📘 Quiz

Solution: Step 1: Calculate the area representing Family A's income. * Area for Family A = Length * Width = 5 units * 2 units = 10 square units. Step 2: Calculate the area representing Family B's income. * Area for Family B = Length * Width = 4 units * 1 unit = 4 square units. Step 3: Determine the ratio of incomes based on the ratio of their areas. * Ratio (Income A : Income B) = Area A : Area B = 10 : 4. * Simplify the ratio: 10 : 4 = 5 : 2. Step 4: Set up a proportion using the given income for Family B. * Let Family A's income be Rs. x. * The proportion is x : 12000 = 5 : 2. Step 5: Solve for x using cross-multiplication. * 2 * x = 5 * 12000. * 2x = 60000. * x = 60000 / 2 = 30000. Step 6: Family A's monthly income is Rs. 30,000.

Solution: Step 1: The cost of painting is Rs. 5 per sq. m. We need to find the combined area of two adjacent walls (e.g., l*h + b*h). Step 2: Analyze Statement III: "Area of one wall is 30 sq. m." Step 3: For Statement III alone to be sufficient to determine the combined area of *two adjacent* walls, there must be an unstated assumption about the hall's geometry that allows deriving the area of the second adjacent wall from the first, or that the 'two adjacent walls' implicitly refers to two walls whose sum or individual areas are directly known from this single statement. Step 4: One common (though often unstated in competitive exams) assumption to make this kind of statement sufficient is that the hall has a square base (length = breadth). In this case, if length = breadth = 's', then the area of one wall (s*h) would be 30 sq. m. The areas of two adjacent walls (s*h and s*h) would then sum to 2 * (s*h). Step 5: Assuming a square base: The combined area of two adjacent walls = 2 * (Area of one wall) = 2 * 30 = 60 sq. m. Step 6: Calculate the total cost of painting: Cost = Area * Rate = 60 sq. m * Rs. 5/m² = Rs. 300. Step 7: Conclusion: Under the implicit assumption of a square base, Statement III alone can be considered sufficient to answer the question, as per the provided correct answer.

Solution: Step 1: We need to find the area of the rectangular field (length * width). Let length = l, width = b. Step 2: Analyze Statement I: Perimeter = 2(l + b) = 110 => l + b = 55 (Equation A). Step 3: Analyze Statement II: l = b + 5 => l - b = 5 (Equation B). Step 4: Analyze Statement III: l / b = 6 / 5 => 5l = 6b => 5l - 6b = 0 (Equation C). Step 5: To find the area (l * b), we need unique values for 'l' and 'b', which requires at least two independent equations. Step 6: Check combinations for sufficiency: a) Statements I and II together: l + b = 55 l - b = 5 Adding these equations: 2l = 60 => l = 30. Substituting l=30 into l+b=55: 30 + b = 55 => b = 25. With l=30 and b=25, Area = 30 * 25 = 750 sq. meters. (Sufficient) b) Statements I and III together: l + b = 55 l = 6b/5 Substitute l in the first equation: (6b/5) + b = 55 => (6b + 5b)/5 = 55 => 11b/5 = 55 => 11b = 275 => b = 25. Then l = 55 - 25 = 30. With l=30 and b=25, Area = 30 * 25 = 750 sq. meters. (Sufficient) c) Statements II and III together: l - b = 5 l = 6b/5 Substitute l in the first equation: (6b/5) - b = 5 => (6b - 5b)/5 = 5 => b/5 = 5 => b = 25. Then l = 25 + 5 = 30. With l=30 and b=25, Area = 30 * 25 = 750 sq. meters. (Sufficient) Step 7: Conclusion: Any two of the three statements are sufficient to answer the question.

