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Data Sufficiency

263 questions

All types263Multiple choice225True / False38
Question 121Question →

A boutique fitness studio offers only yoga and pilates classes. What is the ratio of the number of yoga classes to the number of pilates classes offered each week?

(1) Yoga classes account for 60% of the total number of classes offered each week.
(2) The total number of classes offered each week is 40.

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 correct answer identifies that Statement (1) alone is sufficient because knowing that yoga classes account for 60% of total classes implies pilates classes account for 40%, giving a fixed ratio of 60 to 40 (or 3 to 2). Statement (2) only provides the total number of classes without any relative distribution, making it insufficient on its own.

Step-by-Step Solution

1
Rephrase the target question
We need to find the ratio Y:PY : P, where YY is the number of yoga classes and PP is the number of pilates classes. Note that total classes T=Y+PT = Y + P.
Simplifying the target helps determine whether percentage data or numerical data is needed.
2
Evaluate Statement (1)
If yoga classes are 60% of the total TT, then Y=0.60TY = 0.60T. Since the studio offers only yoga and pilates classes, pilates classes account for the remaining 40%, so P=0.40TP = 0.40T. The ratio Y:P=0.60T0.40T=6040=32Y : P = \frac{0.60T}{0.40T} = \frac{60}{40} = \frac{3}{2}.
Statement (1) yields a unique numerical ratio of 3:23:2. Thus, Statement (1) alone is SUFFICIENT.
3
Evaluate Statement (2)
Statement (2) gives T=40T = 40. We have Y+P=40Y + P = 40, but without knowing the proportion or count of either class type, Y:PY : P can take multiple values (e.g., 20:20=1:120:20 = 1:1 or 30:10=3:130:10 = 3:1).
Statement (2) alone does not give a unique ratio. Thus, Statement (2) alone is INSUFFICIENT.

Key Concept

Data Sufficiency with Ratios and Percentages
Estimated Time:1m 0s
Question 122Question →

If mm is a real number, is ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5?

(1) ∣m−2∣<3|m - 2| < 3
(2) ∣m+1∣<4|m + 1| < 4

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 question stem requires solving ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5. This breaks down into −5<m2−2m−3<5-5 < m^2 - 2m - 3 < 5. The left inequality m2−2m+2>0m^2 - 2m + 2 > 0 holds for all real numbers because its discriminant is negative and the quadratic opens upward. The right inequality m2−2m−8<0m^2 - 2m - 8 < 0 factors as (m−4)(m+2)<0(m - 4)(m + 2) < 0, giving the target range −2<m<4-2 < m < 4.

Evaluating the first statement gives ∣m−2∣<3  ⟹  −1<m<5|m - 2| < 3 \implies -1 < m < 5. This interval extends past 4 (e.g., m=4.5m = 4.5 yields a 'No', while m=1m = 1 yields a 'Yes'), so the first statement alone is not sufficient.

Evaluating the second statement gives ∣m+1∣<4  ⟹  −5<m<3|m + 1| < 4 \implies -5 < m < 3. This interval extends past −2-2 (e.g., m=−4m = -4 yields a 'No', while m=1m = 1 yields a 'Yes'), so the second statement alone is not sufficient.

Combining both statements requires taking the intersection of −1<m<5-1 < m < 5 and −5<m<3-5 < m < 3, which gives −1<m<3-1 < m < 3. Since every number in the range (−1,3)(-1, 3) lies strictly between −2-2 and 44, the target condition is guaranteed to be true. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5 is equivalent to −5<m2−2m−3<5-5 < m^2 - 2m - 3 < 5. Solving the upper bound gives m2−2m−8<0  ⟹  (m−4)(m+2)<0  ⟹  −2<m<4m^2 - 2m - 8 < 0 \implies (m - 4)(m + 2) < 0 \implies -2 < m < 4. The lower bound m2−2m−3>−5  ⟹  m2−2m+2>0  ⟹  (m−1)2+1>0m^2 - 2m - 3 > -5 \implies m^2 - 2m + 2 > 0 \implies (m - 1)^2 + 1 > 0 is true for all real mm. Thus, the question simplifies to: 'Is −2<m<4-2 < m < 4?'
Simplifying the target stem establishes the exact numerical interval required to yield a definitive 'Yes' or 'No' response.
2
Evaluate Statement (1) independently.
Statement (1) states ∣m−2∣<3  ⟹  −3<m−2<3  ⟹  −1<m<5|m - 2| < 3 \implies -3 < m - 2 < 3 \implies -1 < m < 5. If m=1m = 1, −2<1<4-2 < 1 < 4 (Yes). However, if m=4.5m = 4.5, mm is not in (−2,4)(-2, 4) (No). Since Statement (1) allows both 'Yes' and 'No' outcomes, it is NOT sufficient.
A statement is sufficient only if every value in its allowed range produces the same answer to the rephrased question.
3
Evaluate Statement (2) independently.
Statement (2) states ∣m+1∣<4  ⟹  −4<m+1<4  ⟹  −5<m<3|m + 1| < 4 \implies -4 < m + 1 < 4 \implies -5 < m < 3. If m=1m = 1, −2<1<4-2 < 1 < 4 (Yes). However, if m=−4m = -4, mm is not in (−2,4)(-2, 4) (No). Thus, Statement (2) is NOT sufficient.
Checking boundary values reveals that Statement (2) allows values outside the required interval.
4
Evaluate Statement (1) and Statement (2) together.
Combining Statement (1) range (−1<m<5)(-1 < m < 5) and Statement (2) range (−5<m<3)(-5 < m < 3) requires taking their intersection: −1<m<3-1 < m < 3. Since every value in (−1,3)(-1, 3) satisfies −2<m<4-2 < m < 4, the answer to the question is a definitive 'Yes'.
The intersection of the two ranges falls strictly inside the target range, guaranteeing sufficiency.

Key Concept

Data Sufficiency range evaluation for quadratic absolute value inequalities
Estimated Time:2m 0s
Question 123Question →

If xx and yy are real numbers, what is the value of x+yx + y?

(1) x2y+xy2=30x^2 y + xy^2 = 30
(2) x2+y2=19−xyx^2 + y^2 = 19 - xy

Show answer & explanation

Answer: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statements (1) and (2) TOGETHER are NOT sufficient to determine a unique value for x+yx + y.
Evaluating both statements together produces the polynomial relation (x+y)3−19(x+y)−30=0(x + y)^3 - 19(x + y) - 30 = 0. Factoring gives (x+y−5)(x+y+2)(x+y+3)=0(x + y - 5)(x + y + 2)(x + y + 3) = 0. Each of the three roots (55, −2-2, −3-3) produces real numbers xx and yy satisfying both original equations. Because x+yx + y can take three different values, the two statements together are not sufficient to determine a unique value.

