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Data Sufficiency

263 questions

All types263Multiple choice225True / False38
Question 81Question →

For any non-zero real numbers xx and yy, the Data Sufficiency target question 'Is xy>0\frac{x}{y} > 0?' is algebraically equivalent to the rephrased target question 'Do xx and yy have the same sign?'

Show answer & explanation

Answer: True

Answer

True. The target question simplifies directly to asking whether xx and yy share the same sign.
The target question asks if the ratio xy\frac{x}{y} is positive. In arithmetic, the quotient of two real numbers is positive if and only if both numbers have the same sign (both positive or both negative). Thus, rephrasing the target to 'Do xx and yy have the same sign?' is a completely equivalent simplification.

Step-by-Step Solution

1
Analyze the condition for a fraction to be positive
For xy>0\frac{x}{y} > 0, either (x>0x > 0 and y>0y > 0) or (x<0x < 0 and y<0y < 0).
Dividing two numbers with identical signs yields a positive quotient, while dividing numbers with opposite signs yields a negative quotient.
2
Compare the condition to the proposed rephrased question
The condition 'both positive or both negative' is the definition of xx and yy having the same sign.
Rephrasing a Data Sufficiency stem requires finding a target condition that is true under the exact same conditions as the original question.

Key Concept

Question Stem Rephrasing for Quotient Inequalities
Question 82Question →

If kk is a real number, is ∣k2−4∣<3k|k^2 - 4| < 3k?

(1) k2−5k+4<0k^2 - 5k + 4 < 0
(2) ∣k−1∣<2|k - 1| < 2

Which statement(s) provide sufficient information to determine whether the inequality holds?

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The stem inequality ∣k2−4∣<3k|k^2 - 4| < 3k simplifies to 1<k<41 < k < 4. Statement (1) directly yields 1<k<41 < k < 4, providing a definitive Yes answer. Statement (2) yields −1<k<3-1 < k < 3, which includes values inside (1,4)(1, 4) such as k=2k = 2 (Yes) and values outside (1,4)(1, 4) such as k=0.5k = 0.5 (No), making Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality ∣k2−4∣<3k|k^2 - 4| < 3k requires 3k>03k > 0 (so k>0k > 0) and translates to −3k<k2−4<3k-3k < k^2 - 4 < 3k. Solving k2−4<3kk^2 - 4 < 3k yields (k−4)(k+1)<0  ⟹  −1<k<4(k - 4)(k + 1) < 0 \implies -1 < k < 4. Solving −3k<k2−4-3k < k^2 - 4 for k>0k > 0 yields k2+3k−4>0  ⟹  (k+4)(k−1)>0  ⟹  k>1k^2 + 3k - 4 > 0 \implies (k + 4)(k - 1) > 0 \implies k > 1. Combining these gives the rephrased question: 'Is 1<k<41 < k < 4?'
Simplifying the question stem before evaluating the statements prevents misinterpretation of absolute value boundary conditions.
2
Evaluate Statement (1): k2−5k+4<0k^2 - 5k + 4 < 0.
Factoring the quadratic gives (k−1)(k−4)<0(k - 1)(k - 4) < 0, which means 1<k<41 < k < 4.
Since Statement (1) guarantees that kk lies strictly between 11 and 44, the answer to the rephrased question is a definitive Yes. Statement (1) ALONE is sufficient.
3
Evaluate Statement (2): ∣k−1∣<2|k - 1| < 2.
Removing the absolute value gives −2<k−1<2-2 < k - 1 < 2, or −1<k<3-1 < k < 3.
If k=2k = 2, then 1<2<41 < 2 < 4, giving a Yes answer. If k=0.5k = 0.5, then 0.50.5 is in (−1,3)(-1, 3) but not in (1,4)(1, 4), giving a No answer (since ∣0.52−4∣=3.75>3(0.5)=1.5|0.5^2 - 4| = 3.75 > 3(0.5) = 1.5). Because Statement (2) allows both Yes and No outcomes, it is NOT sufficient.

Key Concept

Absolute Value Inequalities and Question Stem Rephrasing

Alternative Method

Instead of solving the quadratic inequality −3k<k2−4<3k-3k < k^2 - 4 < 3k algebraically, test the boundary points k=1k = 1 and k=4k = 4 in ∣k2−4∣=3k|k^2 - 4| = 3k: at k=1k = 1, ∣1−4∣=3=3(1)|1 - 4| = 3 = 3(1); at k=4k = 4, ∣16−4∣=12=3(4)|16 - 4| = 12 = 3(4). Since 3k3k grows linearly while k2−4k^2 - 4 grows quadratically, the inequality holds strictly between these two positive roots 1<k<41 < k < 4.
Estimated Time:2m 0s
Question 83Question →

If kk is a positive integer, is kk divisible by 6?

(1) kk is a multiple of 2.
(2) kk is a multiple of 3.

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
To determine whether an integer is divisible by 6, it must be divisible by both prime factors of 6, which are 2 and 3. Statement (1) indicates kk is divisible by 2, and Statement (2) indicates kk is divisible by 3. Neither statement alone is sufficient because a number can be a multiple of 2 without being a multiple of 3 (e.g., 2, 4) or a multiple of 3 without being a multiple of 2 (e.g., 3, 9). Taking both statements together ensures kk is a multiple of both 2 and 3, which means kk is a multiple of 6. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the question target.
For a positive integer kk to be divisible by 6, it must be divisible by both 2 and 3 because lcm(2,3)=6\text{lcm}(2,3) = 6.
6 is the product of the prime numbers 2 and 3.
2
Evaluate Statement (1) independently.
If k=2k = 2, it is divisible by 2 but not by 6 (No). If k=6k = 6, it is divisible by 2 and by 6 (Yes). Insufficient.
Statement (1) alone allows multiple outcomes for divisibility by 6.
3
Evaluate Statement (2) independently.
If k=3k = 3, it is divisible by 3 but not by 6 (No). If k=6k = 6, it is divisible by 3 and by 6 (Yes). Insufficient.
Statement (2) alone allows multiple outcomes for divisibility by 6.
4
Combine Statement (1) and Statement (2).
Since kk is divisible by 2 and divisible by 3, kk must be a multiple of lcm(2,3)=6\text{lcm}(2,3) = 6. Thus, kk is definitely divisible by 6 (Definitive Yes).
Both conditions combined uniquely answer the question with a definitive Yes.

