator = (w + 6) + 35 + 39 + 42 + (w + 4) + w = 3w + (6 + 35 + 39 + 42 + 4) = 3w + 126.3) Hence A = (3w + 126)/6 = (w/2) + 21.4) The mean depends on w; without a numeric value for w, the average is not uniquely determined.Verification / Alternative check:Pick two different w values (e.g., 0 and 2) to obtain distinct averages (21 and 22), both consistent with I and II—confirming non-uniqueness.Why Other Options Are Wrong:A/B/C/E: Neither statement alone nor together yields a fixed numeric mean; a parameter remains free.Common Pitfalls:Assuming hidden constraints (e.g., integers only still leaves infinitely many values); treating an expression in w as a “determined” average.Final Answer:Both statements are NOT sufficient.

Q: What was the percent profit or loss made by selling an article for ₹ 24,000? I. The ratio of selling price to cost price is 5 : 3. II. The difference between cost price and selling price is ₹ 9,600 (direction not stated).

Introduction / Context:This is a data-sufficiency problem about profit/loss percentage given a fixed selling price of ₹ 24,000. We must decide which statement(s) give enough information to compute the percentage unambiguously, not actually compute it unless needed to justify sufficiency.Given Data / Assumptions:SP = ₹ 24,000 (fixed).Statement I: SP : CP = 5 : 3.Statement II: |CP − SP| = ₹ 9,600 (direction unspecified).Concept / Approach:Profit% = (SP − CP) / CP * 100; Loss% = (CP − SP) / CP * 100. A ratio SP:CP or explicit CP uniquely determines the percentage. A mere “difference” without direction may yield two possible CP values.Step-by-Step Solution:Using I: SP:CP = 5:3 ⇒ Profit% = (5−3)/3 * 100 = 200/3 ≈ 66.67% (unique). Sufficient.Using II alone: |CP − 24,000| = 9,600 ⇒ CP = 14,400 (profit) or 33,600 (loss). Two different percentages (≈66.67% profit or ≈28.57% loss). Insufficient.Combining does not change the fact that I alone already suffices.Why Other Options Are Wrong:II alone is ambiguous; “either” is false since II fails; “both necessary” is false since I alone works; “even both not sufficient” is false.Common Pitfalls:Interpreting “difference” as signed; here it is stated without direction, producing two CP candidates.Final Answer:Statement I alone is sufficient; Statement II alone is not.

Q: What is the difference between two numbers x and y? I. The square of the first number equals 9 times the second: x^2 = 9y. II. The ratio of the first to the second is x : y = 3 : 4. (Assume positive integers to recover solvability.)

Introduction / Context:We must determine whether the difference x − y can be uniquely found from the statements. By Recovery-First, we clarify that x and y are positive integers (a standard minimal assumption for such DS items).Given Data / Assumptions:I: x^2 = 9y.II: x : y = 3 : 4.x, y ∈ positive integers.Concept / Approach:Each statement alone gives a relation but not a unique pair. Together they may pin down a single solution.Step-by-Step Solution:From II: Let x = 3t, y = 4t.Plug into I: (3t)^2 = 9y ⇒ 9t^2 = 9(4t) ⇒ 9t^2 = 36t ⇒ t(9t − 36) = 0 ⇒ t = 4 (discard t = 0 under positive-integer assumption).Thus x = 12, y = 16 ⇒ |x − y| = 4 (difference uniquely 4).Why Other Options Are Wrong:I alone leaves infinite solutions (choose any y that makes x = 3√y integral); II alone leaves a family scaled by t; “either” is false; “even both not sufficient” is false because the pair is unique (x = 12, y = 16).Common Pitfalls:Including the zero solution (t = 0) leads to non-unique difference; Recovery-First clarifies positivity to avoid that ambiguity.Final Answer:Both statements together are necessary.

Q: In how many days can 14 men complete a piece of work? I. 18 women can complete the same work in 24 days. II. 28 children can complete the same work in 56 days. (No rate equivalence among men, women, children is given.)