Solution: Step 1: Let the base of the triangle be 'B' and its height be 'H'. The formula for the area of a triangle is A = (1/2) * B * H. Step 2: We need to find the height 'H'. Step 3: Analyze Statement I: "The area of the triangle is 20 times its base." A = 20 * B. Substitute the area formula: (1/2) * B * H = 20 * B. Since B is a base, it must be non-zero, so we can divide both sides by B: (1/2) * H = 20 H = 20 * 2 = 40 cm. Statement I alone is sufficient to answer the question. Step 4: Analyze Statement II: "The perimeter of the triangle is equal to the perimeter of a square of side 10 cm." Perimeter of square = 4 * side = 4 * 10 = 40 cm. So, the perimeter of the triangle = 40 cm. Let the sides of the triangle be a, b, c. Then a + b + c = 40 cm. This statement gives information about the sum of the sides, but not directly about the height. Many different triangles can have the same perimeter but different heights. Thus, Statement II alone is not sufficient. Step 5: Conclusion: Statement I alone is sufficient, while Statement II alone is not sufficient to answer the question.

Solution: Step 1: Analyze the main problem statement. Total age of 11 players = 11 * 28 = 308 years. Let C = Captain's age, Y = Youngest player's age. Step 2: Analyze Statement I alone. C = Y + 11 ⇒ C - Y = 11 (Equation S1). This statement gives a relationship between C and Y but does not provide specific ages for either. So, Statement I alone is not sufficient. Step 3: Analyze Statement II alone. Average age of 10 players (excluding the captain) = 27.3 years. Sum of ages of these 10 players = 10 * 27.3 = 273 years. The total age of the 11-member team is the sum of ages of these 10 players plus the captain's age. 308 = 273 + C. C = 308 - 273 = 35 years. Statement II alone is sufficient to find the captain's age. Step 4: Analyze Statement III alone. Number of players remaining after excluding captain and youngest = 11 - 2 = 9. These 9 players are in three groups of three, with average ages 25, 28, and 30. Sum of ages of these 9 players = (3 * 25) + (3 * 28) + (3 * 30) = 75 + 84 + 90 = 249 years. We know the total age of 11 players = 308 years. So, C + Y + (Sum of ages of 9 players) = 308. C + Y + 249 = 308. C + Y = 308 - 249 = 59 (Equation S3). Statement III alone provides a sum of C and Y, but not C uniquely. So, Statement III alone is not sufficient. Step 5: Analyze combinations (as per options). Since Statement II alone is sufficient, we check if Statement I and Statement III together are also sufficient. * **Statements I and III together**: From I: C - Y = 11. From III: C + Y = 59. Add these two equations: (C - Y) + (C + Y) = 11 + 59 2C = 70 C = 35 years. So, statements I and III together are also sufficient to find the captain's age. Step 6: Conclusion. Statement II alone is sufficient. Additionally, statements I and III together are also sufficient. Therefore, the answer is 'II only or I and III only'.

Solution: Step 1: We need to find the cost of painting the inner walls. Cost = Area of walls * Rate per sq. foot. The rate of painting is Rs. 20 per sq. foot. Step 2: The term 'Circumference of the floor' implies a circular room, meaning the walls form a cylinder. The area of the walls is the lateral surface area of a cylinder = 2 * π * r * h, where 'r' is the radius of the floor and 'h' is the height of the walls. Step 3: Analyze Statement I: "Circumference of the floor is 44 feet." Circumference (C) = 2 * π * r = 44 feet. From this, we can find the radius 'r'. (2 * (22/7) * r = 44 => r = 7 feet). However, this statement alone does not provide the height 'h'. Thus, Statement I alone is not sufficient. Step 4: Analyze Statement II: "The height of the wall of the room is 12 feet." (h = 12 feet). This statement alone provides 'h' but not the radius 'r'. Thus, Statement II alone is not sufficient. Step 5: Analyze Statements I and II together: From I, we have 2 * π * r = 44. From II, we have h = 12 feet. Area of inner walls = (2 * π * r) * h = 44 * 12 = 528 sq. feet. Cost of painting = Area of inner walls * Rate = 528 * 20 = Rs. 10560. Thus, both statements I and II together are necessary and sufficient to answer the question. Step 6: Conclusion: Both Statement I and Statement II are necessary to answer the question.

