Data Sufficiency Alıştırma Soruları — GMAT — Sayfa 11

Soru 201Soru →

A bio-refinery processes two grades of raw biomass, Grade Alpha and Grade Beta, to produce liquid biofuel. Grade Alpha yields $15\%$ biofuel per ton, while Grade Beta yields $35\%$ biofuel per ton. If a single production batch used a combined total of $1,200$ tons of Grade Alpha and Grade Beta biomass, resulting in $V$ total tons of biofuel, is $V > 300$ ?

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

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

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

EACH statement ALONE is sufficient.

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

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 is the option stating that Statement (1) ALONE is sufficient, but Statement (2) alone is not. Rephrasing the target condition

V > 300

yields

A < 600

. Statement (1) provides

A < 533.33

, which definitively proves

A < 600

(a conclusive YES). Statement (2) provides

A < 700

, which permits values of

A

both above and below

600

, rendering it insufficient.

Adım Adım Çözüm

1

Set up algebraic equations for total biomass and total biofuel yield.

Let

A

be the tons of Grade Alpha biomass and

B

be the tons of Grade Beta biomass. Given

A + B = 1,200

, we have

B = 1,200 - A

. Total biofuel yield is

V = 0.15A + 0.35B

.

Expressing

V

in terms of a single variable

A

simplifies the target question.

2

Rephrase the target question 'Is

V > 300

?' in terms of variable

A

.

Substitute

B = 1,200 - A

into the yield equation:

V = 0.15A + 0.35(1,200 - A) = 420 - 0.20A

. The condition

V > 300

becomes

420 - 0.20A > 300 \implies 120 > 0.20A \implies A < 600

. Thus, the question asks: 'Is

A < 600

?'

Rephrasing reduces the complex weighted average inequality to a straightforward threshold check on

A

.

3

Evaluate Statement (1): The ratio of Grade Alpha to Grade Beta tonnage was less than

4:5

.

\frac{A}{B} < \frac{4}{5} \implies 5A < 4B

. Substituting

B = 1,200 - A

gives

5A < 4(1,200 - A) \implies 9A < 4,800 \implies A < 533.33

. Since

A < 533.33

guarantees that

A < 600

, the answer to 'Is

A < 600

?' is a definitive YES. Statement (1) is SUFFICIENT.

Determining if Statement (1) alone guarantees

A < 600

.

4

Evaluate Statement (2): More than

500

tons of Grade Beta biomass was used in the batch.

B > 500 \implies 1,200 - A > 500 \implies A < 700

. If

A = 500

, then

A < 600

is TRUE (YES). If

A = 650

, then

A < 600

is FALSE (NO). Because both YES and NO outcomes are possible, Statement (2) is NOT SUFFICIENT.

Checking whether Statement (2) produces a single definitive YES/NO answer.

Anahtar Kavram

Data Sufficiency Question Stem Rephrased for Weighted Averages and Mixture Ratios

Soru 202Soru →

If $x$ and $y$ are real numbers, is $x^2 + y^2 < 25$ ?

(1) $x + y = 7$
(2) $(x - 3)^2 + (y - 4)^2 = 0$

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

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

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

EACH statement ALONE is sufficient.

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

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.

Statement (1) allows multiple outcomes for

x^2 + y^2

(both values less than 25 and values greater than or equal to 25), so it is not sufficient. Statement (2) forces

x=3

and

y=4

because the sum of non-negative real squares can only be zero when each square term is zero. Substituting

x=3

and

y=4

yields

x^2 + y^2 = 25

, which conclusively answers 'No' to the question stem (

25 < 25

is false). Hence, Statement (2) alone is sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

Statement (1) gives

x + y = 7

. If

x = 3.5

and

y = 3.5

, then

x^2 + y^2 = 12.25 + 12.25 = 24.5 < 25

(Yes). If

x = 7

and

y = 0

, then

x^2 + y^2 = 49 + 0 = 49 \not< 25

(No).

Since both 'Yes' and 'No' are possible, Statement (1) alone is insufficient.

2

Evaluate Statement (2) independently without carrying over Statement (1)

Statement (2) states

(x - 3)^2 + (y - 4)^2 = 0

. Since the square of a real number is non-negative, the sum of two squared terms can equal zero if and only if each term is zero:

x - 3 = 0 \implies x = 3

and

y - 4 = 0 \implies y = 4

.

This uniquely fixes the values of

x

and

y

.

3

Calculate the target expression using values from Statement (2)

Substitute

x = 3

and

y = 4

into

x^2 + y^2

:

3^2 + 4^2 = 9 + 16 = 25

. The question asks if

x^2 + y^2 < 25

. Since

25 < 25

is false, the answer is a definitive 'No'.

A definitive 'No' answer means the statement provides enough information to answer the question, so Statement (2) alone is sufficient.