Key Concept

Divisibility Rules and Prime Factorization
Question 84Question →

If nn is a real number, is n>5n > 5?

(1) n2+6n+9=0n^2 + 6n + 9 = 0
(2) n2=36n^2 = 36

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) yields n=−3n = -3, which definitively answers 'No' to the question of whether n>5n > 5. In GMAT Data Sufficiency Yes/No questions, a single conclusive 'No' answer makes a statement sufficient. Statement (2) gives n=6n = 6 or n=−6n = -6, leading to both 'Yes' and 'No' answers, making it insufficient. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question stem target.
The target is a Yes/No question: 'Is n>5n > 5?' A statement is sufficient if it allows us to answer either a conclusive 'Yes' for all valid values or a conclusive 'No' for all valid values.
In Data Sufficiency Yes/No questions, both a definitive 'Yes' and a definitive 'No' constitute sufficiency.
2
Evaluate Statement (1) independently.
n2+6n+9=0⇒(n+3)2=0⇒n=−3n^2 + 6n + 9 = 0 \Rightarrow (n + 3)^2 = 0 \Rightarrow n = -3. Testing n=−3n = -3 against n>5n > 5 gives a definitive 'No'. Therefore, Statement (1) ALONE is sufficient.
Since n=−3n = -3 produces a unique answer ('No') to the question stem, Statement (1) is sufficient.
3
Evaluate Statement (2) independently.
n2=36⇒n=6n^2 = 36 \Rightarrow n = 6 or n=−6n = -6. If n=6n = 6, 6>56 > 5 ('Yes'). If n=−6n = -6, −6≯5-6 \ngtr 5 ('No'). Since we get both 'Yes' and 'No', Statement (2) ALONE is not sufficient.
Multiple conflicting answers to a Yes/No question mean the statement cannot conclusively answer the question.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
Question 85Question →

If pp and qq are distinct non-zero real numbers, is p3−q3p−q>p2+q2\frac{p^3 - q^3}{p - q} > p^2 + q^2?

(1) pq+qp>2\frac{p}{q} + \frac{q}{p} > 2

(2) ∣p+q∣>∣p−q∣|p + q| > |p - q|

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
Rephrasing the question stem is the key strategy. Factoring the numerator gives (p−q)(p2+pq+q2)p−q=p2+pq+q2\frac{(p - q)(p^2 + pq + q^2)}{p - q} = p^2 + pq + q^2. Subtracting p2+q2p^2 + q^2 from both sides simplifies the target question to 'Is pq>0pq > 0?'. Statement (1) simplifies to p2+q2pq>2\frac{p^2 + q^2}{pq} > 2; since p2+q2>0p^2 + q^2 > 0, this inequality requires pq>0pq > 0, answering 'Yes'. Statement (2) simplifies by squaring both sides to 4pq>0  ⟺  pq>04pq > 0 \iff pq > 0, also answering 'Yes'. Thus, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem using algebraic factoring.
Since p≠qp \neq q, factor p3−q3=(p−q)(p2+pq+q2)p^3 - q^3 = (p - q)(p^2 + pq + q^2). The target inequality (p−q)(p2+pq+q2)p−q>p2+q2\frac{(p - q)(p^2 + pq + q^2)}{p - q} > p^2 + q^2 simplifies directly to p2+pq+q2>p2+q2p^2 + pq + q^2 > p^2 + q^2, which further simplifies to 'Is pq>0pq > 0?'
Simplifying the target question stem upfront converts a complex cubic rational expression into a simple condition about whether pp and qq have the same sign.
2
Evaluate Statement (1): pq+qp>2\frac{p}{q} + \frac{q}{p} > 2.
Combine fractions over a common denominator: p2+q2pq>2\frac{p^2 + q^2}{pq} > 2. Since pp and qq are distinct non-zero real numbers, (p−q)2>0  ⟹  p2+q2>2pq(p - q)^2 > 0 \implies p^2 + q^2 > 2pq. For p2+q2pq>2\frac{p^2 + q^2}{pq} > 2 to hold, pqpq must be positive (if pq<0pq < 0, the fraction would be negative). Thus, pq>0pq > 0 must be true.
Statement (1) yields a definitive 'Yes' to the rephrased target question 'Is pq>0pq > 0?'. Therefore, Statement (1) alone is sufficient.
3
Evaluate Statement (2): ∣p+q∣>∣p−q∣|p + q| > |p - q|.
Square both non-negative sides: (p+q)2>(p−q)2  ⟹  p2+2pq+q2>p2−2pq+q2  ⟹  4pq>0  ⟹  pq>0(p + q)^2 > (p - q)^2 \implies p^2 + 2pq + q^2 > p^2 - 2pq + q^2 \implies 4pq > 0 \implies pq > 0.
Statement (2) also yields a definitive 'Yes' to the target question 'Is pq>0pq > 0?'. Therefore, Statement (2) alone is sufficient.

Key Concept

Question Stem Simplification and Target Rephrasing
Question 86Question →

Is the real number pp negative?

(1) p2>0p^2 > 0
(2) p3<0p^3 < 0

Show answer & explanation

Answer: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) tells us that p2>0p^2 > 0, which means pp can be any real number except 0. Since pp could be positive (e.g., p=5p = 5) or negative (e.g., p=−5p = -5), Statement (1) alone cannot answer whether pp is negative.

Statement (2) tells us that p3<0p^3 < 0. Because taking an odd power of a real number preserves its sign, p3<0p^3 < 0 implies pp must be negative. This provides a definitive 'Yes' answer to the question stem, making Statement (2) sufficient alone.

Step-by-Step Solution

1
Evaluate Statement (1): p2>0p^2 > 0
pp can be any non-zero real number (e.g., p=2p = 2 or p=−2p = -2).
Squaring any non-zero real number yields a positive value. Thus, pp could be positive or negative, so Statement (1) is not sufficient.
2
Evaluate Statement (2): p3<0p^3 < 0
pp must be negative.
An odd exponent preserves the sign of the base. If p3<0p^3 < 0, then pp must be negative. This provides a definitive 'Yes' answer to whether pp is negative, so Statement (2) is sufficient.

