Examkin

Data Sufficiency

263 soru

Tüm türler263Çoktan seçmeli225Doğru / Yanlış38
Soru 101Soru →

A water reservoir is filled by two pumps, Pump X and Pump Y, each operating continuously at its own constant rate. How many hours does it take for Pump X and Pump Y working together to fill the empty reservoir?

(1) Working alone at its constant rate, Pump X fills the reservoir in 6 hours.
(2) Working alone at its constant rate, Pump Y fills the reservoir in 12 hours.

Cevabı ve açıklamayı göster

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

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct answer is that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) gives only the rate of Pump X, and Statement (2) gives only the rate of Pump Y. Neither statement alone allows calculation of the combined rate. When used together, the combined rate is 16+112=14\frac{1}{6} + \frac{1}{12} = \frac{1}{4} reservoirs per hour, which yields a unique solution of 4 hours.

Adım Adım Çözüm

1
Rephrase the question stem mathematically
Let rXr_X be the hourly rate of Pump X and rYr_Y be the hourly rate of Pump Y. The target combined time TT is given by T=1rX+rYT = \frac{1}{r_X + r_Y}. We need the value of rX+rYr_X + r_Y.
Simplifying the target variable helps determine what information is necessary for sufficiency.
2
Evaluate Statement (1) independently
Statement (1) states rX=16r_X = \frac{1}{6} reservoir per hour, but gives no information about rYr_Y. Therefore, rX+rYr_X + r_Y cannot be calculated.
One variable in a two-variable sum remains unknown.
3
Evaluate Statement (2) independently
Statement (2) states rY=112r_Y = \frac{1}{12} reservoir per hour, but gives no information about rXr_X. Therefore, rX+rYr_X + r_Y cannot be calculated.
One variable in a two-variable sum remains unknown.
4
Evaluate Statement (1) and Statement (2) together
Combining both statements: rX+rY=16+112=312=14r_X + r_Y = \frac{1}{6} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}. Thus, T=11/4=4T = \frac{1}{1/4} = 4 hours. A single, definitive numerical answer is obtained.
Both individual rates are known, allowing exact computation of the combined rate and total time.

Anahtar Kavram

Combined Work Rates in Data Sufficiency
Soru 102Soru →

If xx is a real number, is ∣x+4∣>2x|x + 4| > 2x?

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

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.
Rephrasing the question stem shows that ∣x+4∣>2x|x + 4| > 2x is equivalent to x<4x < 4. Statement (1) establishes that −2<x<4-2 < x < 4, which guarantees x<4x < 4 (definitive Yes). Statement (2) establishes that −2<x<3-2 < x < 3, which also guarantees x<4x < 4 (definitive Yes). Therefore, each statement alone is sufficient.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically.
The inequality ∣x+4∣>2x|x + 4| > 2x simplifies to x<4x < 4.
If x≥−4x \ge -4, then x+4>2x  ⟹  x<4x + 4 > 2x \implies x < 4. If x<−4x < -4, ∣x+4∣=−(x+4)>2x  ⟹  3x<−4  ⟹  x<−4/3|x + 4| = -(x + 4) > 2x \implies 3x < -4 \implies x < -4/3, which holds for all x<−4x < -4. Thus, ∣x+4∣>2x|x + 4| > 2x is true if and only if x<4x < 4.
2
Evaluate Statement (1): ∣x−1∣<3|x - 1| < 3.
Statement (1) yields the range −2<x<4-2 < x < 4.
Unpacking ∣x−1∣<3|x - 1| < 3 gives −3<x−1<3  ⟹  −2<x<4-3 < x - 1 < 3 \implies -2 < x < 4. Since every value in (−2,4)(-2, 4) is strictly less than 44, the answer to 'Is x<4x < 4?' is a definitive 'Yes'. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): x2−x−6<0x^2 - x - 6 < 0.
Statement (2) yields the range −2<x<3-2 < x < 3.
Factoring gives (x−3)(x+2)<0  ⟹  −2<x<3(x - 3)(x + 2) < 0 \implies -2 < x < 3. Since every value in (−2,3)(-2, 3) is strictly less than 44, the answer to 'Is x<4x < 4?' is a definitive 'Yes'. Thus, Statement (2) alone is sufficient.

Anahtar Kavram

Rephrasing absolute value inequalities in Data Sufficiency Yes/No questions
Tahmini Süre:1m 30s
Soru 103Soru →

If pp and qq are numbers, what is the value of the product pqpq?

(1) (p+q)2=49(p + q)^2 = 49
(2) p2+q2=25p^2 + q^2 = 25

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient to determine that pq=12pq = 12, but neither statement alone is sufficient.
The correct option states that both statements together are sufficient, but neither alone is sufficient. Expanding (p+q)2=p2+2pq+q2(p+q)^2 = p^2 + 2pq + q^2 allows substituting (p+q)2=49(p+q)^2 = 49 from statement (1) and p2+q2=25p^2 + q^2 = 25 from statement (2), yielding 49=25+2pq49 = 25 + 2pq, which uniquely determines pq=12pq = 12. Neither statement alone isolates pqpq.

Adım Adım Çözüm

1
Rephrase the target question using algebraic identities
Recall the identity (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2. Rearranging gives 2pq=(p+q)2−(p2+q2)2pq = (p + q)^2 - (p^2 + q^2). Thus, knowing both (p+q)2(p + q)^2 and (p2+q2)(p^2 + q^2) will determine pqpq.
Target rephrasing simplifies evaluating statement sufficiency.
2
Evaluate Statement (1) independently
Statement (1) gives (p+q)2=49(p + q)^2 = 49. If p=7p = 7 and q=0q = 0, then pq=0pq = 0. If p=4p = 4 and q=3q = 3, then (4+3)2=49(4+3)^2 = 49 and pq=12pq = 12. Multiple values for pqpq exist.
A statement is sufficient only if it yields one unique value for the target expression.
3
Evaluate Statement (2) independently
Statement (2) gives p2+q2=25p^2 + q^2 = 25. If p=5p = 5 and q=0q = 0, then pq=0pq = 0. If p=4p = 4 and q=3q = 3, then 42+32=254^2 + 3^2 = 25 and pq=12pq = 12. Multiple values for pqpq exist.
Statement (2) alone does not yield a unique product.
4
Combine Statement (1) and Statement (2)
Substitute (p+q)2=49(p + q)^2 = 49 and p2+q2=25p^2 + q^2 = 25 into (p+q)2=p2+2pq+q2(p + q)^2 = p^2 + 2pq + q^2 to get 49=25+2pq2pq=24pq=1249 = 25 + 2pq 2pq = 24 pq = 12. A single, unique value is found.
Combining the statements provides enough information to determine the value of the target expression pqpq uniquely.