Step-by-Step Solution

1
Analyze Statement (1) independently
Factor Statement (1): x2y+xy2=xy(x+y)=30x^2 y + xy^2 = xy(x + y) = 30.
If (x,y)=(2,3)(x, y) = (2, 3), then xy(x+y)=6(5)=30xy(x + y) = 6(5) = 30, giving x+y=5x + y = 5. If (x,y)=(1,5)(x, y) = (1, 5), then xy(x+y)=5(6)=30xy(x + y) = 5(6) = 30, giving x+y=6x + y = 6. Multiple values exist, so Statement (1) alone is insufficient.
2
Analyze Statement (2) independently
Rearrange Statement (2): x2+y2+xy=19x^2 + y^2 + xy = 19, which equals (x+y)2−xy=19(x + y)^2 - xy = 19.
If (x,y)=(3,2)(x, y) = (3, 2), then 32+22+6=193^2 + 2^2 + 6 = 19, giving x+y=5x + y = 5. If (x,y)=(−5,3)(x, y) = (-5, 3), then (−5)2+32+(−15)=19(-5)^2 + 3^2 + (-15) = 19, giving x+y=−2x + y = -2. Multiple values exist, so Statement (2) alone is insufficient.
3
Evaluate Statements (1) and (2) together
Express xyxy in terms of S=x+yS = x + y: from Statement (1), xy=30Sxy = \frac{30}{S}; from Statement (2), xy=S2−19xy = S^2 - 19.
Equating the two expressions for xyxy yields 30S=S2−19  ⟹  S3−19S−30=0\frac{30}{S} = S^2 - 19 \implies S^3 - 19S - 30 = 0.
4
Solve the cubic polynomial for S=x+yS = x + y and check for real roots (x,y)(x, y)
S3−19S−30=(S−5)(S+2)(S+3)=0S^3 - 19S - 30 = (S - 5)(S + 2)(S + 3) = 0, giving S=5S = 5, S=−2S = -2, or S=−3S = -3.
For S=5S = 5, xy=6xy = 6, giving real solutions {x,y}={2,3}\{x, y\} = \{2, 3\}. For S=−2S = -2, xy=−15xy = -15, giving real solutions {x,y}={−5,3}\{x, y\} = \{-5, 3\}. For S=−3S = -3, xy=−10xy = -10, giving real solutions {x,y}={−5,2}\{x, y\} = \{-5, 2\}. Since three distinct valid real sums exist, the combined statements are not sufficient.

Key Concept

Non-linear systems of equations may yield multiple distinct real solutions. Two equations in two variables do not guarantee a unique solution in Data Sufficiency unless the system is linear or constrained to a single root.
Question 124Question →

A community library cataloged a collection of 120120 historical manuscripts. Each manuscript is written in either Latin, Ancient Greek, or both. How many of the manuscripts are written in both Latin and Ancient Greek?

(1) Exactly 8080 manuscripts are written in Latin.
(2) The number of manuscripts written only in Ancient Greek is twice the number of manuscripts written in both languages.

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.
Neither statement alone provides enough independent relationships to solve for the intersection. Statement (1) determines the count of manuscripts written exclusively in Ancient Greek (120−80=40120 - 80 = 40), while Statement (2) establishes a proportional link between manuscripts exclusively in Ancient Greek and those in both languages (2x2x). Combining both facts produces the linear equation 2x=402x = 40, which uniquely solves to x=20x = 20.

Step-by-Step Solution

1
Rephrase the question stem using overlapping set principles.
Let LL be the set of Latin manuscripts, GG be the set of Ancient Greek manuscripts, and xx be the number of manuscripts in both (L∩GL \cap G). Since every manuscript is in at least one set, Total = (Latin only)+(Greek only)+x=120(\text{Latin only}) + (\text{Greek only}) + x = 120. Also, Total = ∣L∣+(Greek only)=120|L| + (\text{Greek only}) = 120. We need to find the unique value of xx.
Formulating the set relationships establishes the exact algebraic system needed to evaluate sufficiency.
2
Evaluate Statement (1) independently.
Statement (1) gives ∣L∣=80|L| = 80. Substituting into the total formula gives 80+(Greek only)=120  ⟹  Greek only=4080 + (\text{Greek only}) = 120 \implies \text{Greek only} = 40. However, ∣L∣=(Latin only)+x=80|L| = (\text{Latin only}) + x = 80. The overlap xx can range anywhere from 00 to 8080.
Statement (1) does not provide enough information to isolate xx from Latin only.
3
Evaluate Statement (2) independently.
Statement (2) states that Greek only=2x\text{Greek only} = 2x. Substituting into the total equation yields (Latin only)+2x+x=(Latin only)+3x=120(\text{Latin only}) + 2x + x = (\text{Latin only}) + 3x = 120. Multiple integer solutions exist for xx (e.g., if Latin only=90\text{Latin only} = 90, x=10x = 10; if Latin only=60\text{Latin only} = 60, x=20x = 20).
Statement (2) presents one equation with two unknown variables.
4
Evaluate Statements (1) and (2) together.
From Statement (1), Greek only=120−80=40\text{Greek only} = 120 - 80 = 40. From Statement (2), Greek only=2x\text{Greek only} = 2x. Therefore, 2x=40  ⟹  x=202x = 40 \implies x = 20. This gives a single, unique answer.
Combining both statements provides two independent linear equations, uniquely solving for xx.

Key Concept

Overlapping Sets (Two-Group Venn Diagram) Data Sufficiency
Estimated Time:2m 0s
Question 125Question →

If xx and yy are real numbers with x≠0x \neq 0, is x4−y4x3+xy2>x−y\frac{x^4 - y^4}{x^3 + x y^2} > x - y?

(1) y=2xy = 2x
(2) xy>0xy > 0

Show answer & explanation

Answer: Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
Simplifying the numerator (x4−y4)=(x2−y2)(x2+y2)(x^4 - y^4) = (x^2 - y^2)(x^2 + y^2) and denominator x3+xy2=x(x2+y2)x^3 + xy^2 = x(x^2 + y^2) reduces the left side of the inequality to x2−y2x\frac{x^2 - y^2}{x}. Rephrasing the question x2−y2x>x−y\frac{x^2 - y^2}{x} > x - y shows that it is equivalent to asking whether y(x−y)x>0\frac{y(x - y)}{x} > 0.

Under Statement (1), substituting y=2xy = 2x transforms the target into asking 'Is x<0x < 0?'. Because Statement (1) gives no information about the sign of xx, it is insufficient.

Under Statement (2), xy>0xy > 0 means xx and yy share the same sign, but testing values shows both 'Yes' and 'No' are possible (x=2,y=1x=2, y=1 yields Yes, whereas x=1,y=2x=1, y=2 yields No). Thus Statement (2) is insufficient.

Combining both statements gives y=2xy = 2x and xy=2x2>0xy = 2x^2 > 0, which holds true for all non-zero real numbers regardless of sign. Because xx can still be positive or negative, the question 'Is x<0x < 0?' cannot be answered. Hence, both statements together are NOT sufficient, making the option stating that statements (1) and (2) together are not sufficient the correct choice.