Key Concept

Properties of real numbers under even and odd exponents
Estimated Time:1m 0s
Question 87Question →

If xx is a real number, is ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7?

(1) x2−2x−8≤0x^2 - 2x - 8 \le 0
(2) x(x−5)≤0x(x - 5) \le 0

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrasing the question target ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7 shows that it is satisfied if and only if −3≤x≤4-3 \le x \le 4. Statement (1) yields the interval [−2,4][-2, 4], which is completely contained within [−3,4][-3, 4], guaranteeing a definitive 'Yes'. Statement (2) yields [0,5][0, 5], which includes values like 4.54.5 that result in a 'No'. Thus, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question stem inequality ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7.
The target inequality simplifies to −3≤x≤4-3 \le x \le 4.
Interpret ∣x−3∣+∣x+2∣|x - 3| + |x + 2| as the sum of distances from xx to 33 and −2-2 on the real number line. The distance between −2-2 and 33 is 55. For −2≤x≤3-2 \le x \le 3, the sum of distances is constantly 5≤75 \le 7. For x>3x > 3, the sum is 2x−1≤7  ⟹  x≤42x - 1 \le 7 \implies x \le 4. For x<−2x < -2, the sum is 1−2x≤7  ⟹  x≥−31 - 2x \le 7 \implies x \ge -3. Thus, the inequality holds if and only if x∈[−3,4]x \in [-3, 4].
2
Evaluate Statement (1): x2−2x−8≤0x^2 - 2x - 8 \le 0.
Statement (1) gives −2≤x≤4-2 \le x \le 4.
Factoring gives (x−4)(x+2)≤0(x - 4)(x + 2) \le 0, which holds when x∈[−2,4]x \in [-2, 4]. Since [−2,4]⊆[−3,4][-2, 4] \subseteq [-3, 4], every value of xx satisfying Statement (1) guarantees that ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7 is true. This yields a definitive 'Yes'. Statement (1) alone is sufficient.
3
Evaluate Statement (2): x(x−5)≤0x(x - 5) \le 0.
Statement (2) gives 0≤x≤50 \le x \le 5.
The solution set is x∈[0,5]x \in [0, 5]. If x=1x = 1, −3≤1≤4-3 \le 1 \le 4 is true (Yes). If x=4.5x = 4.5, ∣4.5−3∣+∣4.5+2∣=1.5+6.5=8≤7|4.5 - 3| + |4.5 + 2| = 1.5 + 6.5 = 8 \le 7 is false (No). Since Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Key Concept

Question stem rephrasing for distance-based absolute values and set-inclusion sufficiency evaluation
Question 88Question →

If xx is a real number, is ∣x−3∣<2x|x - 3| < 2x?

(1) x2−4x+3>0x^2 - 4x + 3 > 0
(2) ∣x−2∣<1|x - 2| < 1

Which of the following choices correctly describes the sufficiency of the statements?

Show answer & explanation

Answer: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Rephrasing the stem ∣x−3∣<2x|x - 3| < 2x shows it is equivalent to asking whether x>1x > 1. Statement (2) restricts xx to the range 1<x<31 < x < 3, where every value of xx satisfies x>1x > 1, providing a definitive YES answer. Statement (1) permits x<1x < 1 (giving a NO) and x>3x > 3 (giving a YES), making Statement (1) insufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality is ∣x−3∣<2x|x - 3| < 2x. Since absolute values are non-negative (∣x−3∣≥0|x - 3| \ge 0), 2x2x must be strictly positive, so x>0x > 0.
- If x≥3x \ge 3, ∣x−3∣=x−3<2x  ⟹  x>−3|x - 3| = x - 3 < 2x \implies x > -3, which holds for all x≥3x \ge 3.
- If 0<x<30 < x < 3, ∣x−3∣=3−x<2x  ⟹  3x>3  ⟹  x>1|x - 3| = 3 - x < 2x \implies 3x > 3 \implies x > 1.
Thus, the inequality ∣x−3∣<2x|x - 3| < 2x holds if and only if x>1x > 1. The target question simplifies to: 'Is x>1x > 1?'
Simplifying the question stem before analyzing statements prevents misinterpreting the range of valid solutions.
2
Evaluate Statement (1): x2−4x+3>0x^2 - 4x + 3 > 0.
Factoring the quadratic yields (x−1)(x−3)>0(x - 1)(x - 3) > 0, so x<1x < 1 or x>3x > 3.
- If x=0x = 0 (which is <1< 1), ∣0−3∣=3|0 - 3| = 3 and 2(0)=02(0) = 0. Is 3<03 < 0? NO.
- If x=4x = 4 (which is >3> 3), ∣4−3∣=1|4 - 3| = 1 and 2(4)=82(4) = 8. Is 1<81 < 8? YES.
Since Statement (1) allows for both YES and NO outcomes, it is NOT sufficient.
Testing specific values from the disjoint solution set demonstrates that a single definitive answer cannot be reached.
3
Evaluate Statement (2): ∣x−2∣<1|x - 2| < 1.
Solving the absolute value inequality gives −1<x−2<1  ⟹  1<x<3-1 < x - 2 < 1 \implies 1 < x < 3.
For every value of xx in the interval (1,3)(1, 3), xx is strictly greater than 11. Therefore, the answer to 'Is x>1x > 1?' is a definitive YES for all possible values of xx in this statement.
A statement is sufficient if every allowed value of xx yields a consistent YES answer to the rephrased question stem.

Key Concept

Question Stem Rephrasing for Absolute Value Inequalities
Estimated Time:2m 0s
Question 89Question →

In a GMAT Data Sufficiency 'Yes/No' question, a statement that restricts a variable to a domain of values for which the condition in the question stem is false for every value in that domain is insufficient to answer the question.

Show answer & explanation

Answer: False

Answer

False. A statement that consistently yields a 'No' answer for all permissible values provides a definitive answer and is therefore sufficient.
The correct evaluation is False. In GMAT Data Sufficiency 'Yes/No' decision logic, a statement provides sufficient information if it yields a single, conclusive answer to the question stem. If a statement restricts a variable such that the stem condition is false for every possible value, the answer to the question is conclusively 'No'. Because the question is answered with certainty, the statement is sufficient.