Anahtar Kavram

Statement Independence in Data Sufficiency and Definitive Yes/No Decision Rules

Tahmini Süre:2m 0s

Soru 203Soru →

A logistics center uses two sorting systems, Line 1 and Line 2, to process package inventory. Line 1 operates at a constant rate of $r_1$ packages per minute, and Line 2 operates at a constant rate of $r_2$ packages per minute. During a testing run, Line 1 operated for $t_1$ minutes and Line 2 operated for $t_2$ minutes, sorting a total of $P$ packages. Was the combined average sorting rate during the test run—defined as the total packages sorted divided by total machine-minutes worked, $\frac{P}{t_1 + t_2}$ —greater than $40$ packages per minute?

(1) Line 1's sorting rate $r_1$ was $50\%$ greater than Line 2's sorting rate $r_2$ , and Line 1 operated for a duration $t_1$ that was $50\%$ longer than Line 2's operating duration $t_2$ .
(2) If Line 1 and Line 2 were to operate simultaneously for $1$ hour, they would sort a combined total of $3{}000$ packages.

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.

EACH statement ALONE is sufficient.

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

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 the combined average rate is 26 packages per minute, which definitively answers the question with a 'No'. Neither statement alone is sufficient.

The correct option is the one stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) simplifies the weighted average rate to

1.3 r_2

, which is insufficient by itself because

r_2

is unknown. Statement (2) provides

r_1 + r_2 = 50

, which is insufficient alone because individual rates and time proportions are unknown. Combining both statements allows us to solve for

r_2 = 20

and compute the exact weighted average rate of

26

packages per minute. Since

26

is not greater than

40

, we obtain a definitive 'No' answer, establishing sufficiency.

Adım Adım Çözüm

1

Rephrase the target question algebraically.

The combined average sorting rate is

\text{Average Rate} = \frac{P}{t_1 + t_2} = \frac{r_1 t_1 + r_2 t_2}{t_1 + t_2}

. The target question asks: Is

\frac{r_1 t_1 + r_2 t_2}{t_1 + t_2} > 40

?

Establishing the target formula in terms of

r_1, r_2, t_1,

and

t_2

allows direct evaluation of each statement.

2

Evaluate Statement (1) independently.

Statement (1) gives

r_1 = 1.5 r_2

and

t_1 = 1.5 t_2

. Substituting these into the average rate formula gives

\frac{(1.5 r_2)(1.5 t_2) + r_2 t_2}{1.5 t_2 + t_2} = \frac{2.25 r_2 t_2 + r_2 t_2}{2.5 t_2} = \frac{3.25 r_2 t_2}{2.5 t_2} = 1.3 r_2

.

Since the value of

r_2

is unknown, we cannot determine whether

1.3 r_2 > 40

. Statement (1) alone is INSUFFICIENT.

3

Evaluate Statement (2) independently.

Operating simultaneously for

1

hour (

60

minutes) produces

3{}000

packages:

60(r_1 + r_2) = 3000 \implies r_1 + r_2 = 50

packages per minute.

Knowing

r_1 + r_2 = 50

does not fix the ratio of durations

t_1 / t_2

or individual rates

r_1, r_2

. The weighted average could be anywhere between

r_1

and

r_2

. Statement (2) alone is INSUFFICIENT.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

r_1 = 1.5 r_2

. Substituting into Statement (2)'s equation

r_1 + r_2 = 50

yields

1.5 r_2 + r_2 = 50 \implies 2.5 r_2 = 50 \implies r_2 = 20

. Consequently,

r_1 = 30

. Substituting

r_2 = 20

into the expression from Statement (1) gives an average rate of

1.3 \times 20 = 26

packages per minute.

Since

26

is not greater than

40

, we can answer the question with a definitive 'No'. A definitive 'No' means the combined statements are SUFFICIENT.

Anahtar Kavram

Weighted average rate simplification and Yes/No sufficiency determination in Data Sufficiency.

Tahmini Süre:2m 30s

Soru 204Soru →

A renewable energy research laboratory tested four experimental photovoltaic panel coatings (Coating Alpha, Coating Beta, Coating Gamma, and Coating Delta) across three sunlight intensity levels (Low, Medium, and High). The table below displays the measured power output, in watts per square meter ( $\text{W/m}^2$ ), for each coating under each sunlight intensity level:

Coating	Low Intensity	Medium Intensity	High Intensity
Coating Alpha	80	180	280
Coating Beta	90	195	310
Coating Gamma	75	170	$x$
Coating Delta	$y$	200	320

What is the average (arithmetic mean) power output of Coating Gamma across all three sunlight intensity levels?

(1) Under High Intensity, the power output of Coating Gamma is 10% greater than the power output of Coating Alpha under High Intensity.