Step-by-Step Solution

1
Simplify the algebraic expression on the left-hand side of the target inequality.
x4−y4x3+xy2=(x2−y2)(x2+y2)x(x2+y2)=x2−y2x\frac{x^4 - y^4}{x^3 + xy^2} = \frac{(x^2 - y^2)(x^2 + y^2)}{x(x^2 + y^2)} = \frac{x^2 - y^2}{x} since x2+y2>0x^2 + y^2 > 0 for all non-zero real xx.
Factoring the numerator as a difference of squares and factoring out xx in the denominator allows canceling the non-zero common factor (x2+y2)(x^2 + y^2).
2
Rephrase the target inequality in terms of simplified components.
The target question 'Is x2−y2x>x−y\frac{x^2 - y^2}{x} > x - y?' subtracts to x2−y2−x(x−y)x>0  ⟹  y(x−y)x>0\frac{x^2 - y^2 - x(x - y)}{x} > 0 \implies \frac{y(x - y)}{x} > 0.
Simplifying the target question establishes the precise algebraic condition that must be evaluated by the statements.
3
Evaluate Statement (1): y=2xy = 2x.
Substituting y=2xy = 2x into the simplified target yields 2x(x−2x)x>0  ⟹  −2x2x>0  ⟹  −2x>0  ⟹  x<0\frac{2x(x - 2x)}{x} > 0 \implies \frac{-2x^2}{x} > 0 \implies -2x > 0 \implies x < 0.
Statement (1) reduces the question to 'Is x<0x < 0?'. Without information about the sign of xx, this statement alone is NOT sufficient.
4
Evaluate Statement (2): xy>0xy > 0.
If x=2,y=1x = 2, y = 1, 1(2−1)2=0.5>0\frac{1(2-1)}{2} = 0.5 > 0 (Yes). If x=1,y=2x = 1, y = 2, 2(1−2)1=−2≯0\frac{2(1-2)}{1} = -2 \ngtr 0 (No).
Knowing that xx and yy have the same sign does not determine relative magnitudes, so Statement (2) alone is NOT sufficient.
5
Evaluate Statements (1) and (2) together.
Combining y=2xy = 2x and xy>0xy > 0 gives x(2x)=2x2>0x(2x) = 2x^2 > 0, which is true for all non-zero real xx. If x=1,y=2x = 1, y = 2, the target gives 'No'; if x=−1,y=−2x = -1, y = -2, the target gives 'Yes'.
Combining both statements still does not determine whether x<0x < 0 or x>0x > 0, so both statements together are NOT sufficient.

Key Concept

Question Stem Simplification and Target Rephrasing
Question 126Question →

If xx is a real number, is ∣x−5∣<x+1|x - 5| < x + 1?

(1) x2−6x+8<0x^2 - 6x + 8 < 0
(2) ∣x−3∣<2|x - 3| < 2

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 question target ∣x−5∣<x+1|x - 5| < x + 1 simplifies directly to x>2x > 2. Statement (1) bounds xx to 2<x<42 < x < 4, which guarantees that x>2x > 2 is always true, providing a definitive 'Yes'. Statement (2) bounds xx to 1<x<51 < x < 5, allowing values such as 1.51.5 (where x>2x > 2 is false) and 33 (where x>2x > 2 is true), which is insufficient. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question stem by simplifying the absolute value inequality ∣x−5∣<x+1|x - 5| < x + 1.
The inequality holds if and only if −(x+1)<x−5<x+1-(x + 1) < x - 5 < x + 1. Solving −x−1<x−5-x - 1 < x - 5 yields 2x>42x > 4, or x>2x > 2. Solving x−5<x+1x - 5 < x + 1 yields −5<1-5 < 1, which is always true. Thus, the question rephrases to: 'Is x>2x > 2?'
Simplifying absolute value conditions in the stem reduces complex expressions to a single target range.
2
Evaluate Statement (1): x2−6x+8<0x^2 - 6x + 8 < 0.
Factoring gives (x−2)(x−4)<0(x - 2)(x - 4) < 0, which yields the range 2<x<42 < x < 4. For any value of xx in (2,4)(2, 4), xx is strictly greater than 22. This gives a definitive 'Yes' to the rephrased question.
Since every value satisfying Statement (1) satisfies x>2x > 2, Statement (1) alone is sufficient.
3
Evaluate Statement (2): ∣x−3∣<2|x - 3| < 2.
Unfolding the absolute value gives −2<x−3<2-2 < x - 3 < 2, or 1<x<51 < x < 5. If x=1.5x = 1.5, then x>2x > 2 is false ('No'). If x=3x = 3, then x>2x > 2 is true ('Yes'). Because both 'Yes' and 'No' answers are possible, Statement (2) is not sufficient.
Statement (2) permits values both above and below the threshold of 2.

Key Concept

Question Stem Rephrasing for Absolute Value Inequalities
Estimated Time:2m 0s
Question 127Question →

Among a panel of 150150 medical research trials, each trial evaluates at least one of three experimental drugs: Drug XX, Drug YY, or Drug ZZ. If 8080 trials evaluate Drug XX and 7070 trials evaluate Drug YY, how many trials evaluate Drug ZZ only?

(1) Exactly 2525 trials evaluate both Drug XX and Drug YY.
(2) Exactly 4040 trials evaluate at least two of the three drugs, and no trial evaluates all three drugs.

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 correct choice is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. By rephrasing the question, the number of trials evaluating Drug Z only equals the total number of trials (150150) minus the number of trials evaluating Drug X or Drug Y (∣X∪Y∣|X \cup Y|). Statement (1) gives the intersection ∣X∩Y∣=25|X \cap Y| = 25, which immediately yields ∣X∪Y∣=80+70−25=125|X \cup Y| = 80 + 70 - 25 = 125, so Drug Z only =150−125=25= 150 - 125 = 25. Statement (2) only establishes that the double overlap equals 4040, leaving the specific overlap between X and Y unknown.

Step-by-Step Solution

1
Rephrase the target question using set notation.
Let UU be the total set of trials (∣U∣=150|U| = 150). Since every trial evaluates at least one drug, ∣X∪Y∪Z∣=150|X \cup Y \cup Z| = 150. The number of trials evaluating Drug ZZ only is given by ∣Z only∣=∣X∪Y∪Z∣−∣X∪Y∣=150−∣X∪Y∣|Z \text{ only}| = |X \cup Y \cup Z| - |X \cup Y| = 150 - |X \cup Y|.
Simplifying the target target shows that finding ∣X∪Y∣|X \cup Y| is both necessary and sufficient to answer the question.
2
Evaluate Statement (1) independently.
Statement (1) gives ∣X∩Y∣=25|X \cap Y| = 25. Using the standard principle of inclusion-exclusion for two sets: ∣X∪Y∣=∣X∣+∣Y∣−∣X∩Y∣=80+70−25=125|X \cup Y| = |X| + |Y| - |X \cap Y| = 80 + 70 - 25 = 125. Then ∣Z only∣=150−125=25|Z \text{ only}| = 150 - 125 = 25.
Statement (1) yields a single, unique numerical answer (2525), so Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently.
Let a,b,ca, b, c be the double overlaps ∣X∩Y∣|X \cap Y|, ∣Y∩Z∣|Y \cap Z|, and ∣X∩Z∣|X \cap Z|, and d=∣X∩Y∩Z∣=0d = |X \cap Y \cap Z| = 0. Statement (2) states a+b+c=40a + b + c = 40. Total set formula gives 150=80+70+∣Z∣−40  ⟹  ∣Z∣=40150 = 80 + 70 + |Z| - 40 \implies |Z| = 40. However, ∣Z only∣=∣Z∣−(b+c)=40−(40−a)=a=∣X∩Y∣|Z \text{ only}| = |Z| - (b + c) = 40 - (40 - a) = a = |X \cap Y|, which is unknown.
Since the value of ∣X∩Y∣|X \cap Y| can vary, Statement (2) alone does not yield a unique numerical value and is insufficient.