Step-by-Step Solution

1
Define the criterion for sufficiency in GMAT 'Yes/No' Data Sufficiency questions.
A statement is sufficient if it allows the question in the stem to be answered with certainty—either always 'Yes' or always 'No'.
Sufficiency depends on determinism, not on receiving an affirmative result.
2
Analyze the scenario where the stem condition is false for every value in the statement's restricted domain.
Because every permissible value makes the condition false, the answer to the stem's question is guaranteed to be 'No'.
There is no ambiguity or variation in outcome across the allowed values.
3
Evaluate the statement's claim of insufficiency.
The claim that such a statement is 'insufficient' is false because a deterministic 'No' satisfies all requirements for data sufficiency.
Insufficiency only occurs when a statement permits both 'Yes' and 'No' answers under different allowed values.

Key Concept

Definitive Answer Criterion in Yes/No Data Sufficiency Logic
Question 90Question →

If xx is a real number, is x2−4x+3<0x^2 - 4x + 3 < 0?

(1) (x−4)2≤0(x - 4)^2 \le 0
(2) ∣x−2∣<2|x - 2| < 2

Which of the following statements correctly describes the sufficiency of the given statements to answer the question?

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
The choice stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient is correct. Statement (1) forces x=4x = 4. Evaluating x=4x = 4 against the condition x2−4x+3<0x^2 - 4x + 3 < 0 gives 42−4(4)+3=3<04^2 - 4(4) + 3 = 3 < 0, which is conclusively false. A statement that yields a definitive 'No' to a Yes/No question is sufficient. Statement (2) yields the range 0<x<40 < x < 4, which contains values of xx that make x2−4x+3<0x^2 - 4x + 3 < 0 true (such as x=2x = 2) and false (such as x=3.5x = 3.5), making Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the question stem target inequality.
The inequality x2−4x+3<0x^2 - 4x + 3 < 0 factors into (x−1)(x−3)<0(x - 1)(x - 3) < 0, which holds true if and only if 1<x<31 < x < 3. Thus, the question asks: 'Is 1<x<31 < x < 3?'
Simplifying the question stem establishes the exact criteria needed for a 'Yes' vs. 'No' response.
2
Evaluate Statement (1): (x−4)2≤0(x - 4)^2 \le 0.
Since any square of a real number is non-negative, (x−4)2≥0(x - 4)^2 \ge 0. The only way (x−4)2≤0(x - 4)^2 \le 0 can hold is if (x−4)2=0(x - 4)^2 = 0, which means x=4x = 4. Substituting x=4x = 4 into 1<x<31 < x < 3 gives 1<4<31 < 4 < 3, which is false. Therefore, the answer to the stem's question is a definitive 'NO'. In Yes/No Data Sufficiency logic, a definitive 'No' means the statement IS SUFFICIENT.
A statement that yields a consistent and definitive answer—whether 'Yes' or 'No'—provides sufficient information.
3
Evaluate Statement (2): ∣x−2∣<2|x - 2| < 2.
The absolute value inequality ∣x−2∣<2|x - 2| < 2 unravels to −2<x−2<2-2 < x - 2 < 2, or 0<x<40 < x < 4.
- If x=2x = 2 (which is in 0<x<40 < x < 4), then 1<2<31 < 2 < 3 is true ('YES').
- If x=0.5x = 0.5 (which is in 0<x<40 < x < 4), then 1<0.5<31 < 0.5 < 3 is false ('NO').
Since Statement (2) allows both 'Yes' and 'No' answers depending on the value chosen for xx, Statement (2) is NOT SUFFICIENT.
If a statement leads to contradictory answers ('Yes' under some conditions, 'No' under others), it is insufficient.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
Estimated Time:2m 0s
Question 91Question →

If xx is a real number, is x2<4x^2 < 4?

(1) ∣x−1∣<2|x - 1| < 2
(2) ∣x+1∣<3|x + 1| < 3

Show answer & explanation

Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the target question x2<4x^2 < 4 shows that we need to determine if −2<x<2-2 < x < 2. Statement (1) gives −1<x<3-1 < x < 3, which allows values outside (−2,2)(-2, 2) like x=2.5x = 2.5, so it is insufficient alone. Statement (2) gives −4<x<2-4 < x < 2, which allows values outside (−2,2)(-2, 2) like x=−3x = -3, so it is insufficient alone. Taking the intersection of both statements gives −1<x<2-1 < x < 2. Since every number in (−1,2)(-1, 2) satisfies −2<x<2-2 < x < 2, the combined statements definitively answer YES.

Step-by-Step Solution

1
Rephrase the target question
The target question asking whether x2<4x^2 < 4 is equivalent to asking whether −2<x<2-2 < x < 2.
Taking the square root of both sides of x2<4x^2 < 4 yields ∣x∣<2|x| < 2, which expands to −2<x<2-2 < x < 2.
2
Evaluate Statement (1) independently
Statement (1) states ∣x−1∣<2|x - 1| < 2, which expands to −2<x−1<2-2 < x - 1 < 2, or −1<x<3-1 < x < 3.
If x=0x = 0, then −1<0<3-1 < 0 < 3 is true and 02<40^2 < 4 (Yes). If x=2.5x = 2.5, then −1<2.5<3-1 < 2.5 < 3 is true, but 2.52=6.25≮42.5^2 = 6.25 \not< 4 (No). Because we get both Yes and No answers, Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently
Statement (2) states ∣x+1∣<3|x + 1| < 3, which expands to −3<x+1<3-3 < x + 1 < 3, or −4<x<2-4 < x < 2.
If x=0x = 0, then −4<0<2-4 < 0 < 2 is true and 02<40^2 < 4 (Yes). If x=−3x = -3, then −4<−3<2-4 < -3 < 2 is true, but (−3)2=9≮4(-3)^2 = 9 \not< 4 (No). Because we get both Yes and No answers, Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together
Combining −1<x<3-1 < x < 3 and −4<x<2-4 < x < 2 requires xx to satisfy both inequalities simultaneously, yielding the intersection −1<x<2-1 < x < 2.
Every value of xx in the interval (−1,2)(-1, 2) lies strictly inside the required target interval (−2,2)(-2, 2). Thus, x2<4x^2 < 4 is definitively YES. Both statements together are sufficient.