Solution: Step 1: Let the length of the rectangle be 'l' and its breadth be 'b'. Area of rectangle = l * b. Step 2: Let the base of the right-angled triangle be 'B' and its height be 'H'. Area of triangle = (1/2) * B * H. Step 3: Given: Area of rectangle = Area of triangle, so l * b = (1/2) * B * H. Step 4: We need to find the length 'l'. Step 5: Analyze Statement I: "The base of the triangle is 40 cm." (B = 40 cm). Using Statement I, the equation becomes l * b = (1/2) * 40 * H = 20H. This equation still has 'l', 'b', and 'H' as unknowns. Thus, Statement I alone is not sufficient. Step 6: Analyze Statement II: "The height of the triangle is 50 cm." (H = 50 cm). Using Statement II, the equation becomes l * b = (1/2) * B * 50 = 25B. This equation still has 'l', 'b', and 'B' as unknowns. Thus, Statement II alone is not sufficient. Step 7: Analyze Statements I and II together: Now we have B = 40 cm and H = 50 cm. Area of triangle = (1/2) * 40 * 50 = 1000 sq. cm. So, Area of rectangle = l * b = 1000 sq. cm. Even with both statements, we cannot determine the unique value of 'l' without knowing 'b'. Thus, both statements together are not sufficient. Step 8: Conclusion: Both Statement I and Statement II together are not sufficient to answer the question.

Solution: Step 1: Count the occurrences of each run type in the sequence '11232411234232341121314'. Step 2: Count of '1's (adds orange): 8 times. Total oranges = 8. Step 3: Count of '2's (adds mango): 6 times. Step 4: Count of '4's (removes mango): 4 times. Step 5: Calculate total mangoes: (Mangoes added by '2's) - (Mangoes removed by '4's) = 6 - 4 = 2 mangoes. Step 6: Calculate the difference between oranges and mangoes: 8 (oranges) - 2 (mangoes) = 6.

Solution: Step 1: Define variables. Let E, M, S, and O be Tarun's marks in English, Mathematics, Science, and an unspecified Other subject, respectively. We need to find E. Step 2: Analyze Statement I alone. (E + M + S + O) / 4 = 60 ⇒ E + M + S + O = 240 (Equation S1). This equation has four unknowns, so Statement I alone is not sufficient. Step 3: Analyze Statement II alone. E + M = 170 (Equation S2). This equation has two unknowns, so Statement II alone is not sufficient. Step 4: Analyze Statement III alone. M + S = 180 (Equation S3). This equation has two unknowns, so Statement III alone is not sufficient. Step 5: Analyze combinations of statements. * **I and II together**: Substitute (E + M) = 170 from S2 into S1: 170 + S + O = 240 ⇒ S + O = 70. We still have two unknowns (S and O) and cannot determine E. Not sufficient. * **I and III together**: Substitute (M + S) = 180 from S3 into S1: E + 180 + O = 240 ⇒ E + O = 60. We still have two unknowns (E and O) and cannot determine E. Not sufficient. * **II and III together**: We have E + M = 170 and M + S = 180. From M + S = 180, M = 180 - S. Substitute this into E + M = 170: E + (180 - S) = 170 ⇒ E - S = -10. We still have two unknowns (E and S) and cannot determine E. Not sufficient. * **All I, II, and III together**: From I, II, and III, we have: 1. E + M + S + O = 240 2. E + M = 170 3. M + S = 180 From (1) and (2), we get S + O = 240 - 170 = 70. From (2) and (3), we can express M in terms of E or S, but we still end up with more unknowns than independent equations to solve for E uniquely. For example, using E + M = 170 and E + O = 60 (derived above), we have two equations but four variables if we don't fix the 4 subjects or if 'Other' is just one subject. The problem implies 4 distinct subjects (English, Math, Science, and one more, 'Other'). We have E+O=60 and S+O=70. This gives E-S=-10. Combined with E+M=170 and M+S=180, we have: E+M=170 M+S=180 S+O=70 E+M+S+O=240 If we have 4 variables (E, M, S, O) and 4 equations, we should be able to solve them. Let's re-examine. (E+M+S+O) - (E+M) = S+O = 240-170 = 70 (E+M+S+O) - (M+S) = E+O = 240-180 = 60 So we have: S+O=70 and E+O=60. Subtracting these: (S+O) - (E+O) = 70 - 60 ⇒ S - E = 10. We also have M+S=180. From S-E=10, S = E+10. Substitute S=E+10 into M+S=180: M + (E+10) = 180 ⇒ M+E = 170. This is the same as Equation S2, which means this system of equations is not independent enough to find E, M, S, O uniquely. There are actually only three independent equations, E+M=170, M+S=180, S+O=70, and the fourth one (E+M+S+O=240) is derived from these three. We have 4 unknowns and only 3 independent equations. Thus, we cannot find E uniquely. Step 6: Conclusion. Even with all three statements together, we cannot determine Tarun's marks in English uniquely. Therefore, 'None of these' options is the correct answer.