Key Concept

Overlapping Sets and Data Sufficiency Target Rephrasing
Question 128Question →

In GMAT Data Sufficiency 'Value' decision logic, if a question asks for the numerical value of a target expression f(x)f(x), a statement that yields more than one possible value for the variable xx is automatically insufficient.

Show answer & explanation

Answer: False

Answer

The statement is False.
The assertion is false because sufficiency in 'Value' Data Sufficiency questions is determined by whether the overall target expression resolves to one specific numerical value, not whether each constituent variable is uniquely determined.

Step-by-Step Solution

1
Identify the fundamental requirement for sufficiency in a GMAT Data Sufficiency 'Value' question.
A statement is sufficient if and only if it leads to a single, unique numerical value for the requested target expression.
The focus of a Value question is the target expression as a whole, not the isolated variables contained within it.
2
Test whether multiple possible values for a variable xx can still yield a unique value for target expression f(x)f(x).
Consider f(x)=x2f(x) = x^2 with x=2x = 2 or x=−2x = -2. In both cases, f(x)=4f(x) = 4.
Because all permissible values of xx map to the exact same output for f(x)f(x), the target value is uniquely determined.
3
Evaluate the claim made in the statement.
The claim asserts that having multiple values for xx makes a statement 'automatically insufficient', which is disproven by counterexample.
Since a statement can yield multiple values for xx and still be sufficient for f(x)f(x), the assertion is false.

Key Concept

Value Data Sufficiency Decision Logic & Target Expression Sufficiency
Question 129Question →

A corporate cafeteria serves only two types of lunches: hot meals and cold salads. On Tuesday, what was the ratio of the number of hot meals served to the number of cold salads served?

(1) The total number of lunches served on Tuesday was 200.
(2) On Tuesday, the number of hot meals served was 60% of the total number of lunches served.

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.
The correct choice is the option stating Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Statement (1) gives only the total volume of lunches served (200), which allows for multiple possible combinations of hot meals and cold salads, resulting in different ratios. Statement (2) establishes that hot meals comprise 60% of the total, meaning cold salads comprise the remaining 40%. The ratio of hot meals to cold salads is therefore fixed at 60% to 40%, which simplifies to 3:2. Thus, Statement (2) alone provides sufficient information to answer the question uniquely.

Step-by-Step Solution

1
Define variables and target expression
Let HH be the number of hot meals and CC be the number of cold salads. Total lunches T=H+CT = H + C. Target is to find the ratio HC\frac{H}{C}.
Rephrasing the question stem algebraically clarifies what information is needed.
2
Evaluate Statement (1)
Statement (1) states T=H+C=200T = H + C = 200. Without knowing the proportion of either HH or CC, HC\frac{H}{C} could be 19010=19\frac{190}{10} = 19 or 100100=1\frac{100}{100} = 1.
A single total count does not fix the internal ratio of the two components.
3
Evaluate Statement (2)
Statement (2) states H=0.60TH = 0.60T. Since T=H+CT = H + C, then C=T−0.60T=0.40TC = T - 0.60T = 0.40T. The ratio HC=0.60T0.40T=0.600.40=32\frac{H}{C} = \frac{0.60T}{0.40T} = \frac{0.60}{0.40} = \frac{3}{2}.
Statement (2) yields a single, definitive numerical value for the ratio.

Key Concept

Data Sufficiency logic for percentage components and ratio determination
Question 130Question →

In a technology firm of 100100 software engineers, each engineer works on at least one of two projects: Project Alpha or Project Beta. If 7070 engineers work on Project Alpha, what is the average (arithmetic mean) monthly salary of all 100100 engineers at the firm?

(1) The average monthly salary of the engineers who work ONLY on Project Alpha is $6,000\$6,000, and the average monthly salary of all engineers who work on Project Beta is $10,800\$10,800.
(2) The average monthly salary of the engineers who work ONLY on Project Beta is $12,000\$12,000, and the average monthly salary of the engineers who work on BOTH Project Alpha and Project Beta is $9,000\$9,000.

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.
The option stating that BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient is correct. Evaluating either statement independently leaves the overlap count or salary undefined. However, combining both statements allows equating two expressions for the total salary of Project Beta, which uniquely yields 2020 engineers working on both projects, leading to a single deterministic overall average salary of $8,400\$8,400.

Step-by-Step Solution

1
Define variables from the question stem.
Let n1n_1 be the number of engineers working ONLY on Project Alpha, n2n_2 be the number working ONLY on Project Beta, and n3n_3 be the number working on BOTH. Total engineers = n1+n2+n3=100n_1 + n_2 + n_3 = 100. Given 7070 work on Alpha (n1+n3=70n_1 + n_3 = 70), we find n2=100−70=30n_2 = 100 - 70 = 30.
Establishing exact set counts reduces unknowns.
2
Evaluate Statement (1) alone.
Total salary = (n1×6000)+((n2+n3)×10800)=(70−n3)(6000)+(30+n3)(10800)=744,000+4800n3(n_1 \times 6000) + ((n_2 + n_3) \times 10800) = (70 - n_3)(6000) + (30 + n_3)(10800) = 744,000 + 4800 n_3. Since n3n_3 is unknown, overall average cannot be determined.
Statement (1) depends on the unknown overlap n3n_3, so Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone.
We know the average salary for n2=30n_2 = 30 is $12,000\$12,000 and for n3n_3 is $9,000\$9,000, but we have no salary information for the n1n_1 engineers working ONLY on Project Alpha.
Without salary information for Project Alpha only, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
From Statement (2), total salary of Project Beta = (30×12000)+(n3×9000)=360,000+9000n3(30 \times 12000) + (n_3 \times 9000) = 360,000 + 9000 n_3. From Statement (1), total salary of Project Beta = (30+n3)×10800=324,000+10800n3(30 + n_3) \times 10800 = 324,000 + 10800 n_3. Equating both: 360,000+9000n3=324,000+10800n3  ⟹  1800n3=36,000  ⟹  n3=20360,000 + 9000 n_3 = 324,000 + 10800 n_3 \implies 1800 n_3 = 36,000 \implies n_3 = 20.
Finding n3=20n_3 = 20 gives n1=50n_1 = 50, allowing exact calculation of the overall average salary (8,4008,400).

Key Concept

Weighted Averages and Overlapping Sets in Data Sufficiency
Question 131Question →

If mm and nn are non-zero real numbers such that m2≠n2m^2 \neq n^2, is m3−m2n+mn2−n3m2−n2<0\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2} < 0?