Anahtar Kavram

Algebraic Expression Manipulation via Quadratic Identities
Soru 104Soru →

A bakery sells only vanilla cupcakes and chocolate cupcakes. On Monday, what was the ratio of the number of vanilla cupcakes sold to the number of chocolate cupcakes sold?

(1) On Monday, the bakery sold a total of 120 cupcakes.
(2) On Monday, 40% of the cupcakes sold were vanilla cupcakes.

Cevabı ve açıklamayı göster

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

Cevap

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The correct answer states that Statement (2) alone is sufficient while Statement (1) alone is not. Statement (1) gives only the sum of both types of cupcakes, which allows for infinite possible ratio combinations. Statement (2) specifies that vanilla cupcakes represent 40% of the total sales, meaning chocolate cupcakes represent the remaining 60%. The ratio of vanilla to chocolate cupcakes is therefore fixed at 40:60, or 2:3, providing a unique and definitive solution.

Adım Adım Çözüm

1
Rephrase the question stem target algebraically
Let VV be the number of vanilla cupcakes sold and CC be the number of chocolate cupcakes sold. The goal is to find the value of the ratio VC\frac{V}{C}.
Simplifying the target helps determine what information is necessary to answer the question.
2
Evaluate Statement (1) independently
Statement (1) gives V+C=120V + C = 120. With no information about VV or CC individually, the ratio VC\frac{V}{C} could be 1:11:1 (60 of each), 1:21:2 (40 vanilla, 80 chocolate), or many other values. Thus, Statement (1) is NOT sufficient.
A total count alone cannot determine a ratio without additional proportional constraints.
3
Evaluate Statement (2) independently
Statement (2) states V=0.40(V+C)V = 0.40(V + C). Expanding gives V=0.40V+0.40C  ⟹  0.60V=0.40C  ⟹  VC=0.400.60=23V = 0.40V + 0.40C \implies 0.60V = 0.40C \implies \frac{V}{C} = \frac{0.40}{0.60} = \frac{2}{3}. A single, unique ratio of 2:32:3 is determined. Thus, Statement (2) IS sufficient.
Knowing the percentage component of a two-part total fixes the ratio regardless of total volume.

Anahtar Kavram

Determining ratios from component percentages versus absolute quantities in Data Sufficiency
Soru 105Soru →

If xx is a real number and x≠−1x \neq -1, is ∣x−3∣x+1<1\frac{|x - 3|}{x + 1} < 1?

(1) ∣2x−1∣>3|2x - 1| > 3
(2) x(x−1)>0x(x - 1) > 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrasing the question stem shows that ∣x−3∣x+1<1\frac{|x - 3|}{x + 1} < 1 is satisfied when x<−1x < -1 or x>1x > 1. Statement (1) solves to x<−1x < -1 or x>2x > 2. Since (2,∞)(2, \infty) is completely contained inside (1,∞)(1, \infty), every value satisfying Statement (1) yields a definitive YES to the question. Statement (2) solves to x<0x < 0 or x>1x > 1. Choosing x=2x = 2 gives a YES answer, while choosing x=−0.5x = -0.5 gives a NO answer. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Adım Adım Çözüm

1
Rephrase the question stem by analyzing cases for the denominator x+1x + 1.
Case 1 (x>−1x > -1): x+1>0x + 1 > 0, so ∣x−3∣<x+1  ⟺  −(x+1)<x−3<x+1|x - 3| < x + 1 \iff -(x + 1) < x - 3 < x + 1. The right inequality −3<1-3 < 1 is always true, and the left inequality −x−1<x−3-x - 1 < x - 3 simplifies to 2x>2  ⟺  x>12x > 2 \iff x > 1.
Case 2 (x<−1x < -1): x+1<0x + 1 < 0. The numerator ∣x−3∣>0|x - 3| > 0 while the denominator is negative, making the ratio negative, which is always <1< 1.
Combining both cases: The target inequality holds if and only if x<−1x < -1 or x>1x > 1.
Rephrasing the stem target simplifies complex absolute value expressions into clear number line intervals.
2
Evaluate Statement (1): ∣2x−1∣>3|2x - 1| > 3.
2x−1>3  ⟹  2x>4  ⟹  x>22x - 1 > 3 \implies 2x > 4 \implies x > 2, or 2x−1<−3  ⟹  2x<−2  ⟹  x<−12x - 1 < -3 \implies 2x < -2 \implies x < -1.
Range: x∈(−∞,−1)∪(2,∞)x \in (-\infty, -1) \cup (2, \infty). Every value in this range satisfies x<−1x < -1 or x>1x > 1, guaranteeing a definitive YES to the question.
Statement 1 specifies a subset of the valid target range, making it sufficient alone.
3
Evaluate Statement (2): x(x−1)>0x(x - 1) > 0.
Range: x<0x < 0 or x>1x > 1. Testing values within this range:
- If x=2x = 2: 2(1)=2>02(1) = 2 > 0, and ∣2−3∣2+1=13<1\frac{|2 - 3|}{2 + 1} = \frac{1}{3} < 1 (YES).
- If x=−0.5x = -0.5: −0.5(−1.5)=0.75>0-0.5(-1.5) = 0.75 > 0, but ∣−0.5−3∣−0.5+1=3.50.5=7≮1\frac{|-0.5 - 3|}{-0.5 + 1} = \frac{3.5}{0.5} = 7 \not< 1 (NO).
Since both YES and NO outcomes are possible, Statement (2) is insufficient.
Finding a counterexample within the statement's solution set proves insufficiency.

Anahtar Kavram

Data Sufficiency evaluation for rational expressions containing absolute values and rephrasing inequality ranges.
Tahmini Süre:2m 0s
Soru 106Soru →

For all real numbers aa and bb with a≠ba \neq b, the Data Sufficiency Yes/No target question "Is a2−b2(a−b)2>1\frac{a^2 - b^2}{(a - b)^2} > 1?" is algebraically equivalent to the simplified target question "Is a>ba > b?"