(2) The average (arithmetic mean) power output of Coating Delta across the three intensity levels is equal to the average (arithmetic mean) power output of Coating Beta across the three intensity levels.

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

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

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

EACH statement ALONE is sufficient.

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

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 target shows that finding the average power output of Coating Gamma requires finding the unknown value

x

. Statement (1) directly gives a percentage relationship between

x

and Coating Alpha's high-intensity output (

280 \text{ W/m}^2

), allowing us to calculate

x = 308

and thus the exact mean. Statement (2) allows us to solve for

y

(Coating Delta's low-intensity output), but provides no information about

x

. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Adım Adım Çözüm

1

Rephrase the question stem target

The average power output of Coating Gamma is

\frac{75 + 170 + x}{3} = \frac{245 + x}{3}

. To find this average, we only need to determine the value of

x

.

Simplifying the question target clarifies that finding

x

is both necessary and sufficient.

2

Evaluate Statement (1)

Statement (1) states that under High Intensity, Coating Gamma (

x

) is 10% greater than Coating Alpha (280). Thus,

x = 280 \times 1.10 = 308 \text{ W/m}^2

. We can compute the average as

\frac{245 + 308}{3} = 184.33 \text{ W/m}^2

.

Since a single unique numerical value for

x

is found, Statement (1) ALONE is sufficient.

3

Evaluate Statement (2)

Statement (2) gives the mean of Coating Delta as equal to the mean of Coating Beta. Mean of Coating Beta =

\frac{90 + 195 + 310}{3} = \frac{595}{3}

. Setting

\frac{y + 200 + 320}{3} = \frac{595}{3}

yields

y = 75 \text{ W/m}^2

.

Finding

y

gives information about Coating Delta, but provides no information regarding

x

for Coating Gamma. Thus, Statement (2) ALONE is not sufficient.

Anahtar Kavram

Target Rephrasing and Independent Statement Evaluation in Tabular Data Sufficiency

Tahmini Süre:2m 0s

Soru 205Soru →

A pharmaceutical laboratory manufactures a specialized compound by blending two liquid preparations, Solution $X$ and Solution $Y$ . Solution $X$ contains Active Ingredient $P$ and water in the ratio $3:2$ by weight. Solution $Y$ contains Active Ingredient $P$ and water in the ratio $1:4$ by weight. A master batch is created by mixing $x$ grams of Solution $X$ with $y$ grams of Solution $Y$ , where $x > 0$ and $y > 0$ . Is the percentage of Active Ingredient $P$ in the master batch greater than $40\%$ by weight?

(1) $3x - 4y > 0$
(2) After adding $50$ grams of pure water to the master batch, Active Ingredient $P$ accounts for less than $\frac{1}{3}$ of the total weight of the resulting mixture.

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

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

BOTH statements TOGETHER are sufficient, but neither statement ALONE is sufficient.

EACH statement ALONE is sufficient.

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

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 identifies that Statement (1) alone is sufficient while Statement (2) alone is not. By rephrasing the question stem, the target condition 'Is the active ingredient concentration greater than 40%?' simplifies directly to the inequality condition

x > y

. Statement (1) guarantees that

x > \frac{4}{3}y

, which directly implies

x > y

since

y > 0

, providing a definitive 'Yes'. Statement (2) simplifies to

2x - y < 125

, which is satisfied by parameter pairs where

x > y

as well as pairs where

x < y

, making it insufficient.

Adım Adım Çözüm

1

Rephrase the question stem target algebraically.

Solution X is

\frac{3}{3+2} = \frac{3}{5} = 0.60

(or

60\%

) Ingredient P by weight. Solution Y is

\frac{1}{1+4} = \frac{1}{5} = 0.20

(or

20\%

) Ingredient P by weight. The total weight of Ingredient P in the master batch is

0.6x + 0.2y

, and the total weight of the batch is

x + y

. The question asks whether

\frac{0.6x + 0.2y}{x + y} > 0.40

. Since

x > 0

and

y > 0

, multiplying across by

(x + y)

gives

0.6x + 0.2y > 0.4x + 0.4y \implies 0.2x > 0.2y \implies x > y

. Thus, the rephrased question is: 'Is

x > y

?'

Simplifying the question stem before evaluating statements prevents unnecessary system solving and clarifies the exact threshold needed for sufficiency.

2

Evaluate Statement (1):

3x - 4y > 0

.

Rearranging

3x - 4y > 0

gives

3x > 4y \implies x > \frac{4}{3}y

. Because

y > 0

,

\frac{4}{3}y > y

. Therefore, if

x > \frac{4}{3}y

, it must strictly be true that

x > y

. This yields a definitive 'Yes' to the rephrased question. Hence, Statement (1) ALONE is sufficient.