Introduction / Context:We are asked for the completion time for 14 men. The statements give only women- and children-based work rates. No cross-equivalence is provided.Given Data / Assumptions:I: 18 women finish in 24 days ⇒ women’s combined rate known.II: 28 children finish in 56 days ⇒ children’s combined rate known.No relation between a man’s rate and a woman’s/child’s rate.Concept / Approach:To find men’s time, we need men’s rate (or a conversion factor to women/children). Absent any conversion, the unknown scale for men makes the target uncomputable.Step-by-Step Reasoning:I alone: provides only women’s aggregate rate; tells nothing about men.II alone: provides only children’s aggregate rate; tells nothing about men.I + II: still no linkage to men, so men’s rate remains undetermined.Why Other Options Are Wrong:Any option claiming sufficiency is incorrect because a key conversion (man↔woman/child) is missing.Common Pitfalls:Assuming a conventional equivalence like “1 man = 2 women” without it being stated; data sufficiency disallows that.Final Answer:Even both statements together are not sufficient.

Q: What is the average age of the six family members A, B, C, D, E, F? I. Average age of D and E is 14 years. II. Average age of A, B, C and F is 50 years. III. Average age of A, B, D and E is 40 years.

Introduction / Context:We need the overall average for six people using given subset averages. Data-sufficiency asks which combination of statements uniquely determines that average.Given Data / Assumptions:I: (D + E)/2 = 14 ⇒ D + E = 28.II: (A + B + C + F)/4 = 50 ⇒ A + B + C + F = 200.III: (A + B + D + E)/4 = 40 ⇒ A + B + D + E = 160.Concept / Approach:Express the grand total S = A + B + C + D + E + F in terms of the partial sums and check which set of statements suffices to find S.Step-by-Step Solution:From I: D + E = 28.From III: A + B + D + E = 160 ⇒ A + B = 160 − 28 = 132.From II: A + B + C + F = 200 ⇒ C + F = 200 − (A + B) = 68.Total S = (A + B) + (C + F) + (D + E) = 132 + 68 + 28 = 228 ⇒ Average = 228/6 = 38.Why Other Options Are Wrong:Any two statements leave an undetermined pair-sum; all three together fix S uniquely.Common Pitfalls:Trying to average the averages; you must convert to sums first.Final Answer:I, II and III.

Q: What is the ratio x : y? I. 40% of x equals 20% of 50. II. 30% of y equals 25% of 72.

Introduction / Context:We must find x : y. Statements provide one equation for x and one for y separately.Given Data / Assumptions:I: 0.4x = 0.2 * 50 ⇒ x = 25.II: 0.3y = 0.25 * 72 ⇒ 0.3y = 18 ⇒ y = 60.Concept / Approach:Knowing x alone or y alone does not yield the ratio. Together they directly produce x : y.Step-by-Step Solution:From I: x = 25.From II: y = 60.Therefore x : y = 25 : 60 = 5 : 12.Why Other Options Are Wrong:Neither I nor II alone suffices for a ratio; “either” is incorrect; “even both not sufficient” is false.Final Answer:Both statements together are necessary.

Q: In a code language, how is the word ‘‘cost’’ written? I. The phrase ‘‘cost was very high’’ is coded as ‘‘& 6 # 1’’. II. The phrases ‘‘some cost was discount’’ and ‘‘some people like discount’’ are coded as ‘‘1 8 7 #’’ and ‘‘8 7 5 %’’ respectively. (…

Introduction / Context:We must identify the exact code for ‘‘cost.’’Given Data / Assumptions:I: {cost, was, very, high} → {&, 6, #, 1} (order unknown).II: ‘‘some cost was discount’’ → {1, 8, 7, #}; ‘‘some people like discount’’ → {8, 7, 5, %}.Consistent one-to-one word↔code mapping is assumed.Concept / Approach:Use intersections of code sets from sentences sharing common words to isolate candidates.Step-by-Step Reasoning:From II: Common words between the two sentences are ‘‘some’’ and ‘‘discount’’ ⇒ common codes are {8, 7}. Thus in the first sentence of II, the remaining codes {1, #} must correspond to {cost, was} (order unknown).From I: {cost, was, very, high} → {&, 6, #, 1}. This is consistent with {cost, was} being {#, 1}, but still does not distinguish which of or 1 maps to ‘‘cost’’.Even using both statements, ‘‘cost’’ ∈ {#, 1} and remains ambiguous.Why Other Options Are Wrong:Neither I nor II alone isolates a single code; combining still leaves two possibilities. Hence “both necessary” is insufficient and incorrect; only “neither…nor both together” fits.Final Answer:Neither statement alone nor both together are sufficient.:

Q: Is D the mother of S? I. L is the husband of D and has exactly three children. II. N is the brother of S and P. P is the daughter of L.