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is False because rephrasing the target inequality yields 2ba−b>0\frac{2b}{a-b} > 0, which requires analyzing the signs of both bb and a−ba-b, rather than evaluating whether a>ba > b alone.
The statement is False. Correct simplification of a2−b2(a−b)2>1\frac{a^2 - b^2}{(a - b)^2} > 1 leads to 2ba−b>0\frac{2b}{a-b} > 0. This inequality requires 2b2b and a−ba-b to have identical signs, which holds either when b>0b > 0 and a>ba > b or when b<0b < 0 and a<ba < b. Because a>ba > b can be true while b<0b < 0 (making the original inequality false), the proposed rephrasing is invalid.

Adım Adım Çözüm

1
Factor the algebraic expression in the numerator of the target question.
The numerator a2−b2a^2 - b^2 factors as (a−b)(a+b)(a-b)(a+b), yielding (a−b)(a+b)(a−b)2>1\frac{(a-b)(a+b)}{(a-b)^2} > 1.
Factoring allows simplification of common terms between the numerator and denominator.
2
Simplify the fraction by canceling common non-zero terms.
Since a≠ba \neq b, a−b≠0a - b \neq 0, so a+ba−b>1\frac{a+b}{a-b} > 1.
Canceling (a−b)(a-b) is valid as long as a≠ba \neq b.
3
Compare the fraction to zero by subtracting 1 from both sides.
\frac{a+b}{a-b} - 1 > 0 \implies \frac{(a+b) - (a-b)}{a-b} > 0 \implies \frac{2b}{a-b} > 0.
Subtracting 1 avoids multiplying by a variable expression (a−b)(a-b) whose sign is unknown.
4
Determine the conditions under which 2ba−b>0\frac{2b}{a-b} > 0.
The quotient is positive when 2b2b and a−ba-b have the same sign: Case 1 (b>0b > 0 and a>ba > b) OR Case 2 (b<0b < 0 and a<ba < b).
A quotient is strictly positive if and only if its numerator and denominator share the same sign.
5
Test whether "Is a>ba > b?" is equivalent to the derived condition using a counterexample.
If a=2a = 2 and b=−1b = -1, then a>ba > b is true (2>−12 > -1). However, 2(−1)2−(−1)=−23≯0\frac{2(-1)}{2 - (-1)} = -\frac{2}{3} \ngtr 0.
Finding a scenario where a>ba > b is true but the original inequality fails proves the two target questions are not algebraically equivalent.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing
Soru 107Soru →

If xx and yy are non-zero real numbers, is ∣x∣y<1\frac{|x|}{y} < 1?

(1) x2<y2x^2 < y^2
(2) x+y<0x + y < 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) gives x2<y2x^2 < y^2, which means ∣x∣<∣y∣|x| < |y|. If y>0y > 0, ∣y∣=y|y| = y, so ∣x∣<y  ⟹  ∣x∣y<1|x| < y \implies \frac{|x|}{y} < 1. If y<0y < 0, ∣x∣>0|x| > 0 implies ∣x∣y<0<1\frac{|x|}{y} < 0 < 1. Thus, Statement (1) alone yields a definitive 'Yes' and is sufficient. Statement (2) allows x=−3,y=1x = -3, y = 1 (yielding a ratio of 3, which is not less than 1) and x=−1,y=−2x = -1, y = -2 (yielding a ratio of -0.5, which is less than 1), so Statement (2) alone is insufficient.

Adım Adım Çözüm

1
Rephrase the target question
The target question asks whether ∣x∣y<1\frac{|x|}{y} < 1 for non-zero real numbers xx and yy.
Since ∣x∣>0|x| > 0 for any non-zero real number xx, if y<0y < 0, the ratio ∣x∣y\frac{|x|}{y} is strictly negative, which is always less than 1. If y>0y > 0, ∣x∣y<1\frac{|x|}{y} < 1 is equivalent to ∣x∣<y|x| < y.
2
Evaluate Statement (1): x2<y2x^2 < y^2
Taking the principal square root of both sides gives ∣x∣<∣y∣|x| < |y|.
If y>0y > 0, ∣y∣=y|y| = y, so ∣x∣<y|x| < y, which implies ∣x∣y<1\frac{|x|}{y} < 1 (YES). If y<0y < 0, then yy is negative and ∣x∣|x| is positive, so ∣x∣y<0<1\frac{|x|}{y} < 0 < 1 (YES). Since Statement (1) yields a definitive YES in all cases, Statement (1) ALONE is sufficient.
3
Evaluate Statement (2): x+y<0x + y < 0
Test suitable numbers.
Case A: Let x=−3x = -3 and y=1y = 1. Then x+y=−2<0x + y = -2 < 0, but ∣−3∣1=3≮1\frac{|-3|}{1} = 3 \not< 1 (NO). Case B: Let x=−1x = -1 and y=−2y = -2. Then x+y=−3<0x + y = -3 < 0, and ∣−1∣−2=−0.5<1\frac{|-1|}{-2} = -0.5 < 1 (YES). Because Statement (2) can yield both YES and NO, Statement (2) ALONE is not sufficient.

Anahtar Kavram

Data Sufficiency evaluation of absolute values and algebraic inequalities with unknown signs
Tahmini Süre:2m 0s
Soru 108Soru →

If mm is a real number, is m>0m > 0?

(1) ∣m+4∣=−3m|m + 4| = -3m
(2) m2+3m+2=0m^2 + 3m + 2 = 0

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

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.
The choice stating that EACH statement ALONE is sufficient is correct. Statement (1) restricts mm uniquely to m=−1m = -1 due to the absolute value non-negativity constraint, yielding a definitive 'No' to the question m>0m > 0. Statement (2) yields m=−1m = -1 or m=−2m = -2; since both values are negative, Statement (2) also conclusively answers 'No'. Therefore, both statements independently provide sufficient information.

