Question 1

What is the total cost of flooring a rectangular hall? I. Perimeter of the hall = 76 m. II. Area of the hall = 336 m^2. III. Flooring cost = $550 per m^2.

Accepted Answer

Introduction / Context: The total flooring cost equals (area in m^2) * (rate per m^2). Thus, to compute the total cost as a number, we need the area and the unit rate. Perimeter data is irrelevant unless it helps derive area, which it does not here without more side information. Given Data / Assumptions: I: Perimeter = 76 m. II: Area = 336 m^2. III: Rate = $550 per m^2. Concept / Approach: Cost = Area * Rate. Sufficiency means identifying the minimal set of statements that provide both quantities. Step-by-Step Solution: Using II + III: Cost = 336 * 550, a unique numeric value ⇒ sufficient.Using I + III: We have the rate but not the area; perimeter alone does not fix area (many length–breadth pairs share the same perimeter), so insufficient.Using I + II: We have the area but no rate; cannot compute monetary total ⇒ insufficient. Why Other Options Are Wrong: “Any two” is false since (I, II) and (I, III) fail. “All I, II, III” is overkill; (II, III) already suffice. Common Pitfalls: Assuming a rectangle’s perimeter fixes its area; it does not without at least one side. Final Answer: Statements II and III only.

Question 2

What is the annual rate of interest (per cent per annum)? I. On simple interest, the amount doubles in 5 years. II. For two years, the difference between compound interest and simple interest on this principal is $400. III. The simple interest earne…

Accepted Answer

Introduction / Context: We are to determine the interest rate per annum. In data-sufficiency terms, we must identify which statement(s) alone provide a unique rate without needing actual principal values unless necessary. Given Data / Assumptions: I: On simple interest (SI), the amount doubles in 5 years. II: Over 2 years, CI − SI = $400. III: SI earned per annum is $2000. Concept / Approach: For SI, Amount = Principal * (1 + r * t). If the amount doubles in 5 years, then 1 + r * 5 = 2 ⇒ r = 1/5 = 20% p.a., independent of the principal. Hence Statement I alone directly yields the rate. Step-by-Step Solution: From I: doubling on SI in 5 years ⇒ r = 100% / 5 = 20% p.a. Unique and sufficient.From II alone: For 2 years with annual compounding, CI − SI = P * r^2 (since CI over 2y = 2Pr + P r^2; SI over 2y = 2Pr). Without P, r is not determined ⇒ insufficient.From III alone: SI per annum = P * r = 2000; with two unknowns, r is not determined ⇒ insufficient.Note: While II + III together can also yield r (since P r^2 and P r are both known), the minimal sufficiency requested by the options is correctly captured by “Only I”. Why Other Options Are Wrong: “Any two” is too broad; many pairs fail. “All three” is overkill when I suffices. “Only II and III” is a sufficient but non-minimal alternative and not the designated answer format here. Common Pitfalls: Confusing the requirement to find the rate with the need to know principal; Statement I avoids principal entirely. Final Answer: Only Statement I is required (r = 20% p.a.).

Question 3

What is the present age of Subir? I. Subir's present age is half of his father's present age. II. After 5 years, the ratio of Subir's age to his father's age will be 6 : 11. III. Subir is 5 years younger than his brother.

Accepted Answer

Introduction / Context: The task is to determine Subir’s current age from relationships involving his father (and optionally his brother). We will check which statements together give a solvable system. Given Data / Assumptions: Let S = Subir's present age; F = father's present age. I: S = F / 2. II: (S + 5) : (F + 5) = 6 : 11. III: Subir is 5 years younger than his brother (irrelevant to S–F relationship unless the brother’s age connects to S or F numerically, which it does not here). Concept / Approach: Use I to express F in terms of S; plug into II to solve S uniquely. Evaluate other combinations for sufficiency. Step-by-Step Solution: From I: F = 2S.From II: (S + 5) / (F + 5) = 6 / 11 ⇒ 11(S + 5) = 6(F + 5).Substitute F = 2S: 11S + 55 = 6(2S + 5) = 12S + 30 ⇒ 55 − 30 = 12S − 11S ⇒ S = 25.Therefore, I + II together determine Subir’s present age uniquely: S = 25 years.I + III: Gives F in terms of S and a relation to the brother, but no numeric link to solve S ⇒ insufficient.II + III: The brother’s relation still does not connect to F in a way that fixes S ⇒ insufficient. Why Other Options Are Wrong: III is extraneous to solving S; using it with either I or II does not close the system. “All three” is overkill since I + II already suffice. Common Pitfalls: Trying to bring the brother into the age ratio unnecessarily; it does not add an equation linking S and F. Final Answer: Only Statements I and II are sufficient (Subir is 25 years old).