Since statement 1 establishes a lower bound for x relative to y that is strictly greater than 1y, it answers the question 'Is x > y?' definitively.

3

Evaluate Statement (2): Adding 50 grams of water results in P accounting for less than

\frac{1}{3}

of total weight.

Total weight of P remains

0.6x + 0.2y

. The new total weight is

x + y + 50

. The statement gives

\frac{0.6x + 0.2y}{x + y + 50} < \frac{1}{3}

. Multiplying by

3(x + y + 50)

yields

1.8x + 0.6y < x + y + 50 \implies 0.8x - 0.4y < 50 \implies 2x - y < 125

. Testing values:
- Case A: Let

x = 10, y = 50

. Then

2(10) - 50 = -30 < 125

holds. Here

x < y

, so the answer is 'No'.
- Case B: Let

x = 100, y = 90

. Then

2(100) - 90 = 110 < 125

holds. Here

x > y

, so the answer is 'Yes'.
Since Statement (2) permits both 'Yes' and 'No' outcomes, Statement (2) ALONE is not sufficient.

An inequality involving absolute scale (50 grams) fails to determine a purely proportional relationship between x and y without additional constraints on total batch size.

Anahtar Kavram

Question Stem Simplification and Mixture Ratio Inequalities

Soru 206Soru →

A coffee roaster creates a signature blend using only two types of beans: Arabica and Robusta. In a specific batch of this signature blend, what percentage of the total weight of the batch consists of Arabica beans?

(1) If 10 kilograms of Arabica beans were added to the batch, the ratio of the weight of Arabica beans to the weight of Robusta beans in the batch would be $3:2$ .
(2) The total cost of the Arabica beans in the batch is $50\%$ greater than the total cost of the Robusta beans in the batch, and Arabica beans cost $25\%$ more per kilogram than Robusta beans.

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

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

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

EACH statement ALONE is sufficient.

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

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.

Statement (2) alone allows us to express the price per kilogram of Arabica beans in terms of Robusta beans (

p_A = 1.25 p_R

) and set up an equation equating total costs (

A \cdot p_A = 1.50 R \cdot p_R

). Dividing out the unit price

p_R

gives a direct constant value for the weight ratio

\frac{A}{R} = \frac{6}{5}

, which uniquely determines the percentage of Arabica beans in the batch.

Adım Adım Çözüm

1

Define target variables and simplify the question stem.

Let

A

be the weight of Arabica beans (in kg) and

R

be the weight of Robusta beans (in kg). The target percentage is

\frac{A}{A + R} \times 100\%

, which requires finding the ratio

\frac{A}{R}

.

Rephrasing the stem to finding the ratio

\frac{A}{R}

isolates the exact relation needed to answer the question.

2

Evaluate Statement (1) independently.

Statement (1) gives

\frac{A + 10}{R} = \frac{3}{2} \Rightarrow 2A + 20 = 3R \Rightarrow 3R - 2A = 20

. This is one linear equation with two unknowns (

A

and

R

).

Since the ratio

\frac{A}{R}

varies depending on the specific values of

A

and

R

(e.g., if

A=2, R=8

,

\frac{A}{R}=\frac{1}{4}

; if

A=14, R=16

,

\frac{A}{R}=\frac{7}{8}

), Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Let

p_A

and

p_R

be the price per kg of Arabica and Robusta beans, respectively. We are given

p_A = 1.25 p_R = \frac{5}{4} p_R

. Total cost of Arabica =

A \cdot p_A

and total cost of Robusta =

R \cdot p_R

. Statement (2) states

A \cdot p_A = 1.50 (R \cdot p_R) = \frac{3}{2} R \cdot p_R

. Substituting

p_A = \frac{5}{4} p_R

gives

A \left(\frac{5}{4} p_R\right) = \frac{3}{2} R \cdot p_R

. Dividing both sides by

p_R

yields

\frac{5}{4} A = \frac{3}{2} R \Rightarrow \frac{A}{R} = \frac{3/2}{5/4} = \frac{6}{5}

.

Knowing

\frac{A}{R} = \frac{6}{5}

allows us to calculate

\frac{A}{A + R} = \frac{6}{6 + 5} = \frac{6}{11}

, giving a unique percentage of

\frac{600}{11}\% \approx 54.55\%

. Thus, Statement (2) alone IS sufficient.

Anahtar Kavram

Data Sufficiency Evaluation of Weighted Ratios and Multiplicative Pricing Relationships

Tahmini Süre:2m 0s

Soru 207Soru →

If $a$ and $b$ are real numbers such that $a \neq b$ , what is the value of $\frac{a + b}{a - b}$ ?

(1) $a^2 + b^2 = 4ab$
(2) $a > b > 0$

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

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

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

EACH statement ALONE is sufficient.