Adım Adım Çözüm

1
Analyze the question stem
The target is a Yes/No question asking whether m>0m > 0. A statement is sufficient if it conclusively proves m>0m > 0 (Yes) or conclusively proves m≤0m \le 0 (No).
Understanding Yes/No Data Sufficiency logic is essential: a definitive 'No' is just as sufficient as a definitive 'Yes'.
2
Evaluate Statement (1): ∣m+4∣=−3m|m + 4| = -3m
Since an absolute value cannot be negative, −3m≥0  ⟹  m≤0-3m \ge 0 \implies m \le 0. Solving m+4=−3mm + 4 = -3m gives 4m=−4  ⟹  m=−14m = -4 \implies m = -1. Solving −(m+4)=−3m-(m + 4) = -3m gives 2m=4  ⟹  m=22m = 4 \implies m = 2 (extraneous because m≤0m \le 0). Thus, m=−1m = -1 uniquely.
Testing m=−1m = -1 in the target question gives: Is −1>0-1 > 0? No. Because Statement (1) provides a single unique value that yields a definitive 'No', Statement (1) alone is sufficient.
3
Evaluate Statement (2): m2+3m+2=0m^2 + 3m + 2 = 0
Factoring gives (m+1)(m+2)=0(m + 1)(m + 2) = 0, so m=−1m = -1 or m=−2m = -2.
If m=−1m = -1, is m>0m > 0? No. If m=−2m = -2, is m>0m > 0? No. Because all possible values for mm consistently lead to a definitive 'No', Statement (2) alone is sufficient.
4
Combine evaluations to select the correct choice
Since Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct response is that each statement alone is sufficient.
Matches standard GMAT Data Sufficiency Option D.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic
Soru 109Soru →

If aa and bb are non-zero real numbers, what is the value of ab\frac{a}{b}?

(1) 5a−3b=05a - 3b = 0
(2) a+b=8a + b = 8

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient to determine the value of ab\frac{a}{b}, but statement (2) alone is not sufficient.
Statement (1) can be rearranged algebraically to 5a=3b5a = 3b, which directly gives ab=35\frac{a}{b} = \frac{3}{5}, providing a single definitive value. Statement (2), a+b=8a + b = 8, allows infinitely many pairs of (a,b)(a, b) yielding different ratios for ab\frac{a}{b}. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Adım Adım Çözüm

1
Rephrase the question stem target
The target is to find a numerical value for the ratio ab\frac{a}{b}.
Simplifying the target helps determine if an equation directly yields the ratio without needing individual variable values.
2
Evaluate Statement (1) independently: 5a−3b=05a - 3b = 0
Rearranging gives 5a=3b5a = 3b. Dividing both sides by 5b5b (since b≠0b \neq 0) yields ab=35\frac{a}{b} = \frac{3}{5}.
This yields a single, unique numerical value for the target expression. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently: a+b=8a + b = 8
If a=4a = 4 and b=4b = 4, then ab=1\frac{a}{b} = 1. If a=2a = 2 and b=6b = 6, then ab=13\frac{a}{b} = \frac{1}{3}.
Multiple values of the ratio ab\frac{a}{b} are possible. Thus, Statement (2) alone is not sufficient.

Anahtar Kavram

Determining ratios from homogeneous linear equations in Data Sufficiency
Soru 110Soru →

A department has 4040 employees, and each employee speaks at least one of two languages: French or Spanish. How many employees in the department speak Spanish?

(1) 2525 employees speak French.
(2) 1010 employees speak both French and Spanish.

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
The total number of employees who speak at least one language is given by Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}. Neither statement alone provides both the number of French speakers and the number of employees who speak both languages. However, combining both statements gives 40=25+Spanish−1040 = 25 + \text{Spanish} - 10, which uniquely solves to Spanish=25\text{Spanish} = 25. Therefore, both statements together are sufficient, but neither alone is sufficient.

Adım Adım Çözüm

1
Rephrase the question stem using the overlapping sets formula.
Since every employee speaks at least one language, Neither=0\text{Neither} = 0. The relationship is Total=French+Spanish−Both\text{Total} = \text{French} + \text{Spanish} - \text{Both}, which simplifies to 40=French+Spanish−Both40 = \text{French} + \text{Spanish} - \text{Both}.
Establishing the mathematical relationship before evaluating statements clarifies what data is missing.
2
Evaluate Statement (1) independently.
Statement (1) gives French=25\text{French} = 25. Substituting this gives 40=25+Spanish−Both40 = 25 + \text{Spanish} - \text{Both}, or Spanish−Both=15\text{Spanish} - \text{Both} = 15.
Since Both\text{Both} is unknown, Spanish\text{Spanish} could take multiple values. Statement (1) is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives Both=10\text{Both} = 10. Substituting this gives 40=French+Spanish−1040 = \text{French} + \text{Spanish} - 10, or French+Spanish=50\text{French} + \text{Spanish} = 50.
Since French\text{French} is unknown, Spanish\text{Spanish} cannot be uniquely determined. Statement (2) is NOT sufficient.
4
Evaluate Statement (1) and Statement (2) together.
Substitute both values into the equation: 40=25+Spanish−10  ⟹  40=15+Spanish  ⟹  Spanish=2540 = 25 + \text{Spanish} - 10 \implies 40 = 15 + \text{Spanish} \implies \text{Spanish} = 25.
The equation yields a single, unique value for the target variable. Both statements together are sufficient.

Anahtar Kavram

Overlapping Sets (Two Groups)
Tahmini Süre:1m 0s
Soru 111Soru →

If mm and nn are non-zero real numbers, what is the value of m2+n2mn\frac{m^2 + n^2}{mn}?

