Questions & explanations
1. The population of a town increased by a constant percentage each year from 2010 to 2015. What was the population in 2010?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Let P = population in 2010. Statement (1): P * (1 + r)^5 = 1.5P => (1+r)^5 = 1.5, but r unknown, P cancels, cannot find P. Insufficient. Statement (2): r = 0.1, but no absolute population. Insufficient. Together: P * (1.1)^5 = 1.5P => (1.1)^5 ≈ 1.6105, not equal to 1.5., inconsistency?, statement (1) says 1.5 times, statement (2) says 10% each year. Together they give a contradiction? But in DS, we assume statements are true. If both are true, then (1.1)^5 = 1.6105, but statement (1) says factor is 1.5, so no solution?, we must treat statements as given. If both are true, then P * (1.1)^5 = 1.5P => (1.1)^5 = 1.5, which is false. So there is no consistent P? But DS statements are always true. The only way both can be true is if the population in 2015 is 1.5P and also increased by 10% each year, which would require (1.1)^5 = 1.5, which is false. So the statements are contradictory. In official GMAT, contradictory statements are possible?, official DS statements are always consistent with each other and with the question. So this scenario would not appear. revise: Statement (2) should b
2. A bag contains 5 red marbles and 5 blue marbles. Two marbles are drawn without replacement. Is the probability that both marbles are red greater than 0.2?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) alone: given first is red, probability both red = (5/10)*(4/9)=20/90≈0.222>0.2, so YES, sufficient. Statement (2) alone: knowing second is red does not give a unique probability because the first could be red or blue. The probability both red given second is red is P(both red)/P(second red) = (5/10*4/9)/(5/10*4/9 + 5/10*5/9) = (20/90)/(20/90+25/90)=20/45≈0.444>0.2, but this is a conditional probability; the question asks 'Is the probability that both marbles are red greater than 0.2?' Without conditioning, the unconditional probability is (5/10)*(4/9)=20/90≈0.222>0.2, but statement (2) gives information that changes the probability?, the question is ambiguous: it might be asking for the unconditional probability, but statement (2) provides additional information. Typically, in DS, the statements provide facts that affect the probability. If we interpret the question as 'Given the bag composition, is the probability of both red > 0.2?' then the unconditional probability is fixed and both statements are irrelevant. But that would make both statements unnecessary, and answ
3. What is the value of x?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: |2x-3| = |x+1| gives two cases: 2x-3 = x+1 => x=4, or 2x-3 = -(x+1) => 2x-3 = -x-1 => 3x=2 => x=2/3. Two possible values, not sufficient. Statement (2) alone: |x-2|=3 gives x-2=3 => x=5, or x-2=-3 => x=-1. Two values, not sufficient. Together, the common value from both sets is none? 4, 2/3 vs 5, -1: no overlap. So no unique x?, check: from (1) x=4 or 2/3; from (2) x=5 or -1. No common value, so even together no solution? But the question asks for value of x; if no value satisfies both, then together they are not sufficient?, if no value satisfies both, then there is no x, but the question implies existence? Typically DS assumes statements are true; if they are contradictory, then no solution exists, but that still gives a unique answer? No, the question 'What is the value of x?' expects a single numeric value. If statements are contradictory, there is no value, so not sufficient. Thus answer (e). But let's check: maybe I mis-solved. For (1): cases: 2x-3 = x+1 => x=4; 2x-3 = -(x+1) => 2x-3 = -x-1 => 3x=2 => x=2/3. Correct. (2): x-2=3 => x=5; x-2=-3 => x=-1. No ov
4. A company has 10 employees. The average salary is $50,000. Is the median salary greater than $45,000?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1) alone: high outlier pulls mean up, but median could be low. Example: 9 employees earn $30,000 and one earns $200,000 gives mean $47,000 (not $50,000). Need mean $50,000. With highest $200,000, the other 9 sum to $300,000, average $33,333. Median could be $33,333 (if sorted) which is < $45,000. But could also be higher if salaries are more balanced. Not sufficient. Statement (2) alone: lowest $30,000, mean $50,000, other 9 sum $470,000, average $52,222. Median could be $40,000 or $60,000. Not sufficient. Together: still possible median below or above $45,000. For example, set: 30k, 30k, 30k, 30k, 30k, 30k, 30k, 30k, 30k, 200k gives mean $47k (not $50k). Adjust: 30k, 40k, 40k, 40k, 40k, 40k, 40k, 40k, 40k, 200k sum=550k, mean=55k. Need mean 50k, sum=500k. With lowest 30k and highest 200k, remaining 8 sum=270k, average 33.75k. Median is average of 5th and 6th, likely low. Could be <45k. Alternatively, distribute higher: 30k, 45k, 45k, 45k, 45k, 45k, 45k, 45k, 45k, 200k sum=590k, too high. To get sum 500k, many low salaries. Median likely below 45k. But can we get median ab