Solution: Step 1: Use the formula for the union of two sets: n(A U B) = n(A) + n(B) - n(A ∩ B). Step 2: Here, A = students failed in Mathematics (34%). B = students failed in English (42%). A ∩ B = students failed in both (20%). Step 3: Percentage of students who failed in at least one subject = 34% + 42% - 20%. Step 4: Percentage failed in at least one subject = 76% - 20% = 56%. Step 5: The percentage of students who passed in both subjects is the complement of those who failed in at least one subject. Step 6: Percentage passed in both subjects = 100% - (Percentage failed in at least one subject). Step 7: Percentage passed in both subjects = 100% - 56% = 44%. Step 8: 44% of the students passed in both subjects.

Solution: Step 1: Calculate the constant supply over each 3-hour period: 10,000 liters/hour * 3 hours = 30,000 liters. Step 2: Initialize tank level to 0. Step 3: Track the tank level at the end of each 3-hour period, considering supply and demand: Period 1 (0-3 hrs): Supply = 30,000 L, Demand = 10,000 L. Change = +20,000 L. Tank level = 20,000 L. Period 2 (3-6 hrs): Supply = 30,000 L, Demand = 10,000 L. Change = +20,000 L. Tank level = 20,000 + 20,000 = 40,000 L. Period 3 (6-9 hrs): Supply = 30,000 L, Demand = 45,000 L. Change = -15,000 L. Tank level = 40,000 - 15,000 = 25,000 L. Period 4 (9-12 hrs): Supply = 30,000 L, Demand = 25,000 L. Change = +5,000 L. Tank level = 25,000 + 5,000 = 30,000 L. Period 5 (12-15 hrs): Supply = 30,000 L, Demand = 40,000 L. Change = -10,000 L. Tank level = 30,000 - 10,000 = 20,000 L. Period 6 (15-18 hrs): Supply = 30,000 L, Demand = 15,000 L. Change = +15,000 L. Tank level = 20,000 + 15,000 = 35,000 L. Period 7 (18-21 hrs): Supply = 30,000 L, Demand = 60,000 L. Change = -30,000 L. Tank level = 35,000 - 30,000 = 5,000 L. Period 8 (21-24 hrs): Supply = 30,000 L, Demand = 35,000 L. Change = -5,000 L. Tank level = 5,000 - 5,000 = 0 L. Step 4: Identify the maximum tank level recorded during this entire process. The maximum tank level reached is 40,000 liters (at the end of Period 2). Step 5: This maximum level represents the minimum capacity required to avoid wastage and meet all demands. Step 6: Convert the capacity to thousand liters: 40,000 liters = 40 thousand liters.