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

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) allows us to determine that the square of the target expression

(a+b)/(a-b)

equals 3, which gives two possible values:

\sqrt{3}

and

-\sqrt{3}

. Statement (2) provides the condition

a > b > 0

, ensuring that both

a+b

and

a-b

are positive, so their quotient must be positive. Combining both statements eliminates

-\sqrt{3}

, uniquely determining that the expression equals

\sqrt{3}

. Thus, both statements together are sufficient.

Adım Adım Çözüm

1

Express the square of the target expression in terms of basic algebraic components.

Let

E = \frac{a + b}{a - b}

. Squaring both sides yields

E^2 = \frac{(a + b)^2}{(a - b)^2} = \frac{a^2 + 2ab + b^2}{a^2 - 2ab + b^2}

.

Rewriting the ratio in squared form allows substitution of expressions involving

a^2 + b^2

and

ab

.

2

Evaluate Statement (1) independently.

Substitute

a^2 + b^2 = 4ab

into the squared ratio:

E^2 = \frac{4ab + 2ab}{4ab - 2ab} = \frac{6ab}{2ab} = 3

. Taking the square root gives

E = \sqrt{3}

or

E = -\sqrt{3}

.

Because Statement (1) allows two distinct possible values for

E

, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Statement (2) states

a > b > 0

. This implies

a + b > 0

and

a - b > 0

, so

E > 0

, but no numerical value is specified.

Without quantitative equations, Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) combined.

From Statement (1),

E = \pm\sqrt{3}

. From Statement (2),

E > 0

. Combining both rules out

-\sqrt{3}

, leaving uniquely

E = \sqrt{3}

.

The two statements together establish a single, unique value for the target expression.

Anahtar Kavram

Evaluating algebraic ratios via squared identities and applying inequality sign constraints to eliminate redundant roots in Data Sufficiency.

Tahmini Süre:1m 30s

Soru 208Soru →

In a GMAT Data Sufficiency problem asking for the unique value of an integer $x$ , Statement (1) establishes that $x \in \{3, 7\}$ and Statement (2) establishes that $|x - 5| = 2$ . Evaluating both statements together provides sufficient information to determine a unique value for $x$ .

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is False. Combining both statements yields the intersection set {3, 7}, which still contains two possible values for x and therefore does not uniquely determine x.

The statement is false because solving Statement (2)'s absolute value equation

|x - 5| = 2

produces

x = 7

or

x = 3

, which is identical to the candidate set given by Statement (1). Combining two identical solution sets leaves the candidate set as

\{3, 7\}

, which does not yield a single unique value for

x

.

Adım Adım Çözüm

1

Evaluate Statement (1) solution set

Statement (1) restricts

x

to the set

\{3, 7\}

.

Statement (1) explicitly provides two candidate values for

x

.

2

Evaluate Statement (2) solution set independently

Statement (2) yields

|x - 5| = 2 \implies x = 7

or

x = 3

. The solution set is

\{3, 7\}

.

Solving the absolute value equation yields the two roots 3 and 7 without using any information from Statement (1).

3

Combine Statement (1) and Statement (2)

The intersection of the two solution sets is

\{3, 7\} \cap \{3, 7\} = \{3, 7\}

.

When combining statements in Data Sufficiency, valid values must satisfy both statements simultaneously.

4

Determine sufficiency of the combined statements

Since

x

can still be either 3 or 7, a single unique value is not determined. The combined statements are insufficient.

A Value Data Sufficiency question requires a single, unique numerical value to be considered sufficient.

Anahtar Kavram

Statement Combination and Redundancy in Data Sufficiency

Soru 209Soru →

A commercial real estate firm leased office space across two properties, Property X and Property Y. The rental rate per square foot for space in Property X was $20\%$ higher than the rental rate per square foot for space in Property Y. Was the total rental revenue generated from Property X greater than the total rental revenue generated from Property Y?

(1) Property X accounted for more than $45\%$ of the total square feet of office space leased across both properties.
(2) Property Y accounted for less than $54\%$ of the total square feet of office space leased across both properties.

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

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

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

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

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

Cevap

Statement (2) ALONE is sufficient to answer the question with a definitive 'Yes', but Statement (1) ALONE is not sufficient.

The correct response identifies that Statement (2) alone is sufficient because it establishes that Property X's area share relative to Property Y yields a ratio

\frac{S_X}{S_Y} > \frac{23}{27}

. Because

\frac{23}{27} > \frac{5}{6}

, the revenue of Property X is guaranteed to exceed that of Property Y, providing a definitive 'Yes'. Statement (1) alone is insufficient because it permits ratios below

\frac{5}{6}

.

Adım Adım Çözüm

1

Rephrase the question stem in terms of algebraic variables.