Key Concept

Inequality range intersection and absolute value distance expansion in Data Sufficiency
Estimated Time:2m 0s
Question 92Question →

If mm and nn are non-zero real numbers, is ∣m+n∣m>1\frac{|m + n|}{m} > 1?

(1) m<−∣n∣m < -|n|
(2) mn>0mn > 0 and ∣m+n∣=m+n|m + n| = m + n

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
The correct choice is that EACH statement ALONE is sufficient. Statement (1) restricts mm to negative values, guaranteeing that the numerator is non-negative while the denominator is negative, so the ratio can never exceed 1 (yielding a definitive 'No'). Statement (2) forces both mm and nn to be strictly positive, which simplifies the ratio to 1+nm>11 + \frac{n}{m} > 1 (yielding a definitive 'Yes'). Since each statement provides a definitive answer independently, each statement alone is sufficient.

Step-by-Step Solution

1
Analyze the question stem target ∣m+n∣m>1\frac{|m + n|}{m} > 1.
Since absolute value ∣m+n∣|m + n| is always non-negative (∣m+n∣≥0|m + n| \ge 0), if m<0m < 0, the ratio ∣m+n∣m≤0\frac{|m + n|}{m} \le 0, which can never be greater than 1. Thus, if m<0m < 0, the answer is a definitive 'No'. If m>0m > 0, the expression simplifies to ∣m+n∣>m|m + n| > m.
Establishing the behavior of the expression based on the sign of mm simplifies statement evaluation.
2
Evaluate Statement (1): m<−∣n∣m < -|n|.
Since n≠0n \neq 0, ∣n∣>0|n| > 0, which implies −∣n∣<0-|n| < 0. Therefore, m<0m < 0. Because m<0m < 0 and ∣m+n∣≥0|m + n| \ge 0, the fraction ∣m+n∣m\frac{|m + n|}{m} is non-positive (≤0\le 0). Thus, ∣m+n∣m\frac{|m + n|}{m} cannot be greater than 1, yielding a definitive 'No'. Statement (1) ALONE is sufficient.
A statement that yields a definitive 'No' to a Yes/No Data Sufficiency question is sufficient.
3
Evaluate Statement (2): mn>0mn > 0 and ∣m+n∣=m+n|m + n| = m + n.
From mn>0mn > 0, mm and nn must have the same sign (both positive or both negative). The condition ∣m+n∣=m+n|m + n| = m + n requires m+n≥0m + n \ge 0. If both mm and nn were negative, m+nm + n would be negative, contradicting ∣m+n∣≥0|m + n| \ge 0. Hence, both mm and nn must be positive (m>0m > 0 and n>0n > 0). Substituting into the expression: ∣m+n∣m=m+nm=1+nm\frac{|m + n|}{m} = \frac{m + n}{m} = 1 + \frac{n}{m}. Since n>0n > 0 and m>0m > 0, nm>0\frac{n}{m} > 0, so 1+nm>11 + \frac{n}{m} > 1. This yields a definitive 'Yes'. Statement (2) ALONE is sufficient.
Determining that mm and nn must both be positive allows exact evaluation of the inequality.

Key Concept

Data Sufficiency logic for Yes/No questions involving absolute value properties and variable sign analysis.
Question 93Question →

If xx and yy are real numbers such that x≠yx \neq y and x+y≠0x + y \neq 0, is x3−y3x2−y2>x+y\frac{x^3 - y^3}{x^2 - y^2} > x + y?

(1) x>y>0x > y > 0

(2) ∣x+y∣<∣x−y∣|x + y| < |x - y|

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Factoring the numerator and denominator of the target expression transforms the question 'Is x3−y3x2−y2>x+y\frac{x^3 - y^3}{x^2 - y^2} > x + y?' into the simplified target 'Is xyx+y<0\frac{xy}{x + y} < 0?'. Under Statement (1), both xx and yy are positive, so xy>0xy > 0 and x+y>0x + y > 0, making xyx+y>0\frac{xy}{x + y} > 0. This conclusively answers the target question with a definitive 'No', proving Statement (1) is sufficient alone. Statement (2) simplifies to xy<0xy < 0, which allows x+yx + y to be positive or negative, resulting in both 'Yes' and 'No' possibilities, rendering Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the question target algebraically before evaluating the statements.
Simplifying x3−y3x2−y2\frac{x^3 - y^3}{x^2 - y^2} by factoring numerator and denominator yields (x−y)(x2+xy+y2)(x−y)(x+y)=x2+xy+y2x+y\frac{(x - y)(x^2 + xy + y^2)}{(x - y)(x + y)} = \frac{x^2 + xy + y^2}{x + y}. The inequality x2+xy+y2x+y>x+y\frac{x^2 + xy + y^2}{x + y} > x + y simplifies to x2+xy+y2−(x+y)2x+y>0\frac{x^2 + xy + y^2 - (x + y)^2}{x + y} > 0, which reduces to −xyx+y>0\frac{-xy}{x + y} > 0, or equivalently xyx+y<0\frac{xy}{x + y} < 0.
Target rephrasing reveals that the question is asking whether xyxy and x+yx + y have opposite signs.
2
Evaluate Statement (1): x>y>0x > y > 0.
Since x>0x > 0 and y>0y > 0, xy>0xy > 0 and x+y>0x + y > 0. Consequently, xyx+y>0\frac{xy}{x + y} > 0. This yields a definitive 'No' to the rephrased question 'Is xyx+y<0\frac{xy}{x + y} < 0?'. A definitive 'No' is sufficient.
A statement that guarantees a definitive 'No' answer to a Yes/No question is sufficient.
3
Evaluate Statement (2): ∣x+y∣<∣x−y∣|x + y| < |x - y|.
Squaring both sides gives (x+y)2<(x−y)2  ⟹  x2+2xy+y2<x2−2xy+y2  ⟹  4xy<0  ⟹  xy<0(x + y)^2 < (x - y)^2 \implies x^2 + 2xy + y^2 < x^2 - 2xy + y^2 \implies 4xy < 0 \implies xy < 0. However, knowing xy<0xy < 0 leaves the sign of x+yx + y undetermined. For example, if x=3,y=−1x = 3, y = -1, then x+y=2>0x + y = 2 > 0 and xyx+y=−1.5<0\frac{xy}{x+y} = -1.5 < 0 (Yes). If x=1,y=−3x = 1, y = -3, then x+y=−2<0x + y = -2 < 0 and xyx+y=1.5>0\frac{xy}{x+y} = 1.5 > 0 (No).
Since the outcome can be either 'Yes' or 'No', Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency Target Simplification and Yes/No Decision Logic
Question 94Question →

If xx is a real number, is ∣2x−3∣≤5|2x - 3| \le 5?