(1) m+n=−5m + n = -5
(2) m−n=3m - n = 3

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.
Simplifying the target expression by factoring gives (m−n)(m2+n2)(m−n)(m+n)=m2+n2m+n\frac{(m - n)(m^2 + n^2)}{(m - n)(m + n)} = \frac{m^2 + n^2}{m + n}. Because m2+n2>0m^2 + n^2 > 0 for all non-zero real numbers, the quotient is negative if and only if m+n<0m + n < 0. Statement (1) gives m+n=−5<0m + n = -5 < 0, providing a definitive 'Yes' answer. Statement (2) gives m−n=3m - n = 3, which allows m+nm + n to be either positive or negative depending on the specific values of mm and nn. Thus, Statement (1) alone is sufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically
The target expression m3−m2n+mn2−n3m2−n2\frac{m^3 - m^2 n + m n^2 - n^3}{m^2 - n^2} simplifies to m2+n2m+n\frac{m^2 + n^2}{m + n}.
Factor the numerator by grouping: m2(m−n)+n2(m−n)=(m−n)(m2+n2)m^2(m - n) + n^2(m - n) = (m - n)(m^2 + n^2). Factor the denominator: m2−n2=(m−n)(m+n)m^2 - n^2 = (m - n)(m + n). Since m2≠n2m^2 \neq n^2, m−n≠0m - n \neq 0, so cancel (m−n)(m - n).
2
Determine the condition for the simplified expression to be negative
The rephrased target question is 'Is m+n<0m + n < 0?'
Since mm and nn are non-zero real numbers, m2+n2>0m^2 + n^2 > 0 always. Therefore, the sign of m2+n2m+n\frac{m^2 + n^2}{m + n} depends solely on the denominator m+nm + n.
3
Evaluate Statement (1): m+n=−5m + n = -5
Statement (1) is SUFFICIENT.
Statement (1) directly tells us m+n=−5m + n = -5, which is less than 00. This yields a definitive 'Yes' to the rephrased question 'Is m+n<0m + n < 0?'.
4
Evaluate Statement (2): m−n=3m - n = 3
Statement (2) is NOT SUFFICIENT.
Knowing m−n=3m - n = 3 gives m=n+3m = n + 3, so m+n=2n+3m + n = 2n + 3. If n=0.5n = 0.5, m+n=4>0m + n = 4 > 0 ('No'). If n=−5n = -5, m+n=−7<0m + n = -7 < 0 ('Yes'). Since m+nm + n can be positive or negative, Statement (2) is insufficient.

Key Concept

Question Stem Simplification in Data Sufficiency
Estimated Time:2m 0s
Question 132Question →

If uu and vv are real numbers such that u+v≠0u + v \neq 0, what is the value of u−vu+v\frac{u - v}{u + v}?

(1) u2−3uv+2v2=0u^2 - 3uv + 2v^2 = 0
(2) 2u2−5uv+2v2=02u^2 - 5uv + 2v^2 = 0

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.
The correct answer is the option stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) yields two possible ratios (u=vu=v giving 0, or u=2vu=2v giving 1/31/3), so it is not sufficient. Statement (2) also yields two possible ratios (u=v/2u=v/2 giving −1/3-1/3, or u=2vu=2v giving 1/31/3), so it is not sufficient. When combined, u=vu=v and u=v/2u=v/2 would require u=v=0u=v=0, which is forbidden by the condition u+v≠0u+v \neq 0. Therefore, u=2vu=2v is the only valid relation, producing a unique value of 1/31/3.

Step-by-Step Solution

1
Rephrase the target question stem.
Dividing the numerator and denominator of u−vu+v\frac{u - v}{u + v} by vv (assuming v≠0v \neq 0) shows that knowing the ratio uv\frac{u}{v} uniquely determines the value of the expression.
Simplifying the target expression reduces the problem to determining whether a unique ratio uv\frac{u}{v} exists.
2
Evaluate Statement (1) independently.
Factor u2−3uv+2v2=0u^2 - 3uv + 2v^2 = 0 as (u−v)(u−2v)=0(u - v)(u - 2v) = 0. This implies u=vu = v or u=2vu = 2v. If u=vu = v, then u−vu+v=02v=0\frac{u - v}{u + v} = \frac{0}{2v} = 0. If u=2vu = 2v, then u−vu+v=2v−v2v+v=v3v=13\frac{u - v}{u + v} = \frac{2v - v}{2v + v} = \frac{v}{3v} = \frac{1}{3}.
Since Statement (1) allows two distinct numerical values (00 and 13\frac{1}{3}), Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Factor 2u2−5uv+2v2=02u^2 - 5uv + 2v^2 = 0 as (2u−v)(u−2v)=0(2u - v)(u - 2v) = 0. This implies u=12vu = \frac{1}{2}v or u=2vu = 2v. If u=12vu = \frac{1}{2}v, then u−vu+v=12v−v12v+v=−12v32v=−13\frac{u - v}{u + v} = \frac{\frac{1}{2}v - v}{\frac{1}{2}v + v} = \frac{-\frac{1}{2}v}{\frac{3}{2}v} = -\frac{1}{3}. If u=2vu = 2v, then u−vu+v=13\frac{u - v}{u + v} = \frac{1}{3}.
Since Statement (2) allows two distinct numerical values (−13-\frac{1}{3} and 13\frac{1}{3}), Statement (2) alone is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) combined.
Combine the conditions: Statement (1) requires u=vu = v or u=2vu = 2v; Statement (2) requires u=12vu = \frac{1}{2}v or u=2vu = 2v. Testing u=vu = v in Statement (2) yields 2v2−5v2+2v2=−v2=0  ⟹  v=02v^2 - 5v^2 + 2v^2 = -v^2 = 0 \implies v = 0, which implies u=0u = 0, but u+v≠0u + v \neq 0 rules out u=v=0u = v = 0. Testing u=12vu = \frac{1}{2}v in Statement (1) similarly leads to u=v=0u = v = 0. Thus, the only non-zero solution satisfying both equations is u=2vu = 2v, which gives a unique value of 13\frac{1}{3}.
Combining both statements eliminates the ambiguous cases that violate the stem constraint, establishing a unique value.

Key Concept

Homogeneous Quadratic Factoring and Constraint Evaluation in Data Sufficiency
Question 133Question →

If uu and vv are real numbers such that u≠vu \neq v, what is the value of u+vu−v\frac{u + v}{u - v}?

(1) u2−v2=4(u−v)2u^2 - v^2 = 4(u - v)^2
(2) u2+v2=5uvu^2 + v^2 = 5uv

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) allows factoring the difference of squares into (u−v)(u+v)=4(u−v)2(u - v)(u + v) = 4(u - v)^2. Because u≠vu \neq v, we can divide by (u−v)(u - v) to determine directly that u+vu−v=4\frac{u + v}{u - v} = 4, which provides a unique value. Statement (2) gives (u+vu−v)2=73\left(\frac{u + v}{u - v}\right)^2 = \frac{7}{3}, which yields two possible values (±73\pm\sqrt{\frac{7}{3}}) and is therefore not sufficient. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

1
Rephrase the target question and examine constraints.
The target expression is u+vu−v\frac{u + v}{u - v}. The stem specifies u≠vu \neq v, so u−v≠0u - v \neq 0, making the fraction well-defined.
Establishing non-zero denominator conditions allows division by (u−v)(u - v) when evaluating statements.
2
Evaluate Statement (1) independently.
Rewrite the left side of u2−v2=4(u−v)2u^2 - v^2 = 4(u - v)^2 as (u−v)(u+v)=4(u−v)2(u - v)(u + v) = 4(u - v)^2. Since u−v≠0u - v \neq 0, divide both sides by (u−v)(u - v) to get u+v=4(u−v)u + v = 4(u - v). Divide both sides by (u−v)(u - v) again to obtain u+vu−v=4\frac{u + v}{u - v} = 4.
This yields a single, definitive numerical value for the target expression. Statement (1) is sufficient.
3
Evaluate Statement (2) independently.
Expand (u+v)2=u2+v2+2uv(u + v)^2 = u^2 + v^2 + 2uv and (u−v)2=u2+v2−2uv(u - v)^2 = u^2 + v^2 - 2uv. Substituting u2+v2=5uvu^2 + v^2 = 5uv gives (u+v)2=7uv(u + v)^2 = 7uv and (u−v)2=3uv(u - v)^2 = 3uv. Taking the ratio gives (u+vu−v)2=7uv3uv=73\left(\frac{u + v}{u - v}\right)^2 = \frac{7uv}{3uv} = \frac{7}{3} (since uv≠0uv \neq 0). Taking the square root yields u+vu−v=73\frac{u + v}{u - v} = \sqrt{\frac{7}{3}} or −73-\sqrt{\frac{7}{3}}.
Because two distinct numerical values are possible, Statement (2) alone does not yield a unique answer. Statement (2) is not sufficient.