Introduction / Context:We must deduce whether D is S’s mother from family relations.Given Data / Assumptions:I: L is D’s husband; L has exactly three children.II: N is brother of S and P; P is L’s daughter.Standard puzzle convention: “brother/daughter” imply biological child unless otherwise stated.Concept / Approach:Combine sibling info with the fixed count of L’s children and the marriage of L and D.Step-by-Step Reasoning:From II: N, S, P are siblings (N is brother of both S and P). Also P is child of L ⇒ N and S are (typically) children of L as well.From I: L has exactly three children ⇒ the set of L’s children is {N, S, P}.Since L is married to D, and no alternative mother is indicated, D is the mother of L’s children under the usual assumption.Why Other Options Are Wrong:I alone gives no link to S; II alone gives no link to D; both are needed to tie S to L and then to D.Common Pitfalls:Allowing step/adoptive exceptions without textual support; DS uses minimal standard assumptions.Final Answer:Both statements together are necessary.

Question 1

How is the word ‘‘GATES’’ coded in the code language?  I. ‘‘BRICK’’ is coded as ‘‘LDJSC’’ and ‘‘PIN’’ is coded as ‘‘QJQ’’. II. ‘‘WATER’’ is coded as ‘‘SFUBX’’ and ‘‘DIST’’ is coded as ‘‘ITJE’’.

Accepted Answer

Introduction / Context:We must determine a unique coded form of the specific word ‘‘GATES’’ based on sample encodings. Data Sufficiency requires that a single, unambiguous mapping be implied; otherwise the data are insufficient.Given Data / Assumptions:I provides two pairs: BRICK→LDJSC, PIN→QJQ.II provides two pairs: WATER→SFUBX, DIST→ITJE.No explicit rule (e.g., Caesar shift, position-based substitution) is specified; word-level operations, mixed shifts, or polyalphabetic schemes are possible.Concept / Approach:Test for consistent single-letter substitution or uniform shift. The given pairs show inconsistent per-letter shifts across positions (e.g., B→L is +10, R→D is +12 mod 26; W→S is −4, A→F is +5), implying a more complex or variable rule. Without a stated mechanism, multiple encipherments of GATES can fit.Step-by-Step Solution:1) I alone: The heterogeneous shifts across BRICK and PIN do not determine how to transform new letters like G, A, T, E, S. Multiple plausible rules exist (position-wise shifts, alternating patterns, keyed substitution), so I is insufficient.2) II alone: WATER and DIST exhibit different, non-uniform transformations. Extrapolating to GATES remains ambiguous. II is insufficient.3) I + II together: Even combining all four pairs, there is no uniquely implied cipher family; several rule-sets can encode the samples and would encode GATES differently (e.g., per-position shift table, digram rules, or keyed Vigenère-style). Thus, the code for GATES is not uniquely deducible.Verification / Alternative check:Attempt to align letter-wise mappings by position across words of different lengths; conflicts arise, confirming non-uniqueness.Why Other Options Are Wrong:A/B/C/E: None create uniqueness; together or alone, ambiguity persists.Common Pitfalls:Assuming a fixed Caesar shift; enforcing one-to-one letter mapping ignoring position-dependent behavior suggested by the pairs.Final Answer:Both statements together are NOT sufficient.

Question 2

Among schools A, B, C, and D, which school has the highest number of students?  I. School A has fewer students than school D. II. School C has fewer students than school D.