(1) m2n+mn2=6(m+n)m^2 n + m n^2 = 6(m + n)
(2) m2−n2=3(m−n)m^2 - n^2 = 3(m - n)

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
Evaluating each statement alone yields multiple possible numerical outcomes because factoring produces independent cases (m+n=0m+n=0 vs mn=6mn=6 for the first statement, and m−n=0m-n=0 vs m+n=3m+n=3 for the second). However, when combining both statements, three of the four sub-cases fail: two violate non-zero or consistency conditions, and the sub-case where mn=6mn=6 and m+n=3m+n=3 has a negative discriminant (32−4(1)(6)=−15<03^2 - 4(1)(6) = -15 < 0), yielding no real solutions. This leaves only the sub-case mn=6mn=6 and m−n=0m-n=0, which uniquely determines the value of the target expression to be 2. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1
Rephrase the target expression.
The target expression m2+n2mn\frac{m^2 + n^2}{mn} can be rewritten as mn+nm\frac{m}{n} + \frac{n}{m}. We need a unique numerical value for this expression.
Simplifying the target helps evaluate what parameters or relationships are required.
2
Evaluate Statement (1) independently.
Rearrange Statement (1): mn(m+n)−6(m+n)=0  ⟹  (mn−6)(m+n)=0mn(m + n) - 6(m + n) = 0 \implies (mn - 6)(m + n) = 0. This gives two cases: Case 1: m+n=0  ⟹  n=−mm + n = 0 \implies n = -m. Since m,n≠0m, n \neq 0, m2+(−m)2m(−m)=2m2−m2=−2\frac{m^2 + (-m)^2}{m(-m)} = \frac{2m^2}{-m^2} = -2. Case 2: mn=6mn = 6. For instance, if m=2,n=3m = 2, n = 3, then 4+96=136\frac{4+9}{6} = \frac{13}{6}. Multiple values exist (−2-2 and 136\frac{13}{6}), so Statement (1) alone is NOT sufficient.
Factoring instead of dividing by (m+n)(m+n) preserves the root m+n=0m+n=0.
3
Evaluate Statement (2) independently.
Rearrange Statement (2): (m−n)(m+n)−3(m−n)=0  ⟹  (m−n)(m+n−3)=0(m - n)(m + n) - 3(m - n) = 0 \implies (m - n)(m + n - 3) = 0. This gives two cases: Case 1: m−n=0  ⟹  m=nm - n = 0 \implies m = n. Since m,n≠0m, n \neq 0, m2+m2m2=2\frac{m^2 + m^2}{m^2} = 2. Case 2: m+n=3m + n = 3. For instance, if m=1,n=2m = 1, n = 2, then 1+42=52\frac{1+4}{2} = \frac{5}{2}. Multiple values exist (22 and 52\frac{5}{2}), so Statement (2) alone is NOT sufficient.
Factoring preserves the root m−n=0m-n=0.
4
Combine Statement (1) and Statement (2).
We test the four combinations of cases:
- Case A (m+n=0m + n = 0) & Case X (m−n=0m - n = 0): System gives m=0,n=0m = 0, n = 0. Contradicts the condition that m,nm, n are non-zero.
- Case A (m+n=0m + n = 0) & Case Y (m+n=3m + n = 3): 0=30 = 3, impossible.
- Case B (mn=6mn = 6) & Case X (m−n=0m - n = 0): m=n  ⟹  m2=6  ⟹  m=n=±6m = n \implies m^2 = 6 \implies m = n = \pm\sqrt{6}. Here m2+n2mn=6+66=2\frac{m^2+n^2}{mn} = \frac{6+6}{6} = 2.
- Case B (mn=6mn = 6) & Case Y (m+n=3m + n = 3): n=3−m  ⟹  m(3−m)=6  ⟹  m2−3m+6=0n = 3 - m \implies m(3-m) = 6 \implies m^2 - 3m + 6 = 0. The discriminant is b2−4ac=9−24=−15<0b^2 - 4ac = 9 - 24 = -15 < 0, which yields no real solutions for mm and nn.

Only Case B & Case X yields valid real non-zero solutions, providing a unique value of 2.
Checking real constraints eliminates non-real systems and isolates a single valid numerical outcome.

Anahtar Kavram

Factoring non-linear algebraic systems without illegal variable division, and verifying real-number constraints using discriminants in Data Sufficiency.
Tahmini Süre:2m 30s
Soru 112Soru →

If xx and yy are non-zero real numbers, is xy<1\frac{x}{y} < 1?

(1) ∣x+y∣<∣x−y∣|x + y| < |x - y|
(2) x>yx > y

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

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct option states that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Squaring both sides of Statement (1) gives x2+2xy+y2<x2−2xy+y2x^2 + 2xy + y^2 < x^2 - 2xy + y^2, which simplifies directly to 4xy<04xy < 0, meaning xy<0xy < 0. When two numbers have opposite signs, their quotient xy\frac{x}{y} must be negative, making it strictly less than 1. This provides a definitive 'Yes' answer. Statement (2) gives x>yx > y; if x=3x = 3 and y=2y = 2, 32>1\frac{3}{2} > 1 (No), but if x=1x = 1 and y=−2y = -2, 1−2=−0.5<1\frac{1}{-2} = -0.5 < 1 (Yes). Hence Statement (2) is not sufficient.

Adım Adım Çözüm

1
Rephrase the target question
The target question asks if xy<1\frac{x}{y} < 1.
Understanding the algebraic condition required for a fraction to be less than 1 helps evaluate given statements.
2
Evaluate Statement (1): ∣x+y∣<∣x−y∣|x + y| < |x - y|
Squaring both sides yields (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.
Since both sides of the inequality are non-negative distance expressions, squaring preserves the inequality sign.
3
Determine the sufficiency of Statement (1)
Since xy<0xy < 0, xx and yy must have opposite signs. Therefore, xy\frac{x}{y} is negative, which means xy<0<1\frac{x}{y} < 0 < 1. This gives a definitive 'Yes' to the question.
Any negative number is strictly less than 1, so Statement (1) alone is sufficient.
4
Evaluate Statement (2): x>yx > y
If x=3x = 3 and y=2y = 2, then x>yx > y and xy=1.5>1\frac{x}{y} = 1.5 > 1 (No). If x=1x = 1 and y=−2y = -2, then x>yx > y and xy=−0.5<1\frac{x}{y} = -0.5 < 1 (Yes).
Testing cases with positive vs. negative denominators shows that Statement (2) leads to both 'Yes' and 'No' answers.
5
Determine the sufficiency of Statement (2) and select the overall answer choice
Statement (2) alone is not sufficient. Therefore, Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
Only Statement (1) provides a definitive answer to the question stem.

Anahtar Kavram

Distance interpretation of absolute values and algebraic rephrasing of inequalities involving quotient sign analysis.
Tahmini Süre:2m 0s
Soru 113Soru →

A car dealership's inventory consists exclusively of sedans and SUVs. What is the ratio of the number of sedans to the number of SUVs in the inventory?

(1) The total number of sedans and SUVs in the inventory is 150150.
(2) The number of sedans in the inventory is 6060.

