NMAT 2026 Quiz 20: Data Sufficiency

Q1. A committee of 4 people, with at least 2 women, is to be formed from a group of men and women. In how many ways can this be done?
(1) The group has 9 people in total.
(2) The group has 4 women. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option C … Explanation: To count the committees we need both the number of women and the number of men. Statement (1) gives only the total (9), not the gender split, so it is insufficient. Statement (2) gives the number of women (4) but not the number of men, so it is insufficient. Together: 4 women and 5 men. Ways = ⁴C₂·⁵C₂ + ⁴C₃·⁵C₁ + ⁴C₄ = 60 + 20 + 1 = 81, a unique value. Hence, option C. Q2. The marked price and the cost price of a bag are in the ratio of 8:5. Find the profit percentage made on selling it.
(1) If the selling price is 75% more, and the cost price is doubled, the profit remains the same.
(2) The percentage of discount on it is equal to the profit percentage. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option D … Explanation: Let MP = 8k and CP = 5k. Statement (1): 1.75SP − 2CP = SP − CP gives 0.75SP = CP, so SP = (4/3)CP and profit = 33.33%. Sufficient. Statement (2): (8k − SP)/8k = (SP − 5k)/5k gives 40k − 5SP = 8SP − 40k, so SP = 80k/13 and profit = (15k/13)/5k = 3/13 ≈ 23.08%. Sufficient. Each statement alone gives a unique profit percentage. Hence, option D. Q3. A 60-litre mixture contains only milk and water. What was the initial quantity of milk?
(1) When 12 litres of the mixture were removed and replaced by water, the ratio of milk to water became 3:2.
(2) If 15 litres of water is added to the mixture, the ratio of milk and water is reversed. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option D … Explanation: Let the initial milk be m litres. Statement (1): removing 12 litres (1/5 of the mixture) leaves 4m/5 litres of milk in 60 litres. With milk:water = 3:2, milk = 36, so 4m/5 = 36 and m = 45. Sufficient. Statement (2): water = 60 − m, and m : (75 − m) = (60 − m) : m, so m² = (60 − m)(75 − m) = 4500 − 135m + m², giving m = 100/3 ≈ 33.33 litres (ratio 5:4 becomes 4:5). Sufficient. Each statement alone fixes the milk quantity. Hence, option D. Q4. A certain ice cream factory makes two flavors of ice cream: chocolate and caramel. Each flavor comes in two varieties: one with over 20 percent milk and one with under 20 percent milk. Did more than 2/5 of all the ice cream made in July contain over 20 percent milk?
(1) Exactly 80 percent of the caramel ice cream made in July contained over 10 percent milk, and of this amount, 1/4 contained over 20 percent milk.
(2) Exactly 1,000 gallons of the chocolate ice cream made in July contained over 20 percent milk. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option E … Explanation: Statement (1) tells us that 80% × 1/4 = 20% of the caramel ice cream had over 20% milk, but says nothing about chocolate or the total output. Insufficient. Statement (2) gives an absolute quantity (1,000 gallons) of high-milk chocolate ice cream but no total for chocolate or caramel. Insufficient. Together, we still do not know the total chocolate production or how much caramel was made, so the overall fraction cannot be compared with 2/5. Hence, option E. Q5. If a/3 = 4/b, is a less than b?
(1) b ≤ 4.5
(2) b ≥ 3.5 Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option B … Explanation: From a/3 = 4/b, ab = 12, so a = 12/b. Statement (1): if b = 4, a = 3 (a < b, Yes); if b = 1, a = 12 (a > b, No). Insufficient. Statement (2): b ≥ 3.5 means b is positive and b² ≥ 12.25 > 12, so b > 12/b = a. The answer is always Yes. Sufficient. Hence, option B. Q6. Is the positive integer n divisible by 24?
(1) n is divisible by both 6 and 8.
(2) n² is divisible by 144. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option A … Explanation: Statement (1): if n is divisible by both 6 and 8, it is divisible by LCM(6, 8) = 24. Sufficient. Statement (2): n² divisible by 144 = 12² means n is divisible by 12. n = 12 is not divisible by 24, while n = 24 is. Insufficient. Hence, option A. Q7. What is the average marks of all students in sections A and B combined?
(1) Section A has 30 students with an average of 60 marks.
(2) Section B has 20 students with an average of 70 marks. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option C … Explanation: The combined average needs the size and average of both sections. Each statement alone describes only one section, so each is insufficient. Together: combined average = (30 × 60 + 20 × 70)/50 = (1800 + 1400)/50 = 64. Hence, option C. Q8. A geometric progression with |r| < 1 has a sum to infinity of 12. What is its first term?
(1) The common ratio is 1/3.
(2) The second term is 2. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option A … Explanation: Sum to infinity: a/(1 − r) = 12, so a = 12(1 − r). Statement (1): r = 1/3 gives a = 12 × 2/3 = 8. Sufficient. Statement (2): ar = 2 gives 12r(1 − r) = 2, i.e. 6r² − 6r + 1 = 0, so r = (3 ± √3)/6. Both roots lie between 0 and 1, giving two different first terms (≈ 2.54 and ≈ 9.46). Insufficient. Hence, option A. Q9. For positive real numbers a and b, what is the value of a + b?
(1) log₂a + log₂b = 5
(2) log₄(ab) = 2.5 Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option E … Explanation: Statement (1): log₂(ab) = 5, so ab = 32. Statement (2): ab = 4^2.5 = 32. Both statements give the same information, ab = 32, which does not fix a + b (e.g. a = 4, b = 8 gives 12; a = 2, b = 16 gives 18). Even together they are insufficient. Hence, option E. Q10. What is the remainder when the positive integer N is divided by 7?