Accepted Answer

Introduction / Context:We need to identify the unique maximum among four schools given partial inequalities. DS focuses on whether D is necessarily the maximum or whether another school (e.g., B) could exceed D.Given Data / Assumptions:I: A < D.II: C < D.Concept / Approach:Combine inequalities and check if D must be strictly the largest, or if another candidate (e.g., B) can surpass D under the constraints.Step-by-Step Solution:1) I alone: Only A < D is known; B and C could be above or below D. Not sufficient.2) II alone: Only C < D is known; A and B unknown. Not sufficient.3) I + II: We have A < D and C < D. However, there is no information about B vs D. It is possible that B > D, making B the highest; or B < D, making D the highest. The top school is not uniquely determined.Verification / Alternative check:Scenarios: (a) Let counts be A=100, B=400, C=90, D=300 ⇒ highest=B. (b) A=100, B=250, C=90, D=300 ⇒ highest=D. Both satisfy I and II.Why Other Options Are Wrong:A/B/C/E: None force uniqueness; combined data still allow multiple maxima.Common Pitfalls:Assuming transitivity where none exists (no relation given for B). Final Answer:Both statements together are NOT sufficient.

Question 3

What is the perimeter of a semicircle?  I. The radius of the semicircle equals half the side of a square. II. The area of the square is 196 sq cm.

Accepted Answer

Introduction / Context:The goal is to compute a numeric perimeter for a semicircle. Conventionally, the perimeter of a semicircle includes the curved length plus the diameter: P = π*r + 2*r. We must see whether the given statements fix r uniquely.Given Data / Assumptions:I: r = (side of square)/2.II: Area of square = 196 sq cm ⇒ side = 14 cm.Concept / Approach:Use II to get the square’s side numerically; use I to link that side to r; then compute P = π*r + 2*r.Step-by-Step Solution:1) From II: side s = √196 = 14 cm.2) From I: r = s/2 = 14/2 = 7 cm.3) Perimeter of semicircle: P = π*r + 2*r = 7π + 14 cm.4) Sufficiency: I alone gives only a relation; II alone gives only s. Together they yield a unique numeric answer.Verification / Alternative check:Some texts ask for curved length only (π*r); here “perimeter” explicitly includes the straight edge (diameter), which is standard for perimeter.Why Other Options Are Wrong:A/B/C/D: Neither I nor II alone is enough; together they are.Common Pitfalls:Omitting the diameter; misreading II as radius instead of side.Final Answer:Both statements together are sufficient, but NEITHER alone is sufficient.

Question 4

What is the exact average of the six numbers: n, 35, 39, 42, p, and w?  I. n is six more than w. II. w is four less than p.

Accepted Answer

Introduction / Context:We must determine the exact arithmetic mean of six values given only linear relations among three variables (n, p, w). The question is sufficient only if a unique numeric average (not an expression in a variable) results.Given Data / Assumptions:I: n = w + 6.II: w = p − 4 (equivalently, p = w + 4).Constants: 35, 39, 42.Concept / Approach:Combine relations and attempt to express the average purely as a number. If a free variable remains, the data are not sufficient.Step-by-Step Solution:1) Average A = (n + 35 + 39 + 42 + p + w)/6.2) Substitute I and II: n = w + 6 and p = w + 4 ⇒ numerator = (w + 6) + 35 + 39 + 42 + (w + 4) + w = 3w + (6 + 35 + 39 + 42 + 4) = 3w + 126.3) Hence A = (3w + 126)/6 = (w/2) + 21.4) The mean depends on w; without a numeric value for w, the average is not uniquely determined.Verification / Alternative check:Pick two different w values (e.g., 0 and 2) to obtain distinct averages (21 and 22), both consistent with I and II—confirming non-uniqueness.Why Other Options Are Wrong:A/B/C/E: Neither statement alone nor together yields a fixed numeric mean; a parameter remains free.Common Pitfalls:Assuming hidden constraints (e.g., integers only still leaves infinitely many values); treating an expression in w as a “determined” average.Final Answer:Both statements are NOT sufficient.

Question 5

What was the percent profit or loss made by selling an article for ₹ 24,000? I. The ratio of selling price to cost price is 5 : 3. II. The difference between cost price and selling price is ₹ 9,600 (direction not stated).