Key Concept

Factoring difference of squares and avoiding degree miscounts in quadratic ratios
Question 134Question →

If kk is a real number such that k≠−1k \neq -1, is ∣k−3∣k+1<1\frac{|k - 3|}{k + 1} < 1?

(1) ∣k∣>1|k| > 1
(2) k2−2k−3>0k^2 - 2k - 3 > 0

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.
Rephrasing the question stem shows that ∣k−3∣k+1<1\frac{|k - 3|}{k + 1} < 1 holds whenever k<−1k < -1 or k>1k > 1. Statement (1) gives k<−1k < -1 or k>1k > 1, yielding a definitive YES. Statement (2) gives k<−1k < -1 or k>3k > 3, which is a subset of k<−1k < -1 or k>1k > 1, also yielding a definitive YES. Thus, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target inequality
The target inequality ∣k−3∣k+1<1\frac{|k - 3|}{k + 1} < 1 is satisfied if and only if k<−1k < -1 or k>1k > 1.
Analyze by cases depending on the sign of the denominator k+1k + 1.
- Case 1: k+1<0  ⟹  k<−1k + 1 < 0 \implies k < -1. The denominator is negative and the numerator ∣k−3∣≥0|k - 3| \ge 0, so ∣k−3∣k+1≤0<1\frac{|k - 3|}{k + 1} \le 0 < 1 is always true for all k<−1k < -1.
- Case 2: k+1>0  ⟹  k>−1k + 1 > 0 \implies k > -1. Multiplying by k+1>0k + 1 > 0 yields ∣k−3∣<k+1|k - 3| < k + 1, which expands to −(k+1)<k−3<k+1-(k + 1) < k - 3 < k + 1. The left boundary −k−1<k−3-k - 1 < k - 3 reduces to 2k>2  ⟹  k>12k > 2 \implies k > 1. The right boundary k−3<k+1  ⟹  −3<1k - 3 < k + 1 \implies -3 < 1 is universally true. Thus for k>−1k > -1, the condition holds when k>1k > 1.
Combining both cases, the inequality holds whenever k<−1k < -1 or k>1k > 1.
2
Evaluate Statement (1)
Statement (1) alone is SUFFICIENT.
Statement (1) states ∣k∣>1|k| > 1, which unwraps to k<−1k < -1 or k>1k > 1. This matches the target condition exactly. For every value of kk satisfying this statement, the answer to the stem question is a definitive YES.
3
Evaluate Statement (2)
Statement (2) alone is SUFFICIENT.
Statement (2) states k2−2k−3>0k^2 - 2k - 3 > 0, which factors as (k−3)(k+1)>0(k - 3)(k + 1) > 0. Solving gives k<−1k < -1 or k>3k > 3.
- If k<−1k < -1, it falls into the left target region k<−1k < -1.
- If k>3k > 3, it falls into the right target region k>1k > 1.
In all cases allowed by Statement (2), the answer to the stem question is a definitive YES.

Key Concept

Inequalities with absolute values and variables in the denominator require case analysis based on the denominator's sign.
Question 135Question →

An investor purchased a total of NN shares split between Stock XX and Stock YY. What percentage of the total number of shares purchased were shares of Stock XX?

(1) The number of shares of Stock XX purchased was 40%40\% of the number of shares of Stock YY purchased.
(2) The investor purchased 120120 shares of Stock YY.

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) establishes a direct proportional relationship between the number of shares of Stock X and Stock Y (X=0.40YX = 0.40Y). Substituting this into the part-to-whole expression XX+Y\frac{X}{X+Y} yields 0.40Y1.40Y=27\frac{0.40Y}{1.40Y} = \frac{2}{7}, which provides a unique percentage without requiring the actual number of shares. Statement (2) provides only the absolute quantity of Stock Y (120120), leaving the quantity of Stock X completely unknown and preventing the calculation of the percentage. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the target question algebraically.
The target is to find XX+Y×100%\frac{X}{X + Y} \times 100\%.
Determining the ratio XY\frac{X}{Y} is sufficient to find XX+Y\frac{X}{X + Y}.
2
Evaluate Statement (1).
Statement (1) gives X=0.40YX = 0.40Y, which means XY=25\frac{X}{Y} = \frac{2}{5}. Therefore, XX+Y=0.40Y1.40Y=27\frac{X}{X + Y} = \frac{0.40Y}{1.40Y} = \frac{2}{7}, giving a unique percentage of 2007%\frac{200}{7}\%.
Since a single unique value is obtained, Statement (1) is sufficient.
3
Evaluate Statement (2).
Statement (2) gives Y=120Y = 120, but gives no information regarding XX.
Without knowing XX, the total shares X+YX + Y and the fraction XX+Y\frac{X}{X + Y} cannot be determined. Thus, Statement (2) is insufficient.

Key Concept

Part-to-whole percentages can be determined solely from part-to-part ratios without knowing absolute quantities.
Estimated Time:45s
Question 136Question →

If xx and yy are positive real numbers, is xx an integer?

(1) x+yx + y is an integer and xy=12xy = 12.
(2) x−yx - y is an integer and x2+y2=25x^2 + y^2 = 25.