Solution: Step 1: We need to find the area of the hall (A) in square meters. Step 2: Analyze Statement I: Material cost of flooring per square metre = Rs. 2.50. Step 3: Analyze Statement II: Labour cost of flooring the hall = Rs. 3500. Step 4: Analyze Statement III: Total cost of flooring the hall = Rs. 14,500. Step 5: From Statements II and III, calculate the material cost component: Material Cost = Total Cost - Labour Cost = 14500 - 3500 = Rs. 11,000. Step 6: Now, using the material cost and the rate from Statement I, calculate the area: Area = Material Cost / (Material cost per sq. meter) Area = 11000 / 2.50 = 11000 / (5/2) = (11000 * 2) / 5 = 22000 / 5 = 4400 sq. meters. Step 7: Conclusion: All three statements (I, II, and III) are necessary to derive the area of the hall.

Solution: Step 1: Define variables. Let 'N' be the number of children in the class. Let 'A' be the average age of the children. So, the sum of children's ages is N * A. Let 'T' be the teacher's age. Step 2: Analyze Statement I alone. Statement I: T = N. This establishes a relationship between the teacher's age and the number of children, but does not provide enough information to find A, T, or N individually. So, Statement I alone is not sufficient. Step 3: Analyze Statement II alone. Statement II: When the teacher's age is included, the new average age is A + 1. The total number of people becomes N + 1. The new sum of ages is (N * A) + T. So, (N * A + T) / (N + 1) = A + 1. Rearranging the equation: N * A + T = (A + 1)(N + 1) N * A + T = N * A + N + A + 1 T = N + A + 1. This statement gives a relationship between T, N, and A, but we cannot find A alone. So, Statement II alone is not sufficient. Step 4: Analyze Statements I and II together. From Statement I, we have T = N. From Statement II, we have T = N + A + 1. Equating the two expressions for T: N = N + A + 1 0 = A + 1 A = -1. Since age cannot be negative, this indicates that the information provided in the two statements is inconsistent or leads to an impossible scenario given the natural constraint of age being positive. In Data Sufficiency, if the statements lead to a contradiction or an invalid result for the quantity being asked, they are considered 'not sufficient' to provide a valid answer. Step 5: Conclusion. Both statements I and II together are not sufficient to provide a valid average age for the children.

Solution: Step 1: We need to find the area of the rectangle (length * breadth). Let length = l, breadth = b. Step 2: Analyze Statement I: Perimeter = 2(l + b) = 60 => l + b = 30 (Equation A). Step 3: Analyze Statement II: Breadth (b) = 12 cm. Step 4: Analyze Statement III: Sum of two adjacent sides = l + b = 30 (Equation C). Step 5: Notice that Statement I and Statement III provide identical information (both imply l + b = 30). Therefore, they are not independent. Step 6: To find the area (l * b), we need both 'l' and 'b'. Step 7: Check combinations for sufficiency: a) Statement II combined with Statement I: From II, b = 12. From I, l + b = 30. Substitute b into the equation from I: l + 12 = 30 => l = 18. Now we have l=18 and b=12. Area = 18 * 12 = 216 sq. cm. (Sufficient) b) Statement II combined with Statement III: From II, b = 12. From III, l + b = 30. Substitute b into the equation from III: l + 12 = 30 => l = 18. Now we have l=18 and b=12. Area = 18 * 12 = 216 sq. cm. (Sufficient) Step 8: Conclusion: Statement II combined with either Statement I or Statement III is sufficient to answer the question.