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
Statement (1) gives S+U=150S + U = 150, which is insufficient by itself. Statement (2) gives S=60S = 60, which is also insufficient by itself. Together, we can deduce U=90U = 90, giving a unique ratio S:U=60:90=2:3S : U = 60 : 90 = 2 : 3. Therefore, both statements together are sufficient.

Adım Adım Çözüm

1
Rephrase the target question.
Let SS be the number of sedans and UU be the number of SUVs. The target ratio is SU\frac{S}{U}.
Establishing explicit variable definitions simplifies statement analysis.
2
Evaluate Statement (1) independently.
S+U=150S + U = 150.
Knowing only the total does not fix the specific values of SS or UU, so the ratio SU\frac{S}{U} can take multiple values. Statement (1) is insufficient.
3
Evaluate Statement (2) independently.
S=60S = 60.
Without knowing UU, the ratio SU\frac{S}{U} cannot be evaluated. Statement (2) is insufficient.
4
Evaluate Statements (1) and (2) combined.
Substitute S=60S = 60 into S+U=150S + U = 150 to get 60+U=15060 + U = 150, so U=90U = 90. Thus, SU=6090=23\frac{S}{U} = \frac{60}{90} = \frac{2}{3}.
Combining both statements yields a single, definitive ratio value. Statements (1) and (2) together are sufficient.

Anahtar Kavram

Data Sufficiency evaluation for linear systems involving ratios and sums.
Soru 114Soru →

If xx and yy are non-zero real numbers, is x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2?

(1) ∣x+y∣<∣x−y∣|x + y| < |x - y|
(2) x2y3>0x^2 y^3 > 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct response is the option stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient. Rephrasing the stem x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2 by factoring yields xy(x−y)2<0xy(x - y)^2 < 0. Because (x−y)2>0(x - y)^2 > 0 whenever x≠yx \neq y, the inequality holds if and only if xy<0xy < 0. Statement (1) reduces to (x+y)2<(x−y)2(x+y)^2 < (x-y)^2, which simplifies directly to xy<0xy < 0, giving a definitive YES. Statement (2) reduces to y>0y > 0, which leaves the sign of xx (and thus xyxy) unknown.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically.
The target inequality x3y+xy3<2x2y2x^3 y + x y^3 < 2 x^2 y^2 can be rearranged as x3y+xy3−2x2y2<0x^3 y + x y^3 - 2 x^2 y^2 < 0. Factoring out xyxy yields xy(x2−2xy+y2)<0xy(x^2 - 2xy + y^2) < 0, which is xy(x−y)2<0xy(x - y)^2 < 0.
Simplifying the stem isolates the core algebraic condition needed to answer the question.
2
Analyze the conditions under which xy(x−y)2<0xy(x - y)^2 < 0 holds.
Since xx and yy are non-zero, if x=yx = y, xy(x−y)2=0xy(x-y)^2 = 0, so the inequality is false (NO). If x≠yx \neq y, (x−y)2>0(x - y)^2 > 0, so the sign of xy(x−y)2xy(x - y)^2 is determined entirely by xyxy. If xy<0xy < 0, then x≠yx \neq y and xy(x−y)2<0xy(x - y)^2 < 0 (YES). If xy>0xy > 0, then xy(x−y)2≥0xy(x - y)^2 \ge 0 (NO). Thus, the target question simplifies to: 'Is xy<0xy < 0?'
A perfect square of a real number is non-negative, so sign evaluation reduces to checking xy<0xy < 0.
3
Evaluate Statement (1): ∣x+y∣<∣x−y∣|x + y| < |x - y|.
Squaring both sides of ∣x+y∣<∣x−y∣|x + y| < |x - y| 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.
Since Statement (1) directly proves xy<0xy < 0, it gives a definitive YES to the rephrased target question, making Statement (1) sufficient.
4
Evaluate Statement (2): x2y3>0x^2 y^3 > 0.
Since x≠0x \neq 0, x2>0x^2 > 0. Thus x2y3>0x^2 y^3 > 0 simplifies to y3>0y^3 > 0, which means y>0y > 0. However, xx can still be positive (yielding xy>0xy > 0) or negative (yielding xy<0xy < 0).
Knowing only that y>0y > 0 leaves the sign of xyxy undetermined, so Statement (2) is not sufficient.

Anahtar Kavram

Data Sufficiency Target Simplification and Algebraic Rephrasing
Soru 115Soru →

A class of 50 students took tests in both Mathematics and Science. How many students passed both tests?

(1) 35 students passed Mathematics and 30 students passed Science.
(2) 10 students failed both tests.

Cevabı ve açıklamayı göster

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

Cevap

Both statements together are sufficient, but neither statement alone is sufficient.
The correct option is the choice stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) leaves the number of students failing both tests unknown, while Statement (2) leaves the individual subject pass numbers unknown. Combined, the standard set equation Total=Group A+Group B−Both+Neither\text{Total} = \text{Group A} + \text{Group B} - \text{Both} + \text{Neither} yields a unique value of 2525 for students passing both tests.

Adım Adım Çözüm

1
Set up the overlapping sets formula for two groups
Total=Math+Science−Both+Neither\text{Total} = \text{Math} + \text{Science} - \text{Both} + \text{Neither}, which becomes 50=Math+Science−Both+Neither50 = \text{Math} + \text{Science} - \text{Both} + \text{Neither}.
This formula connects all four components of a two-group overlapping set.
2
Evaluate Statement (1) independently
Substituting Math=35\text{Math} = 35 and Science=30\text{Science} = 30 into the formula yields 50=35+30−Both+Neither50 = 35 + 30 - \text{Both} + \text{Neither}, or Both−Neither=15\text{Both} - \text{Neither} = 15.
Since Neither\text{Neither} is unknown, Both\text{Both} cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently
Substituting Neither=10\text{Neither} = 10 yields 50=Math+Science−Both+1050 = \text{Math} + \text{Science} - \text{Both} + 10.
Since Math\text{Math} and Science\text{Science} are unknown, Both\text{Both} cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
Combining all given values gives 50=35+30−Both+10⇒50=75−Both⇒Both=2550 = 35 + 30 - \text{Both} + 10 \Rightarrow 50 = 75 - \text{Both} \Rightarrow \text{Both} = 25.
A single unique value of 25 is obtained for the number of students who passed both tests.