Accepted Answer

Introduction / Context:This is a data-sufficiency problem about profit/loss percentage given a fixed selling price of ₹ 24,000. We must decide which statement(s) give enough information to compute the percentage unambiguously, not actually compute it unless needed to justify sufficiency.Given Data / Assumptions:SP = ₹ 24,000 (fixed).Statement I: SP : CP = 5 : 3.Statement II: |CP − SP| = ₹ 9,600 (direction unspecified).Concept / Approach:Profit% = (SP − CP) / CP * 100; Loss% = (CP − SP) / CP * 100. A ratio SP:CP or explicit CP uniquely determines the percentage. A mere “difference” without direction may yield two possible CP values.Step-by-Step Solution:Using I: SP:CP = 5:3 ⇒ Profit% = (5−3)/3 * 100 = 200/3 ≈ 66.67% (unique). Sufficient.Using II alone: |CP − 24,000| = 9,600 ⇒ CP = 14,400 (profit) or 33,600 (loss). Two different percentages (≈66.67% profit or ≈28.57% loss). Insufficient.Combining does not change the fact that I alone already suffices.Why Other Options Are Wrong:II alone is ambiguous; “either” is false since II fails; “both necessary” is false since I alone works; “even both not sufficient” is false.Common Pitfalls:Interpreting “difference” as signed; here it is stated without direction, producing two CP candidates.Final Answer:Statement I alone is sufficient; Statement II alone is not.

Question 6

What is the difference between two numbers x and y? I. The square of the first number equals 9 times the second: x^2 = 9y. II. The ratio of the first to the second is x : y = 3 : 4. (Assume positive integers to recover solvability.)

Accepted Answer

Introduction / Context:We must determine whether the difference x − y can be uniquely found from the statements. By Recovery-First, we clarify that x and y are positive integers (a standard minimal assumption for such DS items).Given Data / Assumptions:I: x^2 = 9y.II: x : y = 3 : 4.x, y ∈ positive integers.Concept / Approach:Each statement alone gives a relation but not a unique pair. Together they may pin down a single solution.Step-by-Step Solution:From II: Let x = 3t, y = 4t.Plug into I: (3t)^2 = 9y ⇒ 9t^2 = 9(4t) ⇒ 9t^2 = 36t ⇒ t(9t − 36) = 0 ⇒ t = 4 (discard t = 0 under positive-integer assumption).Thus x = 12, y = 16 ⇒ |x − y| = 4 (difference uniquely 4).Why Other Options Are Wrong:I alone leaves infinite solutions (choose any y that makes x = 3√y integral); II alone leaves a family scaled by t; “either” is false; “even both not sufficient” is false because the pair is unique (x = 12, y = 16).Common Pitfalls:Including the zero solution (t = 0) leads to non-unique difference; Recovery-First clarifies positivity to avoid that ambiguity.Final Answer:Both statements together are necessary.

Question 7

In how many days can 14 men complete a piece of work? I. 18 women can complete the same work in 24 days. II. 28 children can complete the same work in 56 days. (No rate equivalence among men, women, children is given.)

Accepted Answer

Introduction / Context:We are asked for the completion time for 14 men. The statements give only women- and children-based work rates. No cross-equivalence is provided.Given Data / Assumptions:I: 18 women finish in 24 days ⇒ women’s combined rate known.II: 28 children finish in 56 days ⇒ children’s combined rate known.No relation between a man’s rate and a woman’s/child’s rate.Concept / Approach:To find men’s time, we need men’s rate (or a conversion factor to women/children). Absent any conversion, the unknown scale for men makes the target uncomputable.Step-by-Step Reasoning:I alone: provides only women’s aggregate rate; tells nothing about men.II alone: provides only children’s aggregate rate; tells nothing about men.I + II: still no linkage to men, so men’s rate remains undetermined.Why Other Options Are Wrong:Any option claiming sufficiency is incorrect because a key conversion (man↔woman/child) is missing.Common Pitfalls:Assuming a conventional equivalence like “1 man = 2 women” without it being stated; data sufficiency disallows that.Final Answer:Even both statements together are not sufficient.