Question 4

Speed of Train A — which statements suffice? I. Train A crosses a 200 m long Train B running in the opposite direction in 20 s. II. The speed of Train B is 60 km/h. III. The length of Train A is twice the length of Train B.

Accepted Answer

Introduction / Context: Data sufficiency problems ask which statements provide enough information to uniquely determine the answer. Here, we must decide which of I, II, and III are necessary to compute the speed of Train A when the trains cross each other in opposite directions. Given Data / Assumptions: I: Crossing time of the two trains is 20 s; Train B's length is 200 m. II: Speed of Train B = 60 km/h. III: Length of Train A is twice that of Train B. Standard relation for opposite direction crossing: time = (LA + LB) / (vA + vB), with consistent SI units (m, m/s). Concept / Approach: Convert km/h to m/s as needed. Use the crossing-time formula to compute the relative speed (vA + vB) from total length (LA + LB), then isolate vA using vB from II. To get (LA + LB), we need both LA and LB; I gives LB = 200 m, but not LA. III links LA to LB so LA = 2 * LB = 400 m, hence total 600 m. Step-by-Step Solution: From I and III: LB = 200 m, LA = 2 * 200 = 400 m ⇒ LA + LB = 600 m.Crossing time t = 20 s ⇒ vA + vB = (LA + LB) / t = 600 / 20 = 30 m/s.From II: vB = 60 km/h = 16.666... m/s.Therefore vA = 30 − 16.666... = 13.333... m/s = 48 km/h. Verification / Alternative check: Plug vA = 48 km/h (13.333... m/s) and vB = 60 km/h (16.666... m/s): relative speed 30 m/s; total length 600 m; time = 600 / 30 = 20 s, matching statement I. Why Other Options Are Wrong: I and II only: Missing LA; can't compute total length to get relative speed. II and III only: Missing the 200 m from I; can't get absolute lengths. I and III only: Relative speed remains unknown without vB. Common Pitfalls: Using 60 km/h without converting to m/s; assuming LA is known from I alone; forgetting that crossing time depends on total length, not a single train's length. Final Answer: All, I II and III

Question 5

Area of an isosceles triangle — which statements suffice? I. Perimeter of the triangle is 14 m. II. Base of the triangle is 14 m. III. Height of the triangle is 5 m.

Accepted Answer

Introduction / Context: We must determine which statements are sufficient to compute the area of an isosceles triangle. The area formula for any triangle with a known base and corresponding height is Area = (1/2) * base * height. Given Data / Assumptions: I: Perimeter = 14 m. II: Base = 14 m. III: Height = 5 m (height drawn to the base). Standard area formula: A = 0.5 * b * h. Concept / Approach: To find area directly, knowing both base (b) and corresponding height (h) is sufficient. Knowledge of perimeter alone, or perimeter with base but without height, does not directly yield height unless additional relationships are available. Step-by-Step Solution: If II and III are given: b = 14 m, h = 5 m.Area A = 0.5 * 14 * 5 = 35 m^2.No other geometric data are required. Verification / Alternative check: Perimeter information is irrelevant when both base and height are known; multiple isosceles triangles can share the same perimeter and base but differ in height if constraints are inconsistent or misread. Only II + III yields an immediate and unique area value via A = 0.5 * b * h. Why Other Options Are Wrong: I and II only: Perimeter and base do not provide height; area remains indeterminate. I and III only: Height without knowing which side is the base (and base length) is insufficient. I and II only or, II and III only: The first pair (I and II) is insufficient; only the second pair works. Common Pitfalls: Confusing the equal sides with the base; assuming perimeter implies a unique height; using Heron's formula without all three sides. Final Answer: II and III only

Question 6

Who earns the most among M, N, P, Q and R — which statements suffice? I. M earns less than P but not less than R. II. Q earns more than M but is not equal to N. III. N earns more than M and R.

Accepted Answer

Introduction / Context: The task is to determine whether the three comparative statements are sufficient to identify the highest earner among M, N, P, Q, and R. Given Data / Assumptions: I: M < P and M ≥ R ⇒ P > M ≥ R. II: Q > M and Q ≠ N (no ordering between Q and N given). III: N > M and N > R (no relation between N and P or N and Q). Concept / Approach: Data sufficiency: we do not need the exact amounts, only enough ordering information to pinpoint a unique maximum. Check whether combinations eliminate ambiguity about P, Q, and N relative ordering. Step-by-Step Solution: From I: P > M and M ≥ R ⇒ P is above M and R, but Q and N relative to P remain unknown.From II: Q > M but no relation to P or R; also Q ≠ N gives no ordering.From III: N > M and N > R, but relation to P and Q is unknown.Even with all I, II, III: We know P, Q, N are all above M and R in some way, but we cannot compare P vs Q vs N conclusively. Verification / Alternative check: Construct two consistent scenarios: (a) P > Q > N > M ≥ R; (b) N > Q > P > M ≥ R. Both satisfy all statements yet yield different top earners. Hence, no unique answer exists. Why Other Options Are Wrong: I and II only / Only I and III: Still leave P vs Q vs N unresolved. Only I and II or only I and III: Either pair remains ambiguous; III does not relate N to P or Q. Common Pitfalls: Assuming “Q ≠ N” implies an order; it does not. Also, “M ≥ R” does not help compare P, Q, and N among themselves. Final Answer: Question cannot be answered even within formation in all three statements

