Question 1

How many students passed the exam?
Statement I: 60% of students passed the exam.
Statement II: 120 students failed the exam.

Accepted Answer

Statement I alone tells us the pass percentage but not the total number of students. Statement II alone tells us the number who failed but not the total or pass percentage. Together: If 60% passed, then 40% failed. If 40% = 120 students, then total = 120/0.40 = 300 students. Number who passed = 60% of 300 = 180. Both statements are necessary and sufficient.

Question 2

What is the length of a rectangle?
Statement I: The perimeter of the rectangle is 50 cm.
Statement II: The breadth of the rectangle is 10 cm.

Accepted Answer

Statement I alone gives perimeter but not the individual dimensions (infinitely many length-breadth combinations yield the same perimeter). Statement II alone gives only breadth. Together: Perimeter = 2(length + breadth). 50 = 2(length + 10). 50 = 2×length + 20. Length = 15 cm. Both statements are necessary and sufficient.

Question 3

What is the average age of three friends A, B, and C?
Statement I: A's age is 24 years.
Statement II: B's age is 6 years more than A's age, and C's age is 3 years less than B's age.

Accepted Answer

Statement I alone gives only A's age (24 years). Statement II provides relationships between all three ages. From Statement II: B = A + 6 and C = B - 3 = A + 3. Combined with Statement I, we can find all three ages: A = 24, B = 30, C = 27. Average = (24 + 30 + 27)/3 = 27 years. Both statements together are necessary and sufficient.

Question 4

Is the number x divisible by 12?
Statement I: x is divisible by 3 and 4.
Statement II: x is divisible by 6 and 8.

Accepted Answer

For a number to be divisible by 12, it must be divisible by both 3 and 4 (since 12 = 3 × 4 and gcd(3,4) = 1). Statement I directly states this. Statement II says x is divisible by 6 and 8. Since 6 = 2 × 3 and 8 = 2³, x must be divisible by lcm(6,8) = 24, which means it is divisible by 12. Both statements are individually sufficient.

Question 5

What is the cost price of an article?
Statement I: The article is sold at a profit of 25%.
Statement II: The selling price of the article is Rs. 500.

Accepted Answer

Statement I alone tells us the profit percentage but not the actual cost price (profit percentage is a ratio, not an absolute value). Statement II alone gives only the selling price without profit information. Together: If profit = 25%, then Selling Price = Cost Price × 1.25. Given SP = 500, we have 500 = CP × 1.25, so CP = 400. Both statements are necessary and sufficient.

Question 6

A rectangular garden has a perimeter of 100 meters. Statement I: The length of the garden is 30 meters. Statement II: The width of the garden is 20 meters. What is the area of the garden?

A) Statement I alone is sufficient
B) Statement II alone is sufficient
C) Both statements together are sufficient
D) Both statements together are not sufficient

Accepted Answer

For a rectangle: Perimeter = 2(L + W) and Area = L × W. Given: Perimeter = 100, so L + W = 50. Statement I: L = 30 → W = 50 - 30 = 20 → Area = 30 × 20 = 600 m². Statement II: W = 20 → L = 50 - 20 = 30 → Area = 30 × 20 = 600 m². Either statement alone is sufficient because the perimeter constraint allows us to derive the missing dimension. Both statements are redundant (they describe the same rectangle).

Question 7

A shopkeeper sells two types of items. The cost price of item A is Rs. 500 and the cost price of item B is Rs. 300. He sells item A at a profit of 20% and item B at a loss of 10%. Statement I: The selling price of item A is Rs. 600. Statement II: The selling price of item B is Rs. 270. What is the profit or loss percentage on the total transaction if he sells 5 units of A and 3 units of B?

A) Statement I alone is sufficient
B) Statement II alone is sufficient
C) Both statements together are sufficient
D) Both statements together are not sufficient

Accepted Answer

To find overall profit/loss percentage, we need: Total Cost Price and Total Selling Price. Statement I gives SP of A = 600 (verifiable: 500 × 1.20 = 600). Statement II gives SP of B = 270 (verifiable: 300 × 0.90 = 270). With both statements, we can calculate: Total CP = (5 × 500) + (3 × 300) = 2500 + 900 = 3400. Total SP = (5 × 600) + (3 × 270) = 3000 + 810 = 3810. Profit = 3810 - 3400 = 410. Profit% = (410/3400) × 100 = 12.06%. Both statements are needed because each provides the selling price of one item type, and we need both to compute total SP.