Show answer & explanation

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

Answer

Both statements together are sufficient to determine that xx must be an integer, but neither statement alone is sufficient.
Evaluating both statements together establishes that (x+y)2=x2+y2+2xy=25+2(12)=49(x+y)^2 = x^2 + y^2 + 2xy = 25 + 2(12) = 49, meaning x+y=7x+y=7. The system x+y=7x+y=7 and xy=12xy=12 yields solutions x=3,y=4x=3, y=4 or x=4,y=3x=4, y=3. In both outcomes, xx is guaranteed to be an integer, yielding a definitive 'Yes' answer.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
Statement (1) gives x+y=kx + y = k for some integer kk, and xy=12xy = 12. If k=7k = 7, the quadratic equation t2−7t+12=0t^2 - 7t + 12 = 0 gives roots t=3t = 3 and t=4t = 4, which are integers. However, if k=9k = 9, the quadratic equation t2−9t+12=0t^2 - 9t + 12 = 0 gives roots t=9±332t = \frac{9 \pm \sqrt{33}}{2}. Here x=9+332x = \frac{9 + \sqrt{33}}{2} is a positive real number, y=9−332>0y = \frac{9 - \sqrt{33}}{2} > 0, x+y=9x+y = 9 (integer), and xy=12xy = 12, but xx is NOT an integer. Thus, xx can be an integer or a non-integer.
Demonstrate that positive real numbers whose sum and product are integers do not necessarily have to be integers themselves.
2
Evaluate Statement (2) independently.
Statement (2) gives x−y=mx - y = m for some integer mm, and x2+y2=25x^2 + y^2 = 25. If m=1m = 1, we can have x=4x = 4 and y=3y = 3, so xx is an integer. If m=2m = 2, then (x−y)2=x2−2xy+y2  ⟹  4=25−2xy  ⟹  2xy=21(x-y)^2 = x^2 - 2xy + y^2 \implies 4 = 25 - 2xy \implies 2xy = 21. Then (x+y)2=x2+y2+2xy=25+21=46  ⟹  x+y=46(x+y)^2 = x^2 + y^2 + 2xy = 25 + 21 = 46 \implies x+y = \sqrt{46}. Thus x=46+22x = \frac{\sqrt{46} + 2}{2}, which is positive and real, but NOT an integer. Thus, Statement (2) alone is not sufficient.
Test whether integer difference and fixed sum of squares guarantees integer values.
3
Evaluate Statements (1) and (2) together.
From Statement (1), xy=12xy = 12 and x+y=kx + y = k (where kk is an integer). From Statement (2), x2+y2=25x^2 + y^2 = 25. Using the algebraic identity x2+y2=(x+y)2−2xyx^2 + y^2 = (x+y)^2 - 2xy, we substitute the known values: 25=k2−2(12)  ⟹  25=k2−24  ⟹  k2=4925 = k^2 - 2(12) \implies 25 = k^2 - 24 \implies k^2 = 49. Since x>0x > 0 and y>0y > 0, x+y=k>0x + y = k > 0, so k=7k = 7. Now we have x+y=7x + y = 7 and xy=12xy = 12. The possible values for xx and yy are the solutions to t2−7t+12=0  ⟹  (t−3)(t−4)=0  ⟹  t∈{3,4}t^2 - 7t + 12 = 0 \implies (t-3)(t-4) = 0 \implies t \in \{3, 4\}. Therefore, xx must be either 3 or 4. In both cases, xx is definitively an integer.
Combine the conditions to solve for x+yx+y and prove xx must be an integer.

Key Concept

Integer constraints vs. real number assumptions in Data Sufficiency algebraic systems
Question 137Question →

For all positive real numbers pp and qq, the Data Sufficiency Yes/No target question "Is p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1?" is algebraically equivalent to the simplified target question "Is p≠qp \neq q?"

Show answer & explanation

Answer: True

Answer

True
The statement is true because clearing the positive denominator 2pq2pq and rearranging terms converts the target inequality p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1 into (p−q)2>0(p - q)^2 > 0. A real squared quantity is strictly positive if and only if its base is non-zero, making the target equivalent to "Is p≠qp \neq q?"

Step-by-Step Solution

1
Multiply both sides of the target inequality p2+q22pq>1\frac{p^2 + q^2}{2pq} > 1 by 2pq2pq.
p2+q2>2pqp^2 + q^2 > 2pq, with the direction of the inequality preserved since p>0p > 0 and q>0q > 0 implies 2pq>02pq > 0.
Clearing positive denominators simplifies fractional inequalities without changing the inequality sign.
2
Subtract 2pq2pq from both sides of the inequality.
p2−2pq+q2>0p^2 - 2pq + q^2 > 0.
Grouping all non-zero terms on one side sets up a recognizable quadratic form.
3
Factor the quadratic expression as a perfect square.
(p−q)2>0(p - q)^2 > 0.
Recognizing p2−2pq+q2p^2 - 2pq + q^2 as the expansion of (p−q)2(p - q)^2 isolates the core variable relationship.
4
Analyze the conditions required for (p−q)2>0(p - q)^2 > 0 to hold for real numbers.
The square of any real number is always non-negative, and it is strictly greater than zero if and only if the base is non-zero (p−q≠0p - q \neq 0, which means p≠qp \neq q).
Simplifying the logical condition reveals the exact target question needed for Data Sufficiency evaluation.

Key Concept

Target Rephrasing via Perfect Square Inequalities
Question 138Question →

If xx and yy are real numbers, what is the value of x+yx + y?

(1) x2+2xy+y2−5x−5y+6=0x^2 + 2xy + y^2 - 5x - 5y + 6 = 0
(2) xy=2xy = 2

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.
The correct response identifies that neither statement alone establishes a single value for x+yx + y, but combining them eliminates x+y=2x + y = 2 because no real numbers xx and yy have a sum of 22 and a product of 22 (discriminant 22−4(2)=−4<02^2 - 4(2) = -4 < 0). Only x+y=3x + y = 3 yields real values for xx and yy (discriminant 32−4(2)=1≥03^2 - 4(2) = 1 \ge 0). Thus, both statements together are sufficient to determine that x+y=3x + y = 3.

Step-by-Step Solution

1
Analyze Statement (1) by grouping terms in terms of (x+y)(x + y).
Rewrite x2+2xy+y2−5(x+y)+6=0x^2 + 2xy + y^2 - 5(x + y) + 6 = 0 as (x+y)2−5(x+y)+6=0(x + y)^2 - 5(x + y) + 6 = 0, which factors into ((x+y)−2)((x+y)−3)=0((x + y) - 2)((x + y) - 3) = 0. Hence, x+y=2x + y = 2 or x+y=3x + y = 3. Since there are two possible values, Statement (1) alone is NOT sufficient.
Determining whether Statement (1) provides a unique value for the target expression.
2
Analyze Statement (2) independently.
Statement (2) gives xy=2xy = 2. Multiple real pairs satisfy this (e.g., (2,1)⇒x+y=3(2, 1) \Rightarrow x + y = 3 or (1,2)⇒x+y=3(1, 2) \Rightarrow x + y = 3 vs. (2,2)⇒x+y=22(\sqrt{2}, \sqrt{2}) \Rightarrow x + y = 2\sqrt{2}). Thus, Statement (2) alone is NOT sufficient.
Testing whether xy=2xy = 2 alone restricts x+yx + y to a single numerical value.
3
Combine Statement (1) and Statement (2) under the real number constraint.
Let S=x+yS = x + y and P=xy=2P = xy = 2. For xx and yy to be real numbers, they must be the real roots of t2−St+P=0t^2 - St + P = 0, requiring discriminant Δ=S2−4P≥0\Delta = S^2 - 4P \ge 0.
- Case 1: If S=2S = 2, then Δ=22−4(2)=−4<0\Delta = 2^2 - 4(2) = -4 < 0. No real solutions exist for xx and yy.
- Case 2: If S=3S = 3, then Δ=32−4(2)=1≥0\Delta = 3^2 - 4(2) = 1 \ge 0. Real solutions exist (e.g., x=2,y=1x = 2, y = 1).
Applying the real-number constraint to eliminate algebraically impossible cases.
4
Conclude total sufficiency.
Only x+y=3x + y = 3 produces real values for xx and yy. Therefore, both statements together uniquely determine x+y=3x + y = 3.
Completing the Data Sufficiency evaluation.

Key Concept

Quadratic expression rephrasing and discriminant constraints for real variable existence in Data Sufficiency
Estimated Time:2m 0s
Question 139Question →

A university research institute evaluated NN technology projects completed last year. Each project received grant funding from at least one of two foundations: Foundation XX or Foundation YY. Exactly 30%30\% of the NN projects received funding from both foundations. If the average (arithmetic mean) grant amount per project among projects funded by Foundation XX was $50,000\$50,000 and the average grant amount per project among projects funded by Foundation YY was $60,000\$60,000, what was the average total grant funding per project across all NN projects?