(1) N leaves a remainder of 3 when divided by 11.
(2) N leaves a remainder of 2 when divided by 7. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option B … Explanation: Statement (1): N could be 3 (remainder 3 on division by 7) or 14 (remainder 0). Insufficient. Statement (2) directly states that the remainder when N is divided by 7 is 2. Sufficient. Hence, option B. Q11. A bag contains only red and blue balls. What is the probability that a ball drawn at random is red?
(1) When two balls are drawn one after another with replacement, the probability that both are red is 4/25.
(2) The number of blue balls is 1.5 times the number of red balls. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option D … Explanation: Let P(red) = p. Statement (1): with replacement, p² = 4/25, so p = 2/5 (probability is non-negative). Sufficient. Statement (2): if red = R, blue = 1.5R, so p = R/2.5R = 2/5. Sufficient. Hence, option D. Q12. A invested Rs 20,000 in a business for the full year. B joined later. What is the difference between the shares of A and B?
(1) B joined after 4 months with an investment of Rs 30,000.
(2) The total profit at the end of the year is Rs 35,000. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option A … Explanation: Statement (1): A's capital-months = 20,000 × 12 = 2,40,000; B's = 30,000 × 8 = 2,40,000. The profit ratio is 1:1, so the shares are equal and the difference is 0, whatever the total profit. Sufficient. Statement (2) gives the total profit but no details of B's investment or duration. Insufficient. Hence, option A. Q13. What is the value of x + y?
(1) x² + 2xy + y² = 49
(2) x > 0 and y > 0 Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option C … Explanation: Statement (1): (x + y)² = 49, so x + y = 7 or −7. Insufficient. Statement (2) only says both are positive and gives no value. Insufficient. Together, x + y must be positive, so x + y = 7. Hence, option C. Q14. z varies directly as x and inversely as y. What is the value of z when x = 6 and y = 4?
(1) z = 5 when x = 4 and y = 3.
(2) z = 9 when x = 10 and y = 7. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option D … Explanation: z = kx/y, so one data point fixes the constant k. Statement (1): 5 = 4k/3 gives k = 15/4, so z = (15/4)(6/4) = 45/8. Sufficient. Statement (2): 9 = 10k/7 gives k = 6.3, so z = 6.3 × 6/4 = 9.45. Sufficient. Each statement alone determines k and hence z. Hence, option D. Q15. Is the positive integer N a perfect square?
(1) N has exactly 3 positive divisors.
(2) N has exactly 9 positive divisors. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option D … Explanation: A positive integer has an odd number of divisors if and only if it is a perfect square. Statement (1): 3 divisors (odd), so N = p², a perfect square. Sufficient. Statement (2): 9 divisors (odd), so N is a perfect square (e.g. p⁸ or p²q²). Sufficient. Hence, option D. Q16. For f(x) = x² + bx + c, what is the value of f(4)?
(1) f(1) = f(3)
(2) f(0) = 5 Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option C … Explanation: f(4) = 16 + 4b + c, so both b and c are needed. Statement (1): 1 + b + c = 9 + 3b + c gives b = −4, but c is unknown. Insufficient. Statement (2): c = 5, but b is unknown. Insufficient. Together: f(4) = 16 − 16 + 5 = 5. Hence, option C. Q17. What is the marked price of an article?
(1) After successive discounts of 20% and 10%, it was sold for Rs 720.
(2) The cost price is Rs 600 and it was sold at a profit of 20%. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option A … Explanation: Statement (1): MP × 0.8 × 0.9 = 720, so MP × 0.72 = 720 and MP = Rs 1,000. Sufficient. Statement (2): SP = 600 × 1.2 = Rs 720, but without the discount information the marked price cannot be found. Insufficient. Hence, option A. Q18. Every man in a certain class either belongs to group A, belongs to group B, or belongs to both groups. 20% of group A consists of men and 65% of group B consists of men. What percentage of the two groups together is made up of men?
(1) Group A contains 50 people.
(2) Group B contains 100 people. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option C … Explanation: The combined percentage is a weighted average of 20% and 65%, so the sizes of both groups are needed. Each statement alone gives the size of only one group. Insufficient. Together, taking the two groups' memberships combined: men = 20% of 50 + 65% of 100 = 10 + 65 = 75 out of 150, i.e. 50%. Hence, option C. Q19. If a and b are consecutive negative integers, is b greater than a?
(1) a + 1 and b – 1 are consecutive negative integers.
(2) b is an odd number. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option A … Explanation: Either b = a + 1 or b = a − 1. Statement (1): if b = a + 1, then b − 1 = a, and a + 1 and a are consecutive (valid). If b = a − 1, then b − 1 = a − 2, which differs from a + 1 by 3 (not consecutive). So b = a + 1 and b > a. Sufficient. Statement (2): b odd says nothing about whether a is b − 1 or b + 1. Insufficient. Hence, option A. Q20. What is the value of a + b?
(1) 'a' and 'b' are the roots of the equation x³ – 3x² + 5x – 9 = 0.
(2) 'a' and 'b' are the roots of the equation x³ + 9x² + 27x + 27 = 0. Statement (1) alone is sufficient, but (2) alone is not sufficient. Statement (2) alone is sufficient, but (1) alone is not sufficient. Both statements together are sufficient, but neither statement alone is sufficient. Each statement alone is sufficient. Statements (1) and (2) together are not sufficient. Correct Option B … Explanation: A cubic has three roots, and a and b are only two of them. Statement (1): the three roots are distinct, and different pairs give different sums, so a + b cannot be determined. Insufficient. Statement (2): x³ + 9x² + 27x + 27 = (x + 3)³, so all three roots equal −3. Whichever two are chosen, a + b = −6. Sufficient. Hence, option B.