Anahtar Kavram

Overlapping Sets (Two-Group Venn Diagram Formula)
Soru 116Soru →

In a technology consulting firm of 100 employees, every employee works in either the Analytics department, the Engineering department, or both. How many employees work in both departments?

(1) Exactly 70 employees work in the Analytics department, and 60 employees work in the Engineering department.
(2) Exactly 40 employees work ONLY in the Analytics department.

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The correct answer states that Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient. Using the standard formula for overlapping sets, Total = Group A + Group B - Both, Statement (1) provides values for Total (100), Group A (70), and Group B (60), allowing us to solve directly for Both = 30. Statement (2) only specifies the number of members belonging strictly to Group A, which leaves the overlap dependent on the unstated size of Group B.

Adım Adım Çözüm

1
Set up the overlapping sets formula for two groups.
Total = N(Analytics) + N(Engineering) - N(Both)
Since every employee belongs to at least one of the two departments, the union of the two sets equals the total number of employees, 100.
2
Evaluate Statement (1) independently.
100 = 70 + 60 - N(Both) => N(Both) = 30.
Statement (1) provides N(Analytics) = 70 and N(Engineering) = 60. Substituting these values into the formula yields a unique value of 30 for N(Both). Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2) independently.
100 = N(Only Analytics) + N(Only Engineering) + N(Both) => 100 = 40 + N(Only Engineering) + N(Both).
We have two unknown variables: N(Only Engineering) and N(Both). Multiple non-negative integer pairs satisfy this equation (e.g., N(Both) could be 0, 10, 20, etc.). Thus, Statement (2) alone is not sufficient.

Anahtar Kavram

Overlapping Sets Formula for Two Groups
Tahmini Süre:1m 30s
Soru 117Soru →

If xx is a real number, is ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8?

(1) ∣x−1∣≤4|x - 1| \le 4
(2) x2−x−6≤0x^2 - x - 6 \le 0

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

Each statement alone is sufficient.
Rephrasing the question stem shows that ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 is equivalent to −3≤x≤5-3 \le x \le 5. Statement (1) gives −3≤x≤5-3 \le x \le 5, which directly yields a definitive 'Yes'. Statement (2) gives −2≤x≤3-2 \le x \le 3, which is a subset of [−3,5][-3, 5], so any value of xx satisfying Statement (2) must also satisfy −3≤x≤5-3 \le x \le 5, also yielding a definitive 'Yes'. Thus, each statement alone is sufficient.

Adım Adım Çözüm

1
Rephrase the question stem target by analyzing critical points x=−2x = -2 and x=4x = 4 for ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8.
The target inequality ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 simplifies to the range −3≤x≤5-3 \le x \le 5.
For x≥4x \ge 4: (x+2)+(x−4)≤8  ⟹  2x−2≤8  ⟹  x≤5(x + 2) + (x - 4) \le 8 \implies 2x - 2 \le 8 \implies x \le 5, giving [4,5][4, 5]. For −2≤x<4-2 \le x < 4: (x+2)+(4−x)=6≤8(x + 2) + (4 - x) = 6 \le 8, which is true for all x∈[−2,4)x \in [-2, 4). For x<−2x < -2: (−x−2)+(4−x)≤8  ⟹  2−2x≤8  ⟹  −2x≤6  ⟹  x≥−3(-x - 2) + (4 - x) \le 8 \implies 2 - 2x \le 8 \implies -2x \le 6 \implies x \ge -3, giving [−3,−2)[-3, -2). Combining all intervals gives −3≤x≤5-3 \le x \le 5.
2
Evaluate Statement (1): ∣x−1∣≤4|x - 1| \le 4.
−4≤x−1≤4  ⟹  −3≤x≤5-4 \le x - 1 \le 4 \implies -3 \le x \le 5.
Statement (1) states that xx is precisely in the range [−3,5][-3, 5]. This provides a definitive 'Yes' answer to the question 'Is −3≤x≤5-3 \le x \le 5?'. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): x2−x−6≤0x^2 - x - 6 \le 0.
(x−3)(x+2)≤0  ⟹  −2≤x≤3(x - 3)(x + 2) \le 0 \implies -2 \le x \le 3.
Since [−2,3][-2, 3] is entirely contained within [−3,5][-3, 5], every value of xx satisfying Statement (2) automatically satisfies −3≤x≤5-3 \le x \le 5. This yields a definitive 'Yes' to the question. Thus, Statement (2) alone is sufficient.
4
Combine evaluations of Statement (1) and Statement (2).
Each statement alone is sufficient.
Because both statements independently provide enough information to give a definitive 'Yes' answer, the correct answer choice is that each statement alone is sufficient.

Anahtar Kavram

Absolute Value Distance Interpretation and Subset Range Sufficiency
Soru 118Soru →

If mm and nn are real numbers, is m2<n2m^2 < n^2?

(1) ∣m∣+n<0|m| + n < 0
(2) m+n>0m + n > 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrased, the question asks whether ∣m∣<∣n∣|m| < |n|. Evaluating the first condition gives n<−∣m∣n < -|m|, which shows that nn must be negative and its absolute value ∣n∣=−n|n| = -n must be strictly greater than ∣m∣|m|. Hence, m2<n2m^2 < n^2 is definitely true. Evaluating the second condition allows both m=2,n=5m=2, n=5 (where m2<n2m^2 < n^2) and m=5,n=2m=5, n=2 (where m2>n2m^2 > n^2), making it insufficient. Therefore, the first condition alone is sufficient while the second condition alone is not.