(1) The number of projects that received funding from Foundation XX only was equal to the number of projects that received funding from Foundation YY only.
(2) The total dollar amount disbursed by Foundation YY exceeded the total dollar amount disbursed by Foundation XX by $1,800,000\$1,800,000.

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not sufficient.
Statement (1) alone is sufficient because equating the counts of projects funded exclusively by Foundation X and exclusively by Foundation Y implies that the total number of projects funded by Foundation X equals the total number funded by Foundation Y. Combined with the principle of overlapping sets where thirty percent of projects received funding from both, this uniquely determines that sixty-five percent of projects were funded by Foundation X and sixty-five percent were funded by Foundation Y, yielding a unique overall average grant of $71,500. Statement (2) alone is insufficient because the equation relating total disbursements depends on the unknown total number of projects N.

Step-by-Step Solution

1
Express the total funding and set proportions in terms of the total number of projects NN.
Let nXn_X be the number of projects funded by Foundation XX and nYn_Y be the number of projects funded by Foundation YY. The number of projects funded by both is 0.30N0.30N. Since every project is funded by at least one foundation, N=nX+nY−0.30N  ⟹  nX+nY=1.30NN = n_X + n_Y - 0.30N \implies n_X + n_Y = 1.30N. Dividing by NN gives the ratio sum nXN+nYN=1.30\frac{n_X}{N} + \frac{n_Y}{N} = 1.30.
Establishing the relationship between the subset counts and total projects simplifies the target average formula.
2
Formulate the expression for the average total grant funding per project.
Total Funding =50,000nX+60,000nY= 50,000 n_X + 60,000 n_Y. Therefore, the average funding per project is Average=50,000nX+60,000nYN=50,000(nXN)+60,000(nYN)\text{Average} = \frac{50,000 n_X + 60,000 n_Y}{N} = 50,000\left(\frac{n_X}{N}\right) + 60,000\left(\frac{n_Y}{N}\right).
Since total grant funding across all projects is the sum of all money disbursed by Foundation XX and Foundation YY, the overall average depends strictly on the ratios nXN\frac{n_X}{N} and nYN\frac{n_Y}{N}.
3
Evaluate Statement (1) independently.
Statement (1) states that the number of projects funded by XX only equals the number funded by YY only: nX−0.30N=nY−0.30N  ⟹  nX=nYn_X - 0.30N = n_Y - 0.30N \implies n_X = n_Y. Since nXN+nYN=1.30\frac{n_X}{N} + \frac{n_Y}{N} = 1.30 and nX=nYn_X = n_Y, we get 2(nXN)=1.30  ⟹  nXN=0.652\left(\frac{n_X}{N}\right) = 1.30 \implies \frac{n_X}{N} = 0.65 and nYN=0.65\frac{n_Y}{N} = 0.65. Substituting these into the average formula yields Average=50,000(0.65)+60,000(0.65)=71,500\text{Average} = 50,000(0.65) + 60,000(0.65) = 71,500. Statement (1) ALONE is sufficient.
Knowing that the two single-foundation set sizes are equal determines the exact proportions of NN funded by each foundation.
4
Evaluate Statement (2) independently.
Statement (2) states that 60,000nY−50,000nX=1,800,00060,000 n_Y - 50,000 n_X = 1,800,000. Dividing by NN gives 60,000(nYN)−50,000(nXN)=1,800,000N60,000\left(\frac{n_Y}{N}\right) - 50,000\left(\frac{n_X}{N}\right) = \frac{1,800,000}{N}. Because NN is unknown, the right-hand side is not fixed, so nXN\frac{n_X}{N} and nYN\frac{n_Y}{N} cannot be uniquely determined. Statement (2) ALONE is not sufficient.
An absolute dollar equation introduces a dependency on the total count NN, preventing a unique calculation of the relative proportions.

Key Concept

Overlapping Sets and Weighted Averages in Data Sufficiency
Estimated Time:2m 0s
Question 140Question →

If xx and yy are real numbers, what is the value of x2−y2x^2 - y^2?

(1) x3−y3=26x^3 - y^3 = 26 and x2+xy+y2=13x^2 + xy + y^2 = 13
(2) x+y=4x + y = 4

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.
Factoring the target expression gives x2−y2=(x+y)(x−y)x^2 - y^2 = (x + y)(x - y). Statement (1) simplifies via the difference of cubes formula x3−y3=(x−y)(x2+xy+y2)x^3 - y^3 = (x - y)(x^2 + xy + y^2) to x−y=2x - y = 2 and xy=3xy = 3. However, solving for x+yx + y yields (x+y)2=16(x + y)^2 = 16, which gives two solutions (x+y=4x + y = 4 or x+y=−4x + y = -4) and thus two values for x2−y2x^2 - y^2 (88 or −8-8). Thus, Statement (1) alone is insufficient. Statement (2) gives x+y=4x + y = 4 without constraining x−yx - y, making it insufficient alone. Combining both statements provides x+y=4x + y = 4 and x−y=2x - y = 2, giving a unique product of 88. Hence, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities
x2−y2=(x+y)(x−y)x^2 - y^2 = (x + y)(x - y)
Determining x2−y2x^2 - y^2 requires knowing either the product (x+y)(x−y)(x + y)(x - y) or the individual values of x+yx + y and x−yx - y.
2
Evaluate Statement (1) independently
Using the difference of cubes identity x3−y3=(x−y)(x2+xy+y2)x^3 - y^3 = (x - y)(x^2 + xy + y^2), substitute the given values: 26=(x−y)(13)  ⟹  x−y=226 = (x - y)(13) \implies x - y = 2. Next, expand (x−y)2=x2−2xy+y2=4(x - y)^2 = x^2 - 2xy + y^2 = 4. Subtracting this from x2+xy+y2=13x^2 + xy + y^2 = 13 gives 3xy=9  ⟹  xy=33xy = 9 \implies xy = 3. Then (x+y)2=(x−y)2+4xy=4+4(3)=16(x + y)^2 = (x - y)^2 + 4xy = 4 + 4(3) = 16, which implies x+y=4x + y = 4 or x+y=−4x + y = -4. Thus x2−y2=(4)(2)=8x^2 - y^2 = (4)(2) = 8 or (−4)(2)=−8(-4)(2) = -8.
Because Statement (1) produces two distinct possible values (88 and −8-8), Statement (1) ALONE is NOT sufficient.
3
Evaluate Statement (2) independently
x+y=4x + y = 4
Without any information about x−yx - y, x2−y2=4(x−y)x^2 - y^2 = 4(x - y) can take infinitely many values. Statement (2) ALONE is NOT sufficient.
4
Evaluate Statements (1) and (2) combined
From Statement (1), x−y=2x - y = 2. From Statement (2), x+y=4x + y = 4. Multiplying these equations yields x2−y2=(x+y)(x−y)=(4)(2)=8x^2 - y^2 = (x + y)(x - y) = (4)(2) = 8.
A single, unique numerical value is determined. Therefore, BOTH statements TOGETHER are sufficient.

Key Concept

Algebraic Factoring and Non-Linear System Ambiguity in Data Sufficiency
Estimated Time:2m 0s
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