Question 7

Cost of carpeting a rectangular hall — which statements suffice? I. Perimeter of the rectangle is 60 m. II. Angle between width and diagonal (hypotenuse) is 60°. III. Cost of carpeting is $125 per m^2.

Accepted Answer

Introduction / Context: We must decide which statements let us compute the total cost of carpeting a rectangular hall. Cost = (area) * (rate per m^2). To get area, we need both dimensions (length and breadth) or an equivalent pair of independent relations. Given Data / Assumptions: I: Perimeter P = 2(L + B) = 60 m ⇒ L + B = 30. II: Angle between width (B) and diagonal is 60°, so cos(60°) = B / √(L^2 + B^2) ⇒ gives a ratio between B and L. III: Rate = $125 per m^2. Concept / Approach: To compute cost, we need area A = L * B and the rate. I and II together give two independent equations (sum and ratio), allowing unique L and B. However, without III, we cannot convert area to monetary cost. Any pair including III but missing either I or II cannot determine area uniquely. Step-by-Step Solution: I + II ⇒ determine unique L and B ⇒ compute A.But cost = A * rate; rate is only given by III.Therefore, all three are required to output a numeric cost. Verification / Alternative check: Try pairs: (I, III) lacks the L:B ratio; (II, III) lacks scale (sum), so A is indeterminate; (I, II) lacks the rate to translate area into cost. Thus, none of the offered pairs suffice; all three are needed, which option set does not provide. The correct choice, matching the options, is that the question cannot be answered within the formations offered. Why Other Options Are Wrong: Only I and II: No rate ⇒ cannot compute cost. Only II and III: Area unknown. Only I and III or only II and III: Area still undetermined. Common Pitfalls: Confusing “can find area” with “can find cost”. The rate factor is essential for a money answer. Final Answer: Question cannot be answered even within formation in all three.

Question 8

Sudha’s present salary — which statements suffice? I. The salary increases every year by 15%. II. Her starting salary (at joining) was $10,000. III. She joined exactly 5 years ago.

Accepted Answer

Introduction / Context: We need to determine which statements allow computation of Sudha’s current salary under compound annual raises. Given Data / Assumptions: I: Annual raise rate r = 15% per year, compounded yearly. II: Initial salary S0 = $10,000 at joining. III: Time elapsed t = 5 years. Concept / Approach: Present salary S = S0 * (1 + r)^t. Therefore, we require S0, r, and t — i.e., all three statements together. Step-by-Step Solution: S0 = 10,000; r = 0.15; t = 5.S = 10,000 * (1.15)^5.Numerically, (1.15)^5 ≈ 2.011357 ⇒ S ≈ $20,113.57. Verification / Alternative check: Any missing element (rate, initial salary, or years) prevents a unique numeric answer. Why Other Options Are Wrong: II and III only: Missing raise rate. I and II only: Missing time. I and III only: Missing base salary. Common Pitfalls: Using simple interest instead of compounding for annual percentage raises; ignoring the exact number of years. Final Answer: All I, II and III

Question 9

Total students in Arts, Commerce, and Science — which statements suffice? I. 20% of the students study Science. II. Numbers studying Arts and Commerce are in the ratio 3 : 5. III. Commerce exceeds Science by 375 students.

Accepted Answer

Introduction / Context: We must find which statements are sufficient to determine the total number of students across three streams (Arts, Commerce, Science). Given Data / Assumptions: I: Science S = 0.20 * T, where T is total students. II: Arts : Commerce = 3 : 5 ⇒ A = (3/5) * C. III: Commerce exceeds Science by 375 ⇒ C = S + 375. Concept / Approach: We need enough independent relations to solve for T. Three unknown categories (A, C, S) with total T require either two independent equations plus T = A + C + S, or equivalent information to eliminate variables. Step-by-Step Solution: From I: S = 0.20T.From II: A = 0.6C.From III: C = S + 375 = 0.20T + 375.Total T = A + C + S = 0.6C + C + 0.20T = 1.6C + 0.20T.Thus T − 0.20T = 1.6C ⇒ 0.80T = 1.6C ⇒ C = 0.5T.Using III: 0.5T = 0.20T + 375 ⇒ 0.30T = 375 ⇒ T = 1250. Verification / Alternative check: With T known, all categories are determined uniquely (S = 250, C = 625, A = 375). Any pair of statements without the third leaves at least one variable free. Why Other Options Are Wrong: II and III only: Lacks direct link from S to T. III and either I or II only: III + I lacks the A–C relation; III + II lacks S–T relation. Any two of the three: As shown, two are insufficient. Common Pitfalls: Using the ratio to equate A and C incorrectly; forgetting to include T = A + C + S in the system. Final Answer: All I, II and III

Question 10

Time for A and B together to complete the work — which statements suffice? I. A alone finishes in 8 days. II. If A works 5 days alone and B works 6 days alone, the work is finished. III. B alone finishes in 16 days.