Let

S_X

and

S_Y

be the square footage leased in Property X and Property Y, respectively, and let

r

be the rate per sq ft for Property Y. The rate for Property X is

1.20r

. Revenue from X is

1.20r S_X

and revenue from Y is

r S_Y

. The question asks whether

1.20r S_X > r S_Y

, which simplifies to asking whether

\frac{S_X}{S_Y} > \frac{1}{1.20} = \frac{5}{6}

.

Simplifying the target question into a ratio threshold

\frac{S_X}{S_Y} > \frac{5}{6} \approx 0.8333

streamlines the statement evaluation.

2

Evaluate Statement (1):

S_X > 0.45(S_X + S_Y)

.

0.55 S_X > 0.45 S_Y \implies \frac{S_X}{S_Y} > \frac{0.45}{0.55} = \frac{9}{11} \approx 0.8182

. Since

\frac{9}{11} < \frac{5}{6}

,

\frac{S_X}{S_Y}

could be

0.82

(giving a 'No' answer to the question) or

1.0

(giving a 'Yes' answer).

Statement (1) allows values both above and below the required threshold of

5/6

, making it insufficient.

3

Evaluate Statement (2):

S_Y < 0.54(S_X + S_Y)

.

0.46 S_Y < 0.54 S_X \implies \frac{S_X}{S_Y} > \frac{0.46}{0.54} = \frac{23}{27} \approx 0.8519

. Comparing

\frac{23}{27}

and

\frac{5}{6}

:

23 \times 6 = 138

and

27 \times 5 = 135

. Since

138 > 135

,

\frac{23}{27} > \frac{5}{6}

. Thus,

\frac{S_X}{S_Y}

must be strictly greater than

\frac{5}{6}

.

Statement (2) forces

\frac{S_X}{S_Y}

to be strictly greater than

5/6

, yielding a definitive 'Yes' to the question stem.

Anahtar Kavram

Question Stem Rephrasing and Threshold Analysis in Data Sufficiency Ratio Problems

Tahmini Süre:2m 0s

Soru 210Soru →

If $u$ and $v$ are non-zero real numbers, what is the value of $\frac{u - 2v}{u + v}$ ?

(1) $3u^2 - 5uv - 2v^2 = 0$
(2) $u > v > 0$

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

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

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

EACH statement ALONE is sufficient.

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

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 identifies that neither statement alone provides a unique value for the expression, but combining them eliminates the negative root case, leaving a unique value of 0 for the target expression.

Adım Adım Çözüm

1

Rephrase the target expression

Dividing numerator and denominator by

v

, the target expression is

\frac{\frac{u}{v} - 2}{\frac{u}{v} + 1}

. Finding a unique value for

\frac{u}{v}

determines the target expression.

Simplifying the question target clarifies what information is necessary to achieve sufficiency.

2

Evaluate Statement (1) independently

Factor

3u^2 - 5uv - 2v^2 = 0

as

(3u + v)(u - 2v) = 0

. This gives

3u + v = 0 \implies u = -\frac{v}{3}

or

u - 2v = 0 \implies u = 2v

. If

u = 2v

, then

\frac{u - 2v}{u + v} = \frac{0}{3v} = 0

. If

u = -\frac{v}{3}

, then

\frac{u - 2v}{u + v} = \frac{-\frac{v}{3} - 2v}{-\frac{v}{3} + v} = \frac{-\frac{7}{3}v}{\frac{2}{3}v} = -\frac{7}{2}

. Two different values are possible.

Since Statement (1) produces two distinct numerical outcomes, it is not sufficient alone.

3

Evaluate Statement (2) independently

Statement (2) states

u > v > 0

. This indicates that both

u

and

v

are positive, but gives no fixed algebraic equality for

\frac{u}{v}

.

Infinitely many positive pairs

(u, v)

satisfy

u > v > 0

while yielding different values for the expression. Statement (2) is not sufficient alone.

4

Evaluate Statements (1) and (2) combined

From Statement (2),

u > 0

and

v > 0

, so

3u + v > 0

. Therefore, the factor

3u + v = 0

is impossible. This leaves

u - 2v = 0

as the only valid relation, so

u = 2v

. Substituting

u = 2v

yields

\frac{2v - 2v}{2v + v} = 0

.

Combining the inequality constraint with the quadratic factorization uniquely determines the value of the target expression.

Anahtar Kavram

Algebraic Equations and Systems in Data Sufficiency

Tahmini Süre:2m 0s

Soru 211Soru →

An investment portfolio consists of three funds: Fund Alpha, Fund Beta, and Fund Gamma. The table below presents the capital committed (in millions of dollars) and the annual yield percentage for each fund during the past fiscal year:

Fund	Capital Committed ($ millions)	Annual Yield (%)
Fund Alpha	40	8%
Fund Beta	60	12%
Fund Gamma	$c$	$y\%$

What was the overall annual yield percentage for the combined portfolio of all three funds?