Question 8

A train travels from City X to City Y. Statement I: The distance between X and Y is 480 km. Statement II: The train's average speed is 60 km/h. What is the time taken by the train to travel from X to Y?

A) Statement I alone is sufficient
B) Statement II alone is sufficient
C) Both statements together are sufficient
D) Both statements together are not sufficient

Accepted Answer

To find time, we use the formula: Time = Distance / Speed. Statement I provides Distance = 480 km. Statement II provides Speed = 60 km/h. Both are required because: Time = 480 / 60 = 8 hours. Without Statement I, we don't know the distance. Without Statement II, we don't know the speed. Neither statement alone is sufficient; both together are necessary and sufficient.

Question 9

A mixture contains milk and water in a certain ratio. Statement I: The mixture contains 60 liters of milk. Statement II: The ratio of milk to water is 3:2. What is the total quantity of the mixture?

A) Statement I alone is sufficient
B) Statement II alone is sufficient
C) Both statements together are sufficient
D) Both statements together are not sufficient

Accepted Answer

To find total mixture, we need to know the quantities of both milk and water (or their ratio and one quantity). Statement I: Milk = 60 liters (one quantity, but ratio unknown). Statement II: Ratio milk:water = 3:2 (ratio known, but absolute quantities unknown). With Statement I alone: We know milk = 60 L, but without the ratio, we cannot determine water quantity. With Statement II alone: We know the ratio, but without an absolute quantity, we cannot determine total. With both statements: Ratio 3:2 means milk = 3x, water = 2x. Given milk = 60, we have 3x = 60 → x = 20. Water = 2(20) = 40. Total = 60 + 40 = 100 liters. Both statements together are necessary and sufficient.

Question 10

A person invests money in two schemes: Scheme A offers 8% simple interest per annum, and Scheme B offers 6% simple interest per annum. Statement I: The person invests Rs. 5,000 in Scheme A. Statement II: The person invests Rs. 3,000 in Scheme B. What is the total interest earned after 2 years from both schemes combined?

A) Statement I alone is sufficient
B) Statement II alone is sufficient
C) Both statements together are sufficient
D) Both statements together are not sufficient

Accepted Answer

Simple Interest formula: SI = (P × R × T) / 100. For Scheme A: SI_A = (5000 × 8 × 2) / 100 = 800 (from Statement I). For Scheme B: SI_B = (3000 × 6 × 2) / 100 = 360 (from Statement II). Total Interest = 800 + 360 = 1160. Statement I alone gives SI from Scheme A only (800 Rs.), which is incomplete. Statement II alone gives SI from Scheme B only (360 Rs.), which is also incomplete. Both statements together provide the principal amounts for both schemes, allowing calculation of total interest from both. Both statements are necessary and sufficient.

Question 11

A shopkeeper sells three types of items: A, B, and C. The cost price of A is Rs. 100. Statement I: The profit percentage on A is 20%, and the profit on B is Rs. 30 more than the profit on A. Statement II: The selling price of C is Rs. 180, and the cost price of C is 25% less than the cost price of B. What is the cost price of B?

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

From Statement I: Profit on A = 20% of 100 = Rs. 20. Profit on B = 20 + 30 = Rs. 50. But we don't know the cost price of B (profit alone doesn't determine cost price). From Statement II: Selling price of C = Rs. 180. Cost price of C = 75% of cost price of B. But we don't know the profit/loss on C, so we can't find the cost price of C, and thus can't find the cost price of B. Using both: From I, profit on B = Rs. 50. From II, CP(C) = 0.75 × CP(B). We still need either the selling price of B or the profit percentage on B to find CP(B). The two statements provide independent information that doesn't interconnect sufficiently.