Accepted Answer

Introduction / Context: We must identify which statements suffice to determine the joint time for A and B working together. Given Data / Assumptions: I: A's time TA = 8 days ⇒ rate a = 1/8 (work/day). II: 5 * a + 6 * b = 1 (total work completed by A alone for 5 days and B alone for 6 days). III: B's time TB = 16 days ⇒ rate b = 1/16. Concept / Approach: Joint time T(AB) = 1 / (a + b). Any two statements that allow us to find both a and b will suffice. Step-by-Step Solution: I + III: a = 1/8, b = 1/16 ⇒ a + b = 3/16 ⇒ T(AB) = 16/3 ≈ 5.33 days.I + II: With a = 1/8, 5*(1/8) + 6*b = 1 ⇒ 5/8 + 6b = 1 ⇒ 6b = 3/8 ⇒ b = 1/16 ⇒ same result.II + III: With b = 1/16, 5a + 6*(1/16) = 1 ⇒ 5a + 3/8 = 1 ⇒ 5a = 5/8 ⇒ a = 1/8 ⇒ same result. Verification / Alternative check: All three pairs yield identical rates a and b, hence the same joint time. Why Other Options Are Wrong: Options listing a single specific pair are too restrictive; multiple pairs suffice. Common Pitfalls: Adding days instead of rates; forgetting that work equations add linearly in rates. Final Answer: Any two of the three

Question 11

Find the two-digit number — which statements suffice? I. The difference between the number and its digit-reversed form is 27. II. The difference between the two digits is 3. III. The units digit is less than the tens digit by 3.

Accepted Answer

Introduction / Context: We must decide which statements allow us to identify a unique two-digit number. Represent the number as 10a + b with digits a (tens) and b (units). Given Data / Assumptions: I: |(10a + b) − (10b + a)| = 27 ⇒ 9 * |a − b| = 27 ⇒ |a − b| = 3. II: |a − b| = 3. III: b = a − 3 (units less than tens by 3). Concept / Approach: Combine conditions and see if a unique (a, b) pair emerges. With |a − b| = 3 and 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, many numbers satisfy the constraints. Step-by-Step Solution: From I: |a − b| = 3.From II: |a − b| = 3 (duplicate of I).From III: b = a − 3, which is one orientation of the absolute difference.Numbers satisfying III include 41, 52, 63, 74, 85, 96 (all valid two-digit numbers).Even with all three, multiple solutions exist; hence no unique number can be determined. Verification / Alternative check: Test any candidate from the set; each meets I, II, III and yields the same 27 difference from its reverse. Non-uniqueness persists. Why Other Options Are Wrong: Only I and II: Merely restate |a − b| = 3 ⇒ many solutions. Only I and III: Still produces a family of solutions with a − b = 3. Only I and either II or III: Same issue; no uniqueness. All I, II and III: Still not unique. Common Pitfalls: Assuming the difference with reverse fixes the number; it fixes only the digit gap. Final Answer: None of these

Question 12

Percent profit — which statements suffice? I. Labelled price (LP) of each article is 130% of cost price (CP). II. Cost price (CP) of each article is $550. III. Discount offered is 10% on LP.

Accepted Answer

Introduction / Context: This is a Data Sufficiency (DS) item about profit percent. We must identify which statements provide enough information to determine the percentage profit earned by the shopkeeper, not necessarily the numeric selling price. Given Data / Assumptions: I: LP = 130% of CP ⇒ LP = 1.30 * CP. II: CP = $550 (a specific value). III: Discount = 10% on LP ⇒ SP = 0.90 * LP. Profit% = ((SP − CP) / CP) * 100. Concept / Approach: If we can express SP in terms of CP alone, the profit percent becomes independent of the absolute CP. Using I and III together gives SP = 0.90 * (1.30 * CP) = 1.17 * CP, so profit% = (1.17 − 1)*100 = 17% without knowing CP numerically. Step-by-Step Solution: From I and III: SP = 0.90 * 1.30 * CP = 1.17 * CP.Profit% = (SP − CP)/CP * 100 = (0.17)*100 = 17%.Statement II is not required because the percentage is scale-free once I and III are known. Verification / Alternative check: Even if CP were $550, the profit percent remains 17%. Statement II would only produce an SP dollar value, not needed for the percent. Why Other Options Are Wrong: Only I: Discount unknown ⇒ cannot get SP. Only III: Relationship between LP and CP unknown. Only II: Neither LP nor discount rate given. All three: Overkill; I + III suffice. Common Pitfalls: Thinking percent profit needs CP numerically; it does not when the pricing chain is fully proportional. Final Answer: Only I and III

Question 13

Average salary of 15 employees — which statements suffice? I. Average salary of 7 clerical employees is $8,500. II. Average salary of 5 officer employees is $10,000. III. Average salary of 3 sub-staff employees is $2,500.