(1) The capital committed to Fund Gamma, $c$ , was equal to the total capital committed to Fund Alpha and Fund Beta combined.
(2) The annual yield percentage for Fund Gamma, $y\%$ , was equal to the weighted average annual yield percentage of Fund Alpha and Fund Beta combined.

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

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

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

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

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

Cevap

Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not sufficient.

The option stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient is correct. Fund Alpha and Fund Beta combined have a weighted average yield of

\frac{40(8\%) + 60(12\%)}{100} = 10.4\%

. Statement (2) tells us that Fund Gamma's yield is also

10.4\%

. Combining any two groups with identical average yields results in an overall average yield equal to

10.4\%

, regardless of the amount of capital

c

in Fund Gamma. Statement (1) provides only the capital amount

c = 100

, leaving the yield

y

unknown and thus insufficient.

Adım Adım Çözüm

1

Formulate the algebraic expression for the overall portfolio yield.

Overall Portfolio Yield =

\frac{40(8\%) + 60(12\%) + c(y\%)}{40 + 60 + c} = \frac{3.2 + 7.2 + \frac{cy}{100}}{100 + c} = \frac{10.4 + \frac{cy}{100}}{100 + c}

The total annual return is the sum of returns from each fund, divided by total capital committed.

2

Calculate the combined weighted average yield of Fund Alpha and Fund Beta.

Sub-portfolio yield =

\frac{40(8) + 60(12)}{40 + 60} = \frac{320 + 720}{100} = 10.4\%

Determining the performance of the known portion of the portfolio simplifies statement evaluation.

3

Evaluate Statement (1) independently:

c = 40 + 60 = 100

.

Overall Yield =

\frac{10.4 + y}{200} \times 100\% = \frac{10.4 + y}{2}\%

. Since

y

is unknown, this value is not unique.

Without knowing Fund Gamma's yield rate

y\%

, the capital amount alone does not provide a definitive overall yield.

4

Evaluate Statement (2) independently:

y\% = 10.4\%

.

Overall Yield =

\frac{100(10.4\%) + c(10.4\%)}{100 + c} = \frac{10.4\%(100 + c)}{100 + c} = 10.4\%

.

When a new component is added to a group with a value equal to the group's current average, the overall average remains unchanged regardless of the size (

c

) of the new component.

Anahtar Kavram

Weighted Average Invariance in Data Sufficiency

Soru 212Soru →

An investment fund allocates its capital between two asset classes, Class $P$ and Class $Q$ . At the beginning of a given year, the ratio of capital in Class $P$ to capital in Class $Q$ was $3 : 2$ . During the year, Class $P$ earned an annual return of $r_p\%$ and Class $Q$ earned an annual return of $r_q\%$ . At the end of the year, a management fee of $2\%$ was deducted from the total ending value of Class $P$ , and a management fee of $5\%$ was deducted from the total ending value of Class $Q$ . Was the total dollar amount of the management fee deducted from Class $P$ greater than the total dollar amount of the management fee deducted from Class $Q$ ?

(1) $r_p - r_q = 10$
(2) At the end of the year, before management fees were deducted, the total value of Class $P$ was $60\%$ greater than the total value of Class $Q$ .

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

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

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

EACH statement ALONE is sufficient.

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

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.

Statement (2) alone allows us to express both fee amounts in terms of the single variable

V_Q

, the ending value of Class

Q

. Because

F_P = 0.02(1.6 V_Q) = 0.032 V_Q

and

F_Q = 0.05 V_Q

, we know with certainty that

F_P < F_Q

. This provides a definitive 'No' answer to whether

F_P > F_Q

, which makes Statement (2) alone sufficient. Statement (1) alone leaves the relationship dependent on the unconstrained value of

r_q

, so it is not sufficient.

Adım Adım Çözüm

1

Algebraically rephrase the question target.

Let initial capital be

P_0

and

Q_0

, where

P_0 = 1.5 Q_0

. Ending values before fees are

V_P = 1.5 Q_0 (1 + \frac{r_p}{100})

and

V_Q = Q_0 (1 + \frac{r_q}{100})

. The fees are

F_P = 0.02 V_P = 0.03 Q_0 (1 + \frac{r_p}{100})

and

F_Q = 0.05 V_Q = 0.05 Q_0 (1 + \frac{r_q}{100})

. The question 'Is

F_P > F_Q

?' simplifies to 'Is

0.03(100 + r_p) > 0.05(100 + r_q)

?' or 'Is

3 r_p - 5 r_q > 200

?'

Rephrased targets clarify the exact mathematical relationship required to answer the Data Sufficiency prompt.