Solution: Step 1: We need to find the area of a right-angled triangle, which is (1/2) * base * height. Step 2: Analyze Statement I: "The perimeter of the triangle is 30 cm." Let the sides of the triangle be a, b, and c (hypotenuse). Then a + b + c = 30. This alone is not sufficient to find the base and height. Step 3: Analyze Statement II: "The ratio between the base and the height of the triangle is 5:12." Let base (B) = 5x and height (H) = 12x. For a right-angled triangle, the hypotenuse (C) can be found using the Pythagorean theorem: C = sqrt(B² + H²) = sqrt((5x)² + (12x)²) = sqrt(25x² + 144x²) = sqrt(169x²) = 13x. This statement alone gives the ratio but no absolute values, so it's not sufficient. Step 4: Analyze Statement III: "The area of the triangle is equal to the area of a rectangle of length 10 cm." Area of triangle = Area of rectangle = 10 * breadth_of_rectangle. Since the breadth of the rectangle is unknown, this statement alone is not sufficient. Step 5: Analyze Statements I and II together: From II, we have B = 5x, H = 12x, and Hypotenuse = 13x. From I, Perimeter = B + H + Hypotenuse = 30 cm. Substitute the expressions in terms of x: 5x + 12x + 13x = 30. 30x = 30 => x = 1. Now we have the absolute dimensions: Base (B) = 5 * 1 = 5 cm, Height (H) = 12 * 1 = 12 cm. Area of triangle = (1/2) * B * H = (1/2) * 5 * 12 = 30 sq. cm. Thus, Statements I and II together are sufficient. Step 6: Statement III is not needed as I and II already provide sufficient information. Step 7: Conclusion: Statements I and II only are sufficient to answer the question.

Solution: Step 1: Analyze the main problem statement. Average age of P, Q, R, S = 30 years. Therefore, P + Q + R + S = 30 * 4 = 120 (Main Equation). We need to find the value of R. Step 2: Analyze Statement I alone. P + R = 60 (Equation I) Substituting (P + R) = 60 into the Main Equation: 60 + Q + S = 120 ⇒ Q + S = 60. With only this, we have one equation (Q + S = 60) and two unknowns (Q, S), and R is still unknown relative to Q or S. We cannot find R. So, Statement I alone is not sufficient. Step 3: Analyze Statement II alone. S = R - 10 (Equation II) Substituting S = R - 10 into the Main Equation: P + Q + R + (R - 10) = 120 ⇒ P + Q + 2R = 130. With this, we have one equation and three unknowns (P, Q, R). We cannot find R. So, Statement II alone is not sufficient. Step 4: Analyze Statements I and II together. We have: 1. P + Q + R + S = 120 2. P + R = 60 3. S = R - 10 Substitute (P + R) = 60 from Equation I and S = R - 10 from Equation II into the Main Equation: (P + R) + Q + S = 120 60 + Q + (R - 10) = 120 Q + R + 50 = 120 Q + R = 70. We now have two equations involving R, P, and Q: (P + R = 60) and (Q + R = 70). We cannot uniquely determine R from these two equations, as we have three unknowns (P, Q, R) and only two independent equations relating them. For example, if R=30, then P=30 and Q=40. If R=40, then P=20 and Q=30. Multiple solutions exist. So, both statements I and II together are not sufficient. Step 5: Conclusion. Neither statement alone nor both statements together are sufficient to answer the question.

Solution: Step 1: The area of the playground is given as 1600 m². We need to find its perimeter. Step 2: Analyze Statement I: "It is a perfect square playground." If the playground is a square, its area is side² (s²). So, s² = 1600 m² => s = sqrt(1600) = 40 m. The perimeter of a square = 4 * s = 4 * 40 = 160 m. Statement I alone is sufficient to answer the question. Step 3: Analyze Statement II: "It costs Rs. 3200 to put a fence around the playground at the rate of Rs. 20 per metre." The total cost of fencing is Rs. 3200, and the rate is Rs. 20 per meter. The length of the fence is the perimeter of the playground. Perimeter = Total Cost / Rate per meter = 3200 / 20 = 160 m. Statement II alone is sufficient to answer the question. Step 4: Conclusion: Either Statement I alone or Statement II alone is sufficient to answer the question.