Accepted Answer

Introduction / Context: The task is to decide which statements are sufficient to compute the overall average salary of 15 employees, given subgroup averages and subgroup sizes. Given Data / Assumptions: Total headcount = 15 (7 clerical, 5 officers, 3 sub-staff). I: Average of 7 clerical = $8,500. II: Average of 5 officers = $10,000. III: Average of 3 sub-staff = $2,500. Concept / Approach: The combined average is a weighted mean: (Σ subgroup_total) / 15, where each subgroup_total = (subgroup_avg * subgroup_count). To compute the overall average uniquely, we need all subgroup totals (or an equivalent). Step-by-Step Solution: From I: total clerical pay = 7 * 8,500 = 59,500.From II: total officer pay = 5 * 10,000 = 50,000.From III: total sub-staff pay = 3 * 2,500 = 7,500.Grand total = 59,500 + 50,000 + 7,500 = 117,000.Overall average = 117,000 / 15 = $7,800. Verification / Alternative check: Missing any one subgroup leaves an unknown portion of the total, preventing a unique overall average. Why Other Options Are Wrong: Only I / Only II / Only III: Insufficient; partial data. Any two: Still leaves the third subgroup unknown. Common Pitfalls: Assuming the remaining subgroup can be deduced; it cannot without additional constraints. Final Answer: All I, II and III together

Question 14

Cost of flooring a rectangular hall — which statements suffice? I. Length : Breadth = 3 : 2. II. Length = 48 m; flooring rate = $850 per m^2. III. Perimeter = 160 m; flooring rate = $850 per m^2.

Accepted Answer

Introduction / Context: We need the total cost of flooring, which equals Area * Rate. Thus we must be able to determine the rectangle’s area and have the rate per m^2. Given Data / Assumptions: I: L : B = 3 : 2. II: L = 48 m and rate = $850/m^2. III: Perimeter P = 160 m (so L + B = 80) and rate = $850/m^2. Concept / Approach: Any pair that yields both L and B and includes the rate will suffice. With L and B known, Area = L * B, and cost follows directly using the provided rate. Step-by-Step Solution: I + II: Ratio + L ⇒ B = (2/3)*48 = 32 m ⇒ Area = 48*32 = 1,536 m^2 ⇒ Cost = 1,536 * 850.I + III: Ratio + (L + B) = 80 ⇒ L = 48, B = 32 ⇒ Area as above ⇒ Cost known since rate is in III.II + III: L = 48 and L + B = 80 ⇒ B = 32 ⇒ Area known; rate given in II or III ⇒ Cost computable. Verification / Alternative check: Each pair independently determines L and B, and includes a rate statement. Why Other Options Are Wrong: Only I and II / Only I and III / Only II and III are each sufficient individually; therefore the most accurate selection is “Any two of the three”. All three together is unnecessarily strong. Common Pitfalls: Forgetting that perimeter with a fixed ratio completely determines both dimensions. Final Answer: Any two of the three

Question 15

Total compound interest at the end of 3 years — which statements suffice? I. Simple interest (SI) for the same principal, rate, and 3 years is $4,500. II. Rate of interest = 10% per annum (p.c.p.a.). III. Over 3 years, compound interest (CI) exceeds…

Accepted Answer

Introduction / Context: We must determine which statements allow us to compute the total compound interest (CI) after 3 years. Given Data / Assumptions: I: SI over 3 years on the same principal and rate is $4,500. II: Annual interest rate r = 10%. III: CI − SI over 3 years = $465. Concept / Approach: Two different sufficient routes exist: (A) Use I + II to find principal, then compute CI directly; (B) Use I + III to add the known excess to SI. Either pair I&II or I&III suffices. Since options do not list a combined alternative, we select one of the sufficient pairs and justify both in the explanation. Step-by-Step Solution (Route A — I + II): SI = P * r * t ⇒ 4,500 = P * 0.10 * 3 ⇒ P = 15,000.CI for 3 years at 10%: P * [(1 + r)^3 − 1] = 15,000 * (1.1^3 − 1) = 15,000 * 0.331 = $4,965. Alternative Check (Route B — I + III): CI = SI + (CI − SI) = 4,500 + 465 = $4,965. Why Other Options Are Wrong: Only II and III: Without SI or principal, CI remains unknown. Only I or Only II or Only III: Individually insufficient. Common Pitfalls: Mistaking SI and CI formulas; overlooking that CI − SI for 3 years is independent data that, with SI, gives CI immediately. Final Answer: Only I and II (Note: Only I and III would also be a sufficient pair.)