5. A store sells two types of pens: type A and type B. The total revenue from selling x pens of type A and y pens of type B is $50. How many pens of type A were sold?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone gives equation 2x + 3y = 50, but x and y are not uniquely determined. Statement (2) alone gives x = 2y, but without prices, revenue cannot be used. Together, substitute x = 2y into 2(2y) + 3y = 50 → 7y = 50 → y = 50/7, not integer, but the question asks for number of pens, which must be integer. However, the problem does not state that x and y are integers; they could be fractional? In real-world context, pens are discrete, so non-integer solution means no valid integer solution, but the question asks 'how many were sold?' implying a unique integer. Since the equations yield a unique value for y (50/7) and x (100/7), which are not integers, there is no valid integer solution. But sufficiency requires a unique answer; here the unique answer is that no integer solution exists, which is a valid unique answer (0 pens? no integer pair satisfies both). However, typically DS expects a numeric answer; if the only solution is non-integer, it is still unique. But careful: the question asks 'How many pens of type A were sold?' If the only solution is non-integer, then there
6. If a, b, and c are consecutive integers, is abc divisible by 8?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1): Positive consecutive integers could be 1,2,3 (product 6, not divisible by 8) or 2,3,4 (product 24, divisible by 8). Not sufficient. Statement (2): If b is even, then the three consecutive integers are either even-odd-even or odd-even-odd. In either case, exactly one of them is a multiple of 4?, among three consecutive integers, if the middle is even, then the three are of the form (k-1, k, k+1) with k even. Then k is divisible by 2, and either k-1 or k+1 is also even?, if k is even, then k-1 and k+1 are odd. So only one even number (k). For product to be divisible by 8, we need three factors of 2. Since k is even, it contributes at least one factor of 2. But we need three. For example, 2,3,4: b=3 (odd) not even., if b is even, e.g., 1,2,3: b=2 even, product=6 not divisible by 8. 2,3,4: b=3 odd. So statement (2) alone? Let's test: consecutive integers with middle even: 1,2,3 (product 6, not divisible by 8); 3,4,5 (product 60, not divisible by 8); 5,6,7 (product 210, not divisible by 8)., if b is even, then the three numbers are odd, even, odd. The even number contribute
7. A chemist has two solutions of acid: Solution X is 20% acid and Solution Y is 50% acid. How many liters of Solution X must be mixed with Solution Y to obtain 10 liters of a 30% acid solution?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: EACH statement ALONE is sufficient.
Statement (1) alone: If Y=6, then X=4 (since total 10). Check: acid from X=0.2*4=0.8, from Y=0.5*6=3, total acid=3.8, concentration=3.8/10=38%, not 30%., the question asks for liters of X needed to achieve 30%?, the question: 'How many liters of Solution X must be mixed... to obtain 10 liters of a 30% acid solution?' This implies a specific mixture. Statement (1) gives Y=6, but then X=4 gives 38%, not 30%. So the mixture would not be 30%, so the scenario is impossible? But the statement is given as fact; it must be consistent., if Y=6, then X=4, but that yields 38% acid, not 30%. So the statement contradicts the goal? In DS, statements are true. So if Y=6, then to get 30% overall, we need X such that 0.2X+0.5*6=0.3*10 => 0.2X+3=3 => X=0. So X=0 liters. That is a unique value. Sufficient. Statement (2) alone: total acid=3 liters. Let X liters of 20% and Y liters of 50%, with X+Y=10 and 0.2X+0.5Y=3. Solve: from first, Y=10-X; substitute: 0.2X+0.5(10-X)=3 => 0.2X+5-0.5X=3 => -0.3X=-2 => X=20/3 ≈6.67 liters. Unique. Sufficient. Answer (d).