(1) ∣x−1∣≤2|x - 1| \le 2
(2) x2−3x−4≤0x^2 - 3x - 4 \le 0

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
Rephrasing the target inequality ∣2x−3∣≤5|2x - 3| \le 5 gives −1≤x≤4-1 \le x \le 4. Statement (1) yields −1≤x≤3-1 \le x \le 3, which is completely contained within [−1,4][-1, 4] and therefore guarantees a definitive 'Yes'. Statement (2) yields −1≤x≤4-1 \le x \le 4, which matches the condition directly and also gives a definitive 'Yes'. Hence, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target inequality algebraically.
−5≤2x−3≤5  ⟹  −2≤2x≤8  ⟹  −1≤x≤4-5 \le 2x - 3 \le 5 \implies -2 \le 2x \le 8 \implies -1 \le x \le 4. Target Question: Is −1≤x≤4-1 \le x \le 4?
Simplifying the question stem converts an absolute value inequality into a clear number range for xx.
2
Evaluate Statement (1): ∣x−1∣≤2|x - 1| \le 2.
−2≤x−1≤2  ⟹  −1≤x≤3-2 \le x - 1 \le 2 \implies -1 \le x \le 3. Since every value in [−1,3][-1, 3] is also in [−1,4][-1, 4], the answer to 'Is −1≤x≤4-1 \le x \le 4?' is a definitive YES.
If a statement's allowed range is a subset of the target range, it guarantees a definitive 'Yes' answer.
3
Evaluate Statement (2): x2−3x−4≤0x^2 - 3x - 4 \le 0.
Factor the quadratic: (x−4)(x+1)≤0  ⟹  −1≤x≤4(x - 4)(x + 1) \le 0 \implies -1 \le x \le 4. This matches the target range exactly, yielding a definitive YES.
Matching the target inequality range provides a definitive 'Yes' answer.

Key Concept

Data Sufficiency Yes/No decision logic for absolute value range constraints and subset range implications.
Estimated Time:1m 30s
Question 95Question →

If xx is a positive integer, is x2−5x+6>0x^2 - 5x + 6 > 0?

(1) (x−2)(x−3)≤0(x - 2)(x - 3) \le 0
(2) x<5x < 5

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
The option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct. Statement (1) restricts the positive integer xx to {2,3}\{2, 3\}, for both of which x2−5x+6=0x^2 - 5x + 6 = 0. This yields a definitive 'No' to the question of whether the expression is strictly greater than 0, making Statement (1) sufficient. Statement (2) permits values such as x=1x = 1 (which yields 'Yes') and x=2x = 2 (which yields 'No'), making it insufficient.

Step-by-Step Solution

1
Rephrase the question stem target algebraically.
The target expression x2−5x+6x^2 - 5x + 6 factors into (x−2)(x−3)(x - 2)(x - 3). The question asks: 'Is (x−2)(x−3)>0(x - 2)(x - 3) > 0?'
Factoring simplifies evaluating the sign of the expression for specific values of xx.
2
Evaluate Statement (1) independently.
Statement (1) gives (x−2)(x−3)≤0(x - 2)(x - 3) \le 0. Since xx is a positive integer, the integer solutions to this inequality are x=2x = 2 and x=3x = 3. For x=2x = 2, (2−2)(2−3)=0(2-2)(2-3) = 0. For x=3x = 3, (3−2)(3−3)=0(3-2)(3-3) = 0. In both cases, x2−5x+6=0x^2 - 5x + 6 = 0, which is NOT >0> 0. Thus, the answer to the stem question is a definitive 'NO'.
In a Yes/No Data Sufficiency question, any statement that yields a definitive 'No' for all possible values of the variable is SUFFICIENT.
3
Evaluate Statement (2) independently.
Statement (2) gives x<5x < 5. Since xx is a positive integer, xx can be 1, 2, 3, or 4. If x=1x = 1, 12−5(1)+6=2>01^2 - 5(1) + 6 = 2 > 0 (Answer: YES). If x=2x = 2, 22−5(2)+6=02^2 - 5(2) + 6 = 0, which is not >0> 0 (Answer: NO). Since Statement (2) allows both 'YES' and 'NO' outcomes, it is NOT SUFFICIENT.
A statement that leads to conflicting answers ('Maybe') is insufficient.

Key Concept

Value vs. Yes/No Data Sufficiency Decision Logic
Question 96Question →

If aa and bb are non-zero real numbers, is ∣a−b∣>∣a+b∣|a - b| > |a + b|?

(1) ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|
(2) ∣a−1∣<∣b−1∣|a - 1| < |b - 1|

Which of the following choices correctly describes the sufficiency of the statements?

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The option stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient is correct. Squaring both sides of the question stem inequality ∣a−b∣>∣a+b∣|a - b| > |a + b| yields a2−2ab+b2>a2+2ab+b2a^2 - 2ab + b^2 > a^2 + 2ab + b^2, which simplifies directly to ab<0ab < 0. Statement (1) states ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|. Squaring both sides yields a2+4ab+4b2<a2−4ab+4b2a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2, which simplifies to 8ab<08ab < 0, or ab<0ab < 0. This gives a definitive YES to the question, so Statement (1) is sufficient. Statement (2) compares the distances of aa and bb from 1 on the number line. Choosing a=0.5a = 0.5 and b=−2b = -2 satisfies ∣0.5−1∣<∣−2−1∣|0.5 - 1| < |-2 - 1| and gives ab=−1<0ab = -1 < 0 (YES), whereas choosing a=0.5a = 0.5 and b=3b = 3 satisfies ∣0.5−1∣<∣3−1∣|0.5 - 1| < |3 - 1| and gives ab=1.5>0ab = 1.5 > 0 (NO). Thus, Statement (2) is insufficient.