Question 12

Question: What is the two-digit number?
Statement I: The number is divisible by 9 and its tens digit is 3.
Statement II: The sum of the digits of the number is 9.

Choose the correct option:
(a) Statement I alone is sufficient but Statement II alone is not sufficient.
(b) Statement II alone is sufficient but Statement I alone is not sufficient.
(c) Both statements together are sufficient but neither alone is sufficient.
(d) Either statement alone is sufficient.

Accepted Answer

Statement I: Divisible by 9 and tens digit is 3 → The number is 36 (3+6=9, divisible by 9). Only one such two-digit number exists. So Statement I alone is sufficient.
Statement II: Sum of digits = 9 → Possible numbers: 18, 27, 36, 45, 54, 63, 72, 81, 90. Multiple answers possible. Statement II alone is NOT sufficient.
Answer: Statement I alone is sufficient but Statement II alone is not sufficient.

Question 13

A rectangular garden has a perimeter of P meters. Statement I: The length is 5 meters more than the width. Statement II: The area of the garden is 84 square meters. What is the perimeter of the garden?

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

From Statement I: Let width = W. Length = W + 5. Perimeter P = 2(W + W + 5) = 2(2W + 5) = 4W + 10. This expresses P in terms of W, but we don't know W. From Statement II: Area = L × W = 84. Without knowing the relationship between L and W, we cannot find specific values. Using both: From I, L = W + 5. From II, (W + 5) × W = 84 → W² + 5W - 84 = 0 → (W + 12)(W - 7) = 0 → W = 7 (since width must be positive). Therefore, L = 12. Perimeter = 2(12 + 7) = 38 meters. Both statements together are sufficient.

Question 14

A mixture contains milk and water. Statement I: The ratio of milk to water is 3:2, and the mixture weighs 100 liters. Statement II: The concentration of milk in the mixture is 60%. What is the quantity of milk in the mixture?

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

From Statement I: Ratio of milk to water = 3:2. Total parts = 5. Milk = (3/5) × 100 = 60 liters. This alone is sufficient to find the quantity of milk. From Statement II: Concentration of milk = 60%. But without knowing the total weight of the mixture, we cannot find the quantity of milk. Statement I alone provides both the ratio and the total quantity, which is sufficient to calculate milk. Statement II alone is insufficient because it only gives the percentage without the total quantity.

Question 15

A company has three employees: X, Y, and Z. Statement I: X's salary is 20% more than Y's salary, and Y's salary is 10% more than Z's salary. Statement II: The total salary of all three is Rs. 1,32,000. What is X's salary?

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

From Statement I: Let Z's salary = Z. Y's salary = 1.10Z. X's salary = 1.20 × 1.10Z = 1.32Z. We have X, Y, Z expressed in terms of Z, but no numerical value. From Statement II: X + Y + Z = 1,32,000. Without knowing the relationship between salaries, we cannot find X. Using both: From I, X = 1.32Z, Y = 1.10Z. From II, 1.32Z + 1.10Z + Z = 1,32,000 → 3.42Z = 1,32,000 → Z = 38,596.49 (approximately). Therefore, X = 1.32 × 38,596.49 ≈ 50,847.27. Both statements together are sufficient.

Question 16

A train travels from City P to City Q. Statement I: The train travels at 60 km/h for the first half of the distance and at 90 km/h for the second half of the distance. Statement II: The total time taken is 5 hours. What is the distance between City P and City Q?

(A) Statement I alone is sufficient
(B) Statement II alone is sufficient
(C) Both statements together are sufficient
(D) Neither statement alone nor together is sufficient

Accepted Answer

From Statement I: Let total distance = D. Time for first half = (D/2)/60 = D/120 hours. Time for second half = (D/2)/90 = D/180 hours. Total time = D/120 + D/180 = 3D/360 + 2D/360 = 5D/360 = D/72 hours. This expresses time in terms of D but doesn't give us a numerical value for D. From Statement II: Total time = 5 hours, but without knowing the speed profile, we cannot find distance. Using both: From I, Total time = D/72. From II, Total time = 5. Therefore, D/72 = 5, so D = 360 km. Both statements are necessary and together sufficient.

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