Question 8

What is the average age of the six family members A, B, C, D, E, F? I. Average age of D and E is 14 years. II. Average age of A, B, C and F is 50 years. III. Average age of A, B, D and E is 40 years.

Accepted Answer

Introduction / Context:We need the overall average for six people using given subset averages. Data-sufficiency asks which combination of statements uniquely determines that average.Given Data / Assumptions:I: (D + E)/2 = 14 ⇒ D + E = 28.II: (A + B + C + F)/4 = 50 ⇒ A + B + C + F = 200.III: (A + B + D + E)/4 = 40 ⇒ A + B + D + E = 160.Concept / Approach:Express the grand total S = A + B + C + D + E + F in terms of the partial sums and check which set of statements suffices to find S.Step-by-Step Solution:From I: D + E = 28.From III: A + B + D + E = 160 ⇒ A + B = 160 − 28 = 132.From II: A + B + C + F = 200 ⇒ C + F = 200 − (A + B) = 68.Total S = (A + B) + (C + F) + (D + E) = 132 + 68 + 28 = 228 ⇒ Average = 228/6 = 38.Why Other Options Are Wrong:Any two statements leave an undetermined pair-sum; all three together fix S uniquely.Common Pitfalls:Trying to average the averages; you must convert to sums first.Final Answer:I, II and III.

Question 9

What is the ratio x : y? I. 40% of x equals 20% of 50. II. 30% of y equals 25% of 72.

Accepted Answer

Introduction / Context:We must find x : y. Statements provide one equation for x and one for y separately.Given Data / Assumptions:I: 0.4x = 0.2 * 50 ⇒ x = 25.II: 0.3y = 0.25 * 72 ⇒ 0.3y = 18 ⇒ y = 60.Concept / Approach:Knowing x alone or y alone does not yield the ratio. Together they directly produce x : y.Step-by-Step Solution:From I: x = 25.From II: y = 60.Therefore x : y = 25 : 60 = 5 : 12.Why Other Options Are Wrong:Neither I nor II alone suffices for a ratio; “either” is incorrect; “even both not sufficient” is false.Final Answer:Both statements together are necessary.

Question 10

In a code language, how is the word ‘‘cost’’ written? I. The phrase ‘‘cost was very high’’ is coded as ‘‘& 6 # 1’’. II. The phrases ‘‘some cost was discount’’ and ‘‘some people like discount’’ are coded as ‘‘1 8 7 #’’ and ‘‘8 7 5 %’’ respectively. (…

Accepted Answer

Introduction / Context:We must identify the exact code for ‘‘cost.’’Given Data / Assumptions:I: {cost, was, very, high} → {&, 6, #, 1} (order unknown).II: ‘‘some cost was discount’’ → {1, 8, 7, #}; ‘‘some people like discount’’ → {8, 7, 5, %}.Consistent one-to-one word↔code mapping is assumed.Concept / Approach:Use intersections of code sets from sentences sharing common words to isolate candidates.Step-by-Step Reasoning:From II: Common words between the two sentences are ‘‘some’’ and ‘‘discount’’ ⇒ common codes are {8, 7}. Thus in the first sentence of II, the remaining codes {1, #} must correspond to {cost, was} (order unknown).From I: {cost, was, very, high} → {&, 6, #, 1}. This is consistent with {cost, was} being {#, 1}, but still does not distinguish which of or 1 maps to ‘‘cost’’.Even using both statements, ‘‘cost’’ ∈ {#, 1} and remains ambiguous.Why Other Options Are Wrong:Neither I nor II alone isolates a single code; combining still leaves two possibilities. Hence “both necessary” is insufficient and incorrect; only “neither…nor both together” fits.Final Answer:Neither statement alone nor both together are sufficient.:

Question 11

Is D the mother of S? I. L is the husband of D and has exactly three children. II. N is the brother of S and P. P is the daughter of L.