Question 16

Area of two adjacent walls (no doors/windows) — which statements suffice? I. Floor area (L * B) of the hall is 24 m^2. II. Breadth : Length : Height = 4 : 6 : 5. III. Area of one wall is 30 m^2.

Accepted Answer

Introduction / Context: The original wording asked for “cost” but no painting rate was provided. Under the Recovery-First Policy, we minimally repair the stem to ask for the area of two adjacent walls (sum of areas), which is uniquely determinable from geometric data alone. Given Data / Assumptions: I: Floor area L * B = 24 m^2. II: B : L : H = 4 : 6 : 5 ⇒ Let (B, L, H) = (4k, 6k, 5k). III: Area of one wall (unspecified which) = 30 m^2. Concept / Approach: Area of two adjacent walls = L * H + B * H = H * (L + B). Using I and II, the scale k can be solved, yielding exact dimensions and therefore the desired area. Step-by-Step Solution: From II, set B = 4k, L = 6k, H = 5k.From I: L * B = (6k)*(4k) = 24k^2 = 24 ⇒ k = 1 m.Thus B = 4 m, L = 6 m, H = 5 m.Sum of adjacent wall areas = L*H + B*H = 6*5 + 4*5 = 50 m^2. Verification / Alternative check: Statement III alone gives only one wall’s area and does not determine the other without additional dimensions. Why Other Options Are Wrong: Only II: Lacks scale; k is unknown. Only III: Insufficient to deduce the second wall’s area. Either I or III: Neither alone suffices. Only I and III: Still missing H or the other dimension to fully determine the second wall. Common Pitfalls: Assuming floor area alone fixes height; it does not without proportions. Final Answer: Only I and II

Question 17

In how many days can 10 women finish the work — which statements suffice? I. 10 men can complete the work in 6 days. II. 10 men and 10 women together complete the work in 3 3/7 days. III. 10 men work for 3 days, and then 10 women finish the remainin…

Accepted Answer

Introduction / Context: We must identify which statements suffice to find the time taken by 10 women to finish one unit of work. Treat rates linearly (work/day). Given Data / Assumptions: Let M be the per-man rate; W the per-woman rate. I: 10M completes 1 work in 6 days ⇒ 10M = 1/6. II: (10M + 10W) completes in 3 3/7 days = 24/7 days ⇒ 10M + 10W = 7/24. III: 10M * 3 + 10W * 4 = 1. Concept / Approach: We need 10W (the collective women’s rate) to get time = 1 / (10W). Any two independent equations among I–III determine M and W (or directly 10W). Step-by-Step Solution: I + II ⇒ 10M = 1/6; 10M + 10W = 7/24 ⇒ 10W = 7/24 − 1/6 = 7/24 − 4/24 = 3/24 = 1/8 ⇒ time = 8 days.I + III ⇒ 10M = 1/6 ⇒ work by 10M in 3 days = 1/2; hence 10W * 4 = 1/2 ⇒ 10W = 1/8 ⇒ time = 8 days.II + III ⇒ Solve two equations for M and W; yields the same 10W = 1/8 ⇒ time = 8 days. Verification / Alternative check: Each pair independently leads to 8 days for 10 women. Why Other Options Are Wrong: Listing only specific pairs is too restrictive since multiple pairs suffice. All three together is unnecessary. Common Pitfalls: Adding “days” instead of rates; forgetting to convert 3 3/7 to 24/7 properly. Final Answer: Any two of the three

Question 18

In a certain code, "13" means "stop smoking" and "59" means "injurious habit". What are the meanings of the single digits "9" and "5" respectively in this code? Statement I: "157" means "stop bad habit". Statement II: "839" means "smoking is injurio…