Step-by-Step Solution

1
Rephrase the question stem target using absolute value properties.
Since both sides of ∣a−b∣>∣a+b∣|a - b| > |a + b| are non-negative, square both sides: (a−b)2>(a+b)2  ⟹  a2−2ab+b2>a2+2ab+b2  ⟹  −2ab>2ab  ⟹  4ab<0  ⟹  ab<0(a - b)^2 > (a + b)^2 \implies a^2 - 2ab + b^2 > a^2 + 2ab + b^2 \implies -2ab > 2ab \implies 4ab < 0 \implies ab < 0. Thus, the question asks: 'Is ab<0ab < 0?' (Do aa and bb have opposite signs?)
Simplifying the target inequality converts an absolute value relationship into a clear sign constraint on the product abab.
2
Evaluate Statement (1): ∣a+2b∣<∣a−2b∣|a + 2b| < |a - 2b|.
Square both sides: (a+2b)2<(a−2b)2  ⟹  a2+4ab+4b2<a2−4ab+4b2  ⟹  4ab<−4ab  ⟹  8ab<0  ⟹  ab<0(a + 2b)^2 < (a - 2b)^2 \implies a^2 + 4ab + 4b^2 < a^2 - 4ab + 4b^2 \implies 4ab < -4ab \implies 8ab < 0 \implies ab < 0. This provides a definitive YES to the question 'Is ab<0ab < 0?'
Statement (1) uniquely guarantees that aa and bb have opposite signs, making Statement (1) ALONE sufficient.
3
Evaluate Statement (2): ∣a−1∣<∣b−1∣|a - 1| < |b - 1|.
Test numbers: Case 1: If a=0.5a = 0.5 and b=−2b = -2, then ∣0.5−1∣=0.5|0.5 - 1| = 0.5 and ∣−2−1∣=3|-2 - 1| = 3. Since 0.5<30.5 < 3, Statement (2) holds, and ab=(0.5)(−2)=−1<0ab = (0.5)(-2) = -1 < 0 (YES). Case 2: If a=0.5a = 0.5 and b=3b = 3, then ∣0.5−1∣=0.5|0.5 - 1| = 0.5 and ∣3−1∣=2|3 - 1| = 2. Since 0.5<20.5 < 2, Statement (2) holds, and ab=(0.5)(3)=1.5>0ab = (0.5)(3) = 1.5 > 0 (NO).
Because Statement (2) permits both ab<0ab < 0 and ab>0ab > 0, it cannot determine whether ab<0ab < 0. Statement (2) ALONE is insufficient.

Key Concept

Properties of Absolute Value Inequalities and Rephrased Data Sufficiency Targets
Estimated Time:2m 0s
Question 97Question →

If xx is a real number, what is the value of (x+3)2(x + 3)^2?

(1) x2+6x=7x^2 + 6x = 7
(2) x=1x = 1

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

Each statement alone is sufficient.
Expanding the target expression (x+3)2(x + 3)^2 gives x2+6x+9x^2 + 6x + 9. Statement (1) provides x2+6x=7x^2 + 6x = 7, which can be directly substituted into the expanded expression to get 7+9=167 + 9 = 16, yielding a unique value. Statement (2) explicitly provides x=1x = 1, which gives (1+3)2=16(1 + 3)^2 = 16. Because each statement independently provides enough information to determine a single value for the expression, the option stating that each statement alone is sufficient is correct.

Step-by-Step Solution

1
Rephrase the target expression using algebraic expansion.
(x+3)2=x2+6x+9(x + 3)^2 = x^2 + 6x + 9
Expanding the binomial allows direct comparison with given algebraic statements.
2
Evaluate Statement (1) using the expanded target expression.
Since x2+6x=7x^2 + 6x = 7, substituting this into x2+6x+9x^2 + 6x + 9 gives 7+9=167 + 9 = 16. Alternatively, solving x2+6x−7=0x^2 + 6x - 7 = 0 yields x=1x = 1 or x=−7x = -7. If x=1x = 1, (1+3)2=16(1 + 3)^2 = 16. If x=−7x = -7, (−7+3)2=(−4)2=16(-7 + 3)^2 = (-4)^2 = 16. Both roots yield the exact same value of 1616.
Statement (1) alone uniquely determines the target value.
3
Evaluate Statement (2) independently.
Given x=1x = 1, substituting directly into the target expression gives (1+3)2=42=16(1 + 3)^2 = 4^2 = 16.
Statement (2) alone uniquely determines the target value.
4
Conclude sufficiency based on independent evaluations.
Both Statement (1) alone and Statement (2) alone are sufficient.
Since each statement independently determines a single value of 16, EACH statement ALONE is sufficient.

Key Concept

Algebraic Expression Expansion and Target Rephrasing in Data Sufficiency
Estimated Time:1m 0s
Question 98Question →

For all distinct non-zero real numbers xx and yy, the Data Sufficiency Yes/No target question "Is x∣y∣−y∣x∣x−y>0\frac{x|y| - y|x|}{x - y} > 0?" is algebraically equivalent to the simplified target question "Is xy<0xy < 0?"

Show answer & explanation

Answer: True

Answer

The statement is true. Simplifying the target question by considering the relative signs of xx and yy proves that the expression is strictly positive if and only if xx and yy have opposite signs (xy<0xy < 0).
Rephrasing a Data Sufficiency target question requires finding a simpler, logically equivalent condition. Testing the two distinct operational domains (xy>0xy > 0 vs. xy<0xy < 0) demonstrates that the given algebraic fraction yields a positive result if and only if xx and yy carry opposite signs (xy<0xy < 0). Therefore, the proposed rephrasing is completely accurate and equivalent.