8. A company has two types of employees: managers and associates. The total annual salary budget for all employees is $1,200,000. Each manager earns an annual salary of $80,000, and each associate earns an annual salary of $40,000. How many managers does the company have?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Let m = number of managers, a = number of associates. From the stem: 80,000m + 40,000a = 1,200,000, or 2m + a = 30. (1) gives m + a = 20. Together with stem, we have two equations: 2m + a = 30 and m + a = 20. Solving gives m = 10, a = 10. So (1) alone is not sufficient because infinite pairs satisfy 2m + a = 30. (2) gives a = 3m. Substituting into stem: 2m + 3m = 30 => 5m = 30 => m = 6. So (2) alone is sufficient. But wait: (2) alone gives m=6, a=18, total employees 24, which satisfies stem. So (2) alone is sufficient. However, (1) alone is not sufficient. Thus answer is (b). Correction: (2) alone gives unique m=6, so (2) sufficient. (1) alone gives m + a = 20, with stem 2m + a = 30 => m=10, a=10, also unique. So each alone sufficient? Check: (1) alone: from stem and (1) we get m=10, a=10. (2) alone: from stem and (2) we get m=6, a=18. Both give unique m, but different values. So each alone is sufficient to determine m uniquely. Thus answer is (d).
9. A set of five distinct positive integers has a mean of 10. What is the median?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: range=10, mode=12. Many sets possible (e.g., {5,12,12,12,15} mean=11.2, not 10; need mean 10. Could be {6,8,12,12,16} mean=10.8; not unique median. Not sufficient. Statement (2) alone: smallest=5, largest=15, mean=10. Sum=50, so remaining three sum=30. Many possibilities (e.g., {5,6,10,14,15} median=10; {5,8,9,13,15} median=9). Not sufficient. Together: from (2) sum of middle three=30, from (1) mode=12 so 12 appears at least twice. Since distinct?, mode=12 implies at least two 12s, but set must be distinct? distinct integers, so mode cannot be 12 unless there are two 12s, contradicting distinctness. Thus no valid set exists? But problem says distinct, so statements together impossible? However, in DS, if statements contradict, answer is (e) because no unique solution. But careful: if statements together lead to no possible set, then the question cannot be answered, so not sufficient. Thus answer (e).
10. Set T consists of 6 integers. Is the median of T greater than 10?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: min=5, max=20, median could be any number between 5 and 20 depending on other values; not sufficient. Statement (2) alone: sum=90, mean=15, but median could be less than 10 (e.g., {1,1,1,1,1,85} median=1) or greater; not sufficient. Together: with min=5, max=20, sum=90, the remaining four integers sum to 65 (since 5+20=25, 90-25=65). To have median ≤10, the third and fourth smallest must be ≤10. The smallest two are 5 and something ≤10, but then the sum of the four middle numbers would be at most 5+10+10+20=45? we need to check possibility. If median ≤10, then the third smallest ≤10 and fourth smallest ≤10 (since even number, median is average of 3rd and 4th). So the four smallest numbers are at most 5, ≤10, ≤10, ≤10, sum ≤5+10+10+10=35, plus max 20 gives total ≤55, but sum is 90, impossible. So median must be >10. Sufficient. Answer (c).
11. What is the median of a set of 5 distinct integers?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1) alone: mean=12 gives sum=60, but median unknown. Not sufficient. Statement (2) alone: smallest+largest=20, but median unknown. Not sufficient. Together: let numbers a<b<c<d<e. a+e=20, sum=60 => b+c+d=40. Median is c. Many possibilities: e.g., a=5, e=15, then b+c+d=40, with b,c,d distinct between 5 and 15. Could be (6,14,20) invalid since e=15, so b,c,d <15. Try (6,13,21) no. need b,c,d <15 and >5. For example, a=4, e=16, then b+c+d=40, possible (10,13,17) but 17>16 invalid. So constraints but still multiple medians. For instance, set {4,9,13,14,16}: sum=56 not 60. Need sum 60. Let a=5, e=15, then b+c+d=40. Possible {5,10,13,17,15} no. Try {5,11,12,17,15} sum=60? 5+11+12+17+15=60, median=12. Another: {5,10,14,16,15} sum=60, median=14. Different medians. Not unique. Thus insufficient. Answer (e).
12. If x is an integer, is x > 10?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (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.
Answer: Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1): x can be 5, 10, or 15. Not all >10 (5 and 10 are not >10). Not sufficient. Statement (2): x can be 7, 11, 13, etc. Not all >10 (7 is not >10). Not sufficient. Together: x must be a prime multiple of 5 between 5 and 20, so x=5? No, 5 is prime and multiple of 5, but 5 is not >5? x>5 from (2) and x<20 from (1), so x=5 is excluded because x>5? (2) says x>5, so x=5 is invalid. Then possible x: 5? No, 5 is not >5. Next prime multiple of 5 is 5? multiples of 5 that are prime: only 5. Since x>5, no integer satisfies both. Thus no solution, but the question is 'is x>10?' With no possible x, the answer is not determined (the premise is contradictory). In DS, if statements are inconsistent, the answer is (e) because together they do not yield a unique answer. So (e).