Accepted Answer

Introduction / Context: This is a data sufficiency question based on a simple code language for words and digits. You are not asked to find the full code dictionary, but only to decide whether the given statements provide enough information to determine the meanings of the digits 9 and 5 individually. In data sufficiency, the focus is on judging sufficiency, not doing unnecessary computation, so you must be precise about what each statement allows you to deduce. Given Data / Assumptions: - From the main information: "13" means "stop smoking". - From the main information: "59" means "injurious habit". - We must find the meanings of "9" and "5" respectively. - Statement I: "157" means "stop bad habit". - Statement II: "839" means "smoking is injurious". - Each digit corresponds to exactly one word, and each word corresponds to exactly one digit within this code system. Concept / Approach: The logical method is to use intersections: whenever two codes share a digit, the corresponding messages must share a word, and that digit must stand for that word. By comparing the messages for different codes, we can progressively map digits to words. For data sufficiency, we check what can be deduced using the original information plus statement I alone, and then using the original information plus statement II alone. If either set is enough by itself, the correct option is the one that reflects that level of sufficiency. Step-by-Step Solution: Step 1: Use statement I with the base information. "13" → "stop smoking" and "157" → "stop bad habit". The common word is "stop", and the common digit is "1". Thus, digit 1 stands for "stop". Step 2: From the main information "59" → "injurious habit", we know digits 5 and 9 represent "injurious" and "habit" in some order. Step 3: In "157" → "stop bad habit", the remaining words are "bad" and "habit", and the remaining digits are 5 and 7. Since "habit" appears in both "59" and "157", but only digit 5 is common to those two codes, digit 5 must stand for "habit". Step 4: Once 5 is mapped to "habit", in "59" the only remaining word is "injurious", so digit 9 must stand for "injurious". Thus, using statement I with the base information, we can determine both meanings, so statement I alone is sufficient. Step 5: Now test statement II separately with the base information. "13" → "stop smoking" and "839" → "smoking is injurious". The common word is "smoking", and the common digit is 3, so 3 stands for "smoking". Step 6: Compare "59" → "injurious habit" and "839" → "smoking is injurious". The common word is "injurious", and the common digit is 9, so 9 stands for "injurious". Once again, in "59" the remaining word is "habit" and the remaining digit is 5, so 5 stands for "habit". Statement II alone, combined with the base information, is also sufficient. Verification / Alternative check: In both analyses, we derived the same mapping: 9 corresponds to "injurious" and 5 corresponds to "habit". This consistency confirms that each individual statement, when used with the general information, is independently strong enough to answer the original question. There is no contradiction between the statements, and using both together does not add any new necessary information beyond what each already gives on its own. Why Other Options Are Wrong: - Option a is wrong because statement I is indeed sufficient, but so is statement II; it is not the case that only I works. - Option b is wrong for the symmetric reason: statement II is sufficient, but so is statement I. - Option d is wrong because both statements together are not needed; each is already enough by itself. - Option e is clearly wrong because the data are more than sufficient to answer which words digits 9 and 5 represent. Common Pitfalls: One common mistake is to forget that the base information about "13" and "59" is always available, even when you are judging each statement separately. Another pitfall is to assume that you must use both statements together just because they are provided, rather than testing each one independently. It is also easy to mix up which digit corresponds to which word if intersections are not carefully tracked. Final Answer: Each statement, taken separately with the given code information, is sufficient to determine that 9 means "injurious" and 5 means "habit". Therefore, the correct data sufficiency choice is option C, which states that either statement alone is sufficient.

Question 19

On which day in January did Subhas leave for Germany? Statement I: Subhas has so far spent 10 years in Germany. Statement II: Subhas's friend Anil left for Germany on 15th February and joined Subhas 20 days after Subhas's arrival.

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Introduction / Context: This problem is a data sufficiency question about dates and timing. The question asks for the exact day in January when Subhas left for Germany. Two statements give information about how long Subhas has been in Germany and when his friend Anil joined him. Your job is not to compute an approximate number but to decide whether the given information is enough to pinpoint the exact date in January. Given Data / Assumptions: - The question: On which day in January did Subhas leave for Germany? - Statement I: Subhas has so far spent 10 years in Germany. - Statement II: Anil left for Germany on 15th February and joined Subhas 20 days after Subhas's arrival. - There is no explicit reference year; only relative timing is described. - The length of a year and month follows the standard calendar, but specific years are not provided. Concept / Approach: For data sufficiency, we check each statement separately with the common background, then consider them together if necessary. The key is whether the information allows us to determine a unique calendar date for Subhas's departure in January. A mere relationship between events, like someone arriving a certain number of days earlier or later, does not necessarily give an absolute date unless at least one fixed date is clearly anchored to Subhas's travel. Step-by-Step Solution: Step 1: Use statement I alone. It tells us that Subhas has spent 10 years in Germany so far. However, it does not mention the current year or when that 10 year period was measured. Without knowing the present year or the year of his departure, we cannot identify the exact date in January on which he left. Step 2: Therefore, statement I alone is not sufficient. Step 3: Use statement II alone. It says that Anil left for Germany on 15th February and joined Subhas 20 days after Subhas arrived. This tells us that Subhas must have reached Germany 20 days before Anil joined him, but the statement does not connect this directly to January or provide which year is being discussed. Step 4: Even with the fixed date of 15th February for Anil's departure, we do not know how long Anil's journey took or the exact arrival date of Anil in Germany, so we cannot count back precisely to Subhas's arrival date, let alone his departure date from the home country. Step 5: Therefore, statement II alone is also not sufficient. Step 6: Now combine statements I and II. The first gives a total period of 10 years but still no starting calendar date, while the second only relates Anil's journey to Subhas's arrival by 20 days. Even together, they do not connect Subhas's starting date in January to any fixed known year or date beyond a general period of 10 years. Verification / Alternative check: We can try to imagine different possible timelines. For example, if Subhas left on 10th January of some year, he might still have spent 10 years in Germany by now depending on the current date. If he left on 25th January of some year, the 10 year period can also hold with a suitable choice of current date. Statement II adjusts Anil's journey relative to Subhas but does not fix Subhas's departure. Because many different January dates are consistent with both statements, they are not sufficient to answer the question uniquely. Why Other Options Are Wrong: - Option a is wrong because statement I alone does not provide any specific date information, only a duration. - Option b is wrong since statement II alone also fails to pin down the exact date in January. - Option c is incorrect because neither I nor II alone is sufficient. - Option d is incorrect because even using both statements together still leaves multiple possible January dates that fit all the conditions. Common Pitfalls: A frequent mistake is to assume that knowing a total period like 10 years automatically allows you to backtrack to a unique starting date, but this requires knowing exactly when that period is measured from. Another pitfall is to over-interpret relational information such as "20 days after" as if it determines an absolute date, when in fact it only relates two events. Careful reading is essential in data sufficiency questions, and you must resist the urge to infer extra information that is not explicitly stated. Final Answer: Even when both statements are used together, the exact January date of Subhas's departure cannot be determined uniquely. Therefore, the correct data sufficiency choice is option E, meaning the data in both statements together are still not sufficient.