Step-by-Step Solution

1
Analyze the expression when xx and yy have the same sign (xy>0xy > 0).
If x>0,y>0x > 0, y > 0, then ∣x∣=x|x| = x and ∣y∣=y|y| = y, so x∣y∣−y∣x∣=xy−yx=0x|y| - y|x| = xy - yx = 0. If x<0,y<0x < 0, y < 0, then ∣x∣=−x|x| = -x and ∣y∣=−y|y| = -y, so x∣y∣−y∣x∣=x(−y)−y(−x)=−xy+xy=0x|y| - y|x| = x(-y) - y(-x) = -xy + xy = 0. Thus, when xy>0xy > 0, x∣y∣−y∣x∣x−y=0\frac{x|y| - y|x|}{x - y} = 0, yielding a 'No' to the question 'Is the expression >0> 0?'.
Determining the behavior of absolute value expressions under identical signs simplifies the numerator to zero.
2
Analyze the expression when xx and yy have opposite signs (xy<0xy < 0).
Case A: If x>0x > 0 and y<0y < 0, then ∣x∣=x|x| = x and ∣y∣=−y|y| = -y. The numerator is x(−y)−y(x)=−2xy>0x(-y) - y(x) = -2xy > 0. The denominator is x−y>0x - y > 0. The quotient is positivepositive>0\frac{\text{positive}}{\text{positive}} > 0.
Case B: If x<0x < 0 and y>0y > 0, then ∣x∣=−x|x| = -x and ∣y∣=y|y| = y. The numerator is x(y)−y(−x)=2xy<0x(y) - y(-x) = 2xy < 0. The denominator is x−y<0x - y < 0. The quotient is negativenegative>0\frac{\text{negative}}{\text{negative}} > 0.
Evaluating absolute values under opposite signs demonstrates that the numerator and denominator always have matching signs.
3
Compare the conditions for a 'Yes' answer.
The target expression is strictly positive if and only if xx and yy have opposite signs, which is defined by the inequality xy<0xy < 0.
Establishing biconditional equivalence confirms that rephrasing the target stem to 'Is xy<0xy < 0?' preserves all logical outcomes.

Key Concept

Data Sufficiency target rephrasing using piecewise definition of absolute value and sign analysis.
Question 99Question →

If rr and ss are positive integers, is the sum r+sr + s an odd integer?

(1) r⋅sr \cdot s is an odd integer.
(2) rr is an even integer.

Show answer & explanation

Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) gives enough information to determine that r+sr + s must be even, resulting in a definitive 'No' answer to the question. In Data Sufficiency, a definitive 'No' indicates sufficiency. Statement (2) does not fix the parity of ss, so r+sr + s could be either even or odd, making it insufficient.

Step-by-Step Solution

1
Analyze the question stem
We are given that rr and ss are positive integers and need to answer the Yes/No question: Is r+sr + s odd?
Establishing the target question and constraints is essential before evaluating the statements.
2
Evaluate Statement (1): r⋅sr \cdot s is an odd integer
The product of two integers is odd if and only if both integers are odd. Therefore, rr is odd and ss is odd. The sum of two odd integers is always an even integer (odd + odd = even). Thus, r+sr + s is definitely not odd.
A definitive 'No' answer to a Yes/No Data Sufficiency question means the statement is SUFFICIENT.
3
Evaluate Statement (2): rr is an even integer
If r=2r = 2 and s=1s = 1, then r+s=3r + s = 3 (odd, answer is Yes). If r=2r = 2 and s=2s = 2, then r+s=4r + s = 4 (even, answer is No). Because both Yes and No are possible, Statement (2) is INSUFFICIENT.
Since Statement (2) allows multiple outcomes for the target question, it does not provide sufficient information.

Key Concept

Parity rules for addition and multiplication of integers in Data Sufficiency Yes/No decision logic.
Question 100Question →

If xx and yy are positive real numbers, is x+y2−xy>1\frac{x + y}{2} - \sqrt{xy} > 1?

(1) (x+y)2=9(\sqrt{x} + \sqrt{y})^2 = 9 and xy=4xy = 4
(2) x+y=8x + y = 8 and xy=9xy = 9

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
Rephrasing the question stem shows that x+y2−xy>1\frac{x + y}{2} - \sqrt{xy} > 1 is equivalent to (x−y)2>2(\sqrt{x} - \sqrt{y})^2 > 2. Statement (1) establishes that (x−y)2=1(\sqrt{x} - \sqrt{y})^2 = 1, giving a definitive 'No' to the target question. Statement (2) establishes that (x−y)2=2(\sqrt{x} - \sqrt{y})^2 = 2, which also gives a definitive 'No' since 2 is not strictly greater than 2. Thus, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target question 'Is x+y2−xy>1\frac{x + y}{2} - \sqrt{xy} > 1?' simplifies to 'Is x+y−2xy>2x + y - 2\sqrt{xy} > 2?', which is equivalent to 'Is (x−y)2>2(\sqrt{x} - \sqrt{y})^2 > 2?'
Since xx and yy are positive real numbers, x+y−2xyx + y - 2\sqrt{xy} can be rewritten as the perfect square (x−y)2(\sqrt{x} - \sqrt{y})^2.
2
Evaluate Statement (1) using the rephrased target.
From (x+y)2=9(\sqrt{x} + \sqrt{y})^2 = 9 and xy=4xy = 4, we have xy=2\sqrt{xy} = 2. Using the algebraic identity (x−y)2=(x+y)2−4xy(\sqrt{x} - \sqrt{y})^2 = (\sqrt{x} + \sqrt{y})^2 - 4\sqrt{xy}, we find (x−y)2=9−4(2)=1(\sqrt{x} - \sqrt{y})^2 = 9 - 4(2) = 1. Asking 'Is 1>21 > 2?' yields a definitive 'No'.
A statement that yields a definitive 'No' to a Yes/No question is sufficient.
3
Evaluate Statement (2) using the rephrased target.
From x+y=8x + y = 8 and xy=9xy = 9, we have xy=3\sqrt{xy} = 3. Expanding (x−y)2=x+y−2xy(\sqrt{x} - \sqrt{y})^2 = x + y - 2\sqrt{xy} gives 8−2(3)=28 - 2(3) = 2. Asking 'Is 2>22 > 2?' yields a definitive 'No'.
Since 2 is not strictly greater than 2, Statement (2) provides a definitive 'No' and is therefore sufficient.
4
Conclude the final sufficiency decision.
Because each statement independently provides a definitive 'No' answer, each statement alone is sufficient.
Both statements satisfy the Data Sufficiency requirements independently.

Key Concept

Question Stem Simplification and Target Rephrasing
Estimated Time:2m 0s
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