Accepted Answer

Introduction / Context:We must deduce whether D is S’s mother from family relations.Given Data / Assumptions:I: L is D’s husband; L has exactly three children.II: N is brother of S and P; P is L’s daughter.Standard puzzle convention: “brother/daughter” imply biological child unless otherwise stated.Concept / Approach:Combine sibling info with the fixed count of L’s children and the marriage of L and D.Step-by-Step Reasoning:From II: N, S, P are siblings (N is brother of both S and P). Also P is child of L ⇒ N and S are (typically) children of L as well.From I: L has exactly three children ⇒ the set of L’s children is {N, S, P}.Since L is married to D, and no alternative mother is indicated, D is the mother of L’s children under the usual assumption.Why Other Options Are Wrong:I alone gives no link to S; II alone gives no link to D; both are needed to tie S to L and then to D.Common Pitfalls:Allowing step/adoptive exceptions without textual support; DS uses minimal standard assumptions.Final Answer:Both statements together are necessary.

Question 12

How many students are in the class? I. The number is more than 20 but less than 27. II. The number is more than 24 but less than 31, and the class can be partitioned into equal groups of 5.

Accepted Answer

Introduction / Context:We must determine a unique class size using two overlapping ranges and a divisibility condition.Given Data / Assumptions:I: 20 < n < 27 ⇒ n ∈ {21, 22, 23, 24, 25, 26}.II: 24 < n < 31 and n is a multiple of 5 ⇒ n ∈ {25, 30}.Concept / Approach:Intersect the candidate sets from I and II.Step-by-Step Solution:From I: {21,22,23,24,25,26}.From II: {25,30}.Intersection ⇒ {25} only. Thus n = 25.Why Other Options Are Wrong:Neither statement alone yields a unique value; together they do.Final Answer:Both statements together are necessary.

Question 13

Among five friends J, K, L, M, N, each of different height, who is the second tallest? I. N is taller than M and K; K is shorter than M. II. L is taller than M and N; J is not the tallest.

Accepted Answer

Introduction / Context:We must uniquely identify the second tallest using partial order information.Given Data / Assumptions:I: N > M > K; no info on L, J.II: L > M and L > N; J is not tallest.Concept / Approach:Combine inequalities and see whether a unique second tallest emerges.Step-by-Step Reasoning:From I: N > M > K; positions of L and J unknown.From II: L is taller than both M and N; J is not tallest, but J’s relation to L and N is unknown.Together: L > N > M > K; J is somewhere below L but could be above or below N. If J is between L and N, second tallest is J; if J is below N, second tallest is N. Non-unique.Why Other Options Are Wrong:Neither I nor II alone suffices, and even together ambiguity remains.Final Answer:Neither statement alone nor both together are sufficient.

Question 14

Five letters A, E, G, N, R are arranged left to right under certain conditions. Which letter occupies the third position? I. G is second to the right of A; E is immediately to the right of G; exactly two letters lie between R and G. II. N is exactly…

Accepted Answer

Introduction / Context:We must find the letter in the 3rd position of a 5-slot linear arrangement.Reasoning from Statement I:Let positions be 1–5. “G second right of A” ⇒ G = A + 2. “E immediately right of G” ⇒ E = G + 1 = A + 3. “Exactly two between R and G” ⇒ |R − G| = 3.Try A = 2 ⇒ G = 4, E = 5, R = 1 (only valid). Remaining letter N occupies position 3. This is the unique realization; hence I alone suffices and the 3rd letter is N.Reasoning from Statement II:N exactly between A and G and neither A nor G is at an end still allows multiple valid configurations; the 3rd position is not fixed uniquely by II alone.Why Other Options Are Wrong:II alone is not enough; “either” is false; “neither” is false because I alone is sufficient.Final Answer:Statement I alone is sufficient; Statement II alone is not.

Question 15

In how many days will B alone complete the work? I. A and B together finish in 8 days. II. B and C together finish in 10 days. III. A and C together finish in 12 days.