Question 20

What is Reena's rank in the class? Statement I: There are 26 students in the class. Statement II: There are 9 students who have scored less than Reena.

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Introduction / Context: This is a ranking based data sufficiency question. You are asked to determine Reena's rank in a class, which means identifying her exact position among all students, usually counted from the top. Two statements provide partial information about the total number of students and the number of students who scored less than Reena. Your goal is to decide which combination of these statements, if any, is enough to determine Reena's exact rank. Given Data / Assumptions: - The question: What is Reena's rank in the class? - Statement I: There are 26 students in the class. - Statement II: There are 9 students who have scored less than Reena. - All students are assumed to have distinct scores for a unique ranking unless otherwise mentioned. - Rank is interpreted in the usual way as the position counted from the top (best performer has rank 1). Concept / Approach: For ranking problems, the position of a student from the top can be derived if we know both the total number of students and the number of students below that student. Specifically, if there are N students in total and k students with lower scores, then the rank from the top is N - k. In data sufficiency, we first determine what can be derived using statement I alone, then statement II alone, and finally both together. Step-by-Step Solution: Step 1: Use statement I alone. It tells us there are 26 students in the class. However, this gives no information about where Reena stands among them. Her rank could be anywhere from 1 to 26. Therefore, statement I alone is not sufficient. Step 2: Use statement II alone. It states that 9 students scored less than Reena. While this tells us her position from the bottom (10th from the bottom), we do not know how many students are in the class, so we cannot convert this into a rank from the top. Therefore, statement II alone is also not sufficient. Step 3: Now combine statements I and II. From statement I, total students N = 26. From statement II, the number of students scoring less than Reena is 9. Step 4: Reena is therefore ahead of those 9 students. The number of students who scored greater than or equal to Reena is 26 - 9 = 17. Since all ranks are distinct, Reena must be 17th from the top. Step 5: Thus, with both statements together, we can uniquely determine Reena's rank as 17. Verification / Alternative check: We can verify by imagining the class list. If there are 26 students and exactly 9 of them scored less than Reena, there are 16 students with scores greater than or equal to hers, including Reena herself. That places her at position 17 from the top. There is no other possible rank consistent with both a fixed total and a fixed number below her. Therefore the combination of I and II is sufficient, and neither statement alone is enough. Why Other Options Are Wrong: - Option a is wrong because statement I alone leaves Reena's position completely undetermined. - Option b is wrong because statement II alone only provides a partial rank from the bottom and cannot be translated into a top rank without knowing the total class size. - Option c is incorrect, as neither statement alone can answer the question; both are needed together. - Option e is wrong because we have seen that the two statements together are indeed sufficient to find Reena's rank. Common Pitfalls: Students often misinterpret rank information, confusing rank from the top with rank from the bottom. Another common mistake is to assume a total class size based on typical examples instead of using the value given in the statement. In data sufficiency problems, it is crucial to separate the analysis of each statement and not assume extra data that has not been explicitly provided. Final Answer: Both statements I and II together are needed to determine that Reena is 17th in the class. Therefore, the correct data sufficiency choice is option D.

Data Sufficiency Questions