Adım Adım Çözüm

1
Rephrase the target question stem m2<n2m^2 < n^2.
The target condition m2<n2m^2 < n^2 is equivalent to ∣m∣2<∣n∣2|m|^2 < |n|^2, which is true if and only if ∣m∣<∣n∣|m| < |n|.
Since both m2m^2 and n2n^2 are non-negative, taking the square root of both sides preserves the inequality order for non-negative magnitudes.
2
Evaluate Statement (1): ∣m∣+n<0|m| + n < 0.
Rearranging gives n<−∣m∣n < -|m|. Since ∣m∣≥0|m| \ge 0, this implies nn is strictly negative. Taking absolute values of both sides of n<−∣m∣n < -|m| gives ∣n∣=−n>∣m∣|n| = -n > |m|. Therefore, ∣n∣>∣m∣|n| > |m|, which means n2>m2n^2 > m^2 or m2<n2m^2 < n^2.
Statement (1) yields a definitive 'Yes' to the target question. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): m+n>0m + n > 0.
Case 1: Let m=2m = 2 and n=5n = 5. Then 2+5=7>02 + 5 = 7 > 0, and 22=4<25=522^2 = 4 < 25 = 5^2 (Yes). Case 2: Let m=5m = 5 and n=2n = 2. Then 5+2=7>05 + 2 = 7 > 0, but 52=25>4=225^2 = 25 > 4 = 2^2 (No).
Because Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Anahtar Kavram

Data Sufficiency evaluation of absolute values and algebraic inequalities
Soru 119Soru →

If xx and yy are real numbers such that x+y≠0x + y \neq 0, what is the value of x3+y3x+y\frac{x^3 + y^3}{x + y}?

(1) x2−xy+y2=12x^2 - xy + y^2 = 12
(2) x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.
Factoring the numerator using the sum of cubes identity x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2) allows canceling (x+y)(x + y), since x+y≠0x + y \neq 0. Thus, the question asks for the value of x2−xy+y2x^2 - xy + y^2. Statement (1) directly states that x2−xy+y2=12x^2 - xy + y^2 = 12, which is sufficient. Statement (2) gives x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8, so x2−xy+y2=(x2+y2)−xy=20−8=12x^2 - xy + y^2 = (x^2 + y^2) - xy = 20 - 8 = 12, which is also sufficient. Since each statement independently yields a unique value, the correct choice states that each statement alone is sufficient.

Adım Adım Çözüm

1
Simplify and rephrase the target expression in the question stem.
Since x+y≠0x + y \neq 0, factor the numerator using the sum of cubes formula x3+y3=(x+y)(x2−xy+y2)x^3 + y^3 = (x + y)(x^2 - xy + y^2). The expression simplifies to (x+y)(x2−xy+y2)x+y=x2−xy+y2\frac{(x + y)(x^2 - xy + y^2)}{x + y} = x^2 - xy + y^2. The question target is equivalent to finding the value of x2−xy+y2x^2 - xy + y^2.
Simplifying the target expression before analyzing the statements eliminates unnecessary variables and reveals the exact algebraic value needed.
2
Evaluate Statement (1) independently.
Statement (1) gives x2−xy+y2=12x^2 - xy + y^2 = 12. Since this matches the simplified question target directly, the value is uniquely determined as 12.
Statement (1) provides the exact value of the rephrased target expression.
3
Evaluate Statement (2) independently.
Statement (2) provides x2+y2=20x^2 + y^2 = 20 and xy=8xy = 8. Substituting these into the target expression x2−xy+y2=(x2+y2)−xyx^2 - xy + y^2 = (x^2 + y^2) - xy gives 20−8=1220 - 8 = 12. The value is uniquely determined as 12.
Statement (2) supplies component values that combine to form the target expression uniquely.
4
Determine the final Data Sufficiency decision.
Because Statement (1) alone is sufficient and Statement (2) alone is sufficient, the correct choice is that EACH statement ALONE is sufficient.
Both statements independently yield a single, consistent answer to the rephrased question.

Anahtar Kavram

Question Stem Rephrasing and Algebraic Identity Simplification in Data Sufficiency
Soru 120Soru →

If xx and yy are real numbers such that x≠0x \neq 0, is ∣x−y∣x>1\frac{|x - y|}{x} > 1?

(1) x<0x < 0
(2) y<0y < 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) tells us x<0x < 0. Since ∣x−y∣≥0|x - y| \ge 0 for all real numbers, dividing a non-negative numerator by a negative denominator guarantees that ∣x−y∣x≤0\frac{|x - y|}{x} \le 0. Consequently, ∣x−y∣x\frac{|x - y|}{x} can never be greater than 11, giving a definitive 'No' answer to the question. Because Statement (1) provides a single definitive answer, it is sufficient. Statement (2) tells us y<0y < 0, but leaves the sign and value of xx unconstrained. Choosing x=−1x = -1 with y=−2y = -2 gives a value of −1-1 (answering 'No'), while choosing x=1x = 1 with y=−2y = -2 gives a value of 33 (answering 'Yes'). Since Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient. Thus, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct.

Adım Adım Çözüm

1
Analyze the target question stem and algebraic constraints.
We are asked whether ∣x−y∣x>1\frac{|x - y|}{x} > 1. Notice that for any real numbers xx and yy, the absolute value in the numerator ∣x−y∣≥0|x - y| \ge 0. The sign of the denominator xx determines the sign of the entire quotient.
Understanding the algebraic behavior of non-negative numerators over signed denominators simplifies statement evaluation.
2
Evaluate Statement (1): x<0x < 0.
Since ∣x−y∣≥0|x - y| \ge 0 and x<0x < 0, the ratio ∣x−y∣x\frac{|x - y|}{x} is a non-negative number divided by a negative number. Thus, ∣x−y∣x≤0\frac{|x - y|}{x} \le 0. Since a number ≤0\le 0 can never be greater than 11, the answer to the question is a definitive 'No'. A definitive 'No' means the statement IS sufficient.
In Data Sufficiency, any statement that allows us to answer the question with a single, unambiguous 'Yes' or 'No' is sufficient.
3
Evaluate Statement (2): y<0y < 0.
Test values for xx while holding y<0y < 0 (e.g., y=−2y = -2). If x=−1x = -1, then ∣−1−(−2)∣−1=1−1=−1≯1\frac{|-1 - (-2)|}{-1} = \frac{1}{-1} = -1 \ngtr 1 (Answer: No). If x=1x = 1, then ∣1−(−2)∣1=31=3>1\frac{|1 - (-2)|}{1} = \frac{3}{1} = 3 > 1 (Answer: Yes). Because both 'Yes' and 'No' are possible, Statement (2) is not sufficient.
Getting conflicting answers from permissible test cases proves a statement is insufficient.

Anahtar Kavram

Definitive Yes/No logic and sign properties of absolute value quotients in Data Sufficiency
ÖncekiSayfa 6 / 14Sonraki