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Introduction / Context:Given three pairwise times, find B’s solo time.Concept / Approach:Let a, b, c be daily rates. Then a + b = 1/8, b + c = 1/10, a + c = 1/12. Solve for b.Step-by-Step Solution:Add first two and subtract the third: (a + b) + (b + c) − (a + c) = 1/8 + 1/10 − 1/12.Left side simplifies to 2b ⇒ 2b = 15/120 + 12/120 − 10/120 = 17/120 ⇒ b = 17/240.Therefore B alone finishes in 1/b = 240/17 days.Why Other Options Are Wrong:Any two equations leave three unknowns inadequately constrained for a unique b; all three are required.Final Answer:I, II and III.

Question 16

What is the annual rate of interest (per annum)? I. An amount doubles at simple interest in 10 years. II. The difference between compound interest and simple interest on ₹ 15,000 for 2 years is ₹ 150. III. Compound interest after 8 years exceeds the…

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Introduction / Context:We must identify which statements suffice to find the unique annual rate r.Reasoning:I alone: At simple interest A = P(1 + r t). Doubling in 10 years ⇒ 2P = P(1 + 10r) ⇒ r = 0.1 = 10% p.a. Sufficient.II alone: For 2 years, CI − SI = P*(r/100)^2. So 150 = 15000*(r^2)/10000 ⇒ r^2 = 100 ⇒ r = 10% (taking positive rate). Sufficient.III: “CI after 8 years exceeds principal” only implies r > 0, which does not fix r. Insufficient.Why Other Options Are Wrong:Neither “Only I” nor “Only II” captures that both I or II, individually, suffice; combining I with III is unnecessary.Final Answer:Either I or II is sufficient (III is irrelevant).

Question 17

What marks did Abhijit score in English? I. Marks in Mathematics exceed marks in Science by 20. II. Total marks in Mathematics, Science, English = 197. III. Marks in Science exceed marks in English by 12.

Accepted Answer

Introduction / Context:We must uniquely determine the English score using linear relations among three subjects.Step-by-Step Solution:Let E = English, S = Science, M = Math.I: M = S + 20.II: M + S + E = 197.III: S = E + 12.Substitute III into I ⇒ M = (E + 12) + 20 = E + 32.Use II: (E + 32) + (E + 12) + E = 197 ⇒ 3E + 44 = 197 ⇒ E = 51.Sufficiency Check:Any two statements leave one degree of freedom; all three are needed for a unique E.Final Answer:I, II and III.

Question 18

In how many days will Y alone complete the work? I. X and Y together finish in 8 days. II. Y and Z together finish in 10 days. III. X and Z together finish in 12 days.

Accepted Answer

Introduction / Context:Identical structure to the prior rates problem; now solve for Y’s solo rate.Solution Sketch:Let x, y, z be daily rates. We have x + y = 1/8, y + z = 1/10, x + z = 1/12. Then 2y = (1/8 + 1/10 − 1/12) = 17/120 ⇒ y = 17/240 ⇒ Y alone takes 240/17 days.Sufficiency:All three equations are required to isolate y uniquely.Final Answer:I, II and III.

Question 19

What is Supriya’s present age? I. Supriya is 3 years older than Priya. II. The ratio of Priya’s and Reshma’s ages is 3 : 4.

Accepted Answer

Introduction / Context:We need Supriya’s exact age.Reasoning:I: S = P + 3 (S = Supriya, P = Priya). Infinite possibilities without absolute values.II: P : R = 3 : 4 (R = Reshma). No link to S or absolute ages.I + II: Still no absolute age; the pair gives only relations, not totals.Final Answer:Both statements together are not sufficient.

Question 20

What is the annual rate of interest (per annum)? I. ₹ 6,200 earns ₹ 1,736 simple interest in 2 years. II. ₹ 4,500 earns ₹ 1,348.20 compound interest in 2 years.

Accepted Answer

Introduction / Context:Determine r uniquely from either SI or CI data over 2 years.Calculations:I: r = SI / (P * t) = 1736 / (6200 * 2) = 0.14 ⇒ 14% p.a.II: For 2 years, CI = P[(1 + r/100)^2 − 1]. Solve (1 + r/100)^2 − 1 = 1348.20 / 4500 = 0.2996 ⇒ (1 + r/100)^2 = 1.2996 ⇒ r ≈ 14% (unique, positive).Final Answer:Either statement alone is sufficient.

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