2

Evaluate Statement (1):

r_p - r_q = 10

.

Substituting

r_p = r_q + 10

into the inequality yields

3(r_q + 10) - 5 r_q = 30 - 2 r_q > 200

, which simplifies to

r_q < -85

. Since

r_q

can be any real number (e.g.,

r_q = 0

yields 'No', while

r_q = -100

yields 'Yes'), Statement (1) alone is NOT sufficient.

Without specific values for

r_q

, the inequality cannot be answered definitively.

3

Evaluate Statement (2):

V_P = 1.6 V_Q

.

We are given that

V_P = 1.6 V_Q

. Substituting this directly into the fee expressions gives

F_P = 0.02 V_P = 0.02 (1.6 V_Q) = 0.032 V_Q

. Since

F_Q = 0.05 V_Q

and portfolio value

V_Q > 0

,

0.032 V_Q

is strictly less than

0.05 V_Q

. Thus,

F_P > F_Q

is definitively FALSE.

A definitive 'No' answer is sufficient in Data Sufficiency decision logic.

Anahtar Kavram

Data Sufficiency Yes/No decision logic and algebraic target rephrasing for percentage adjustments and fee calculations.

Tahmini Süre:2m 0s

Soru 213Soru →

If $a$ and $b$ are positive integers, is $a^2b$ divisible by $12$ ?

(1) $ab^2$ is divisible by $18$ .
(2) $a^3b$ is divisible by $72$ .

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

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

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

EACH statement ALONE is sufficient.

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

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.

Statement (2) alone guarantees that

a^3b

contains at least three factors of

2

and two factors of

3

. Analyzing the exponent bounds for non-negative integers proves that

a^2b

must contain at least two factors of

2

and one factor of

3

, ensuring

a^2b

is divisible by

12

. Statement (1) alone is insufficient because

a=3, b=6

makes

ab^2=108

(divisible by

18

) but

a^2b=54

(not divisible by

12

). Thus, Statement (2) ALONE is sufficient.

Adım Adım Çözüm

1

Rephrase the target question in terms of prime factorization

For

a^2b

to be divisible by

12 = 2^2 \times 3^1

, we need

2 v_2(a) + v_2(b) \ge 2

and

2 v_3(a) + v_3(b) \ge 1

, where

v_p(n)

is the exponent of prime

p

in the prime factorization of

n

.

Decomposing divisibility into prime factor exponent inequalities allows definitive evaluation of sufficiency.

2

Evaluate Statement (1):

ab^2

is divisible by

18 = 2^1 \times 3^2

This gives

v_2(a) + 2 v_2(b) \ge 1

and

v_3(a) + 2 v_3(b) \ge 2

. Test counterexample

a=3, b=6

:

ab^2 = 3 \times 36 = 108

(divisible by 18), but

a^2b = 9 \times 6 = 54

, which is not divisible by 12.

A single valid counterexample proves Statement (1) is NOT sufficient.

3

Evaluate Statement (2):

a^3b

is divisible by

72 = 2^3 \times 3^2

This gives

3 v_2(a) + v_2(b) \ge 3

and

3 v_3(a) + v_3(b) \ge 2

. If

v_2(a) = 0

, then

v_2(b) \ge 3 \implies 2 v_2(a) + v_2(b) \ge 3 \ge 2

. If

v_2(a) \ge 1

, then

2 v_2(a) + v_2(b) \ge 2(1) + 0 = 2

. Similarly for prime 3: if

v_3(a) = 0

,

v_3(b) \ge 2 \implies 2 v_3(a) + v_3(b) \ge 2 \ge 1

; if

v_3(a) \ge 1

,

2 v_3(a) + v_3(b) \ge 2 \ge 1

. Thus

a^2b

is always divisible by 12.

Statement (2) strictly guarantees that the prime factor counts for 2 and 3 in

a^2b

meet or exceed the required thresholds, yielding a definitive 'Yes'.

Anahtar Kavram

Divisibility analysis using prime factor exponent inequalities in Data Sufficiency.

Soru 214Soru →

A retailer sold standard boxes at $\$s$ each and premium boxes at $\$p$ each yesterday, generating a total revenue of $\$900$ . If the retailer sold $x$ standard boxes and $y$ premium boxes, where $x$ and $y$ are positive integers, did the retailer sell more standard boxes than premium boxes?

(1) Standard boxes cost $\$15$ each and premium boxes cost $\$25$ each.
(2) The average (arithmetic mean) price of all boxes sold yesterday was $\$18$ , and premium boxes cost $\$10$ more per box than standard boxes.

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

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

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

EACH statement ALONE is sufficient.

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

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 one stating that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) allows multiple positive integer pairs for

(x, y)

such that

x > y

in some cases and

x < y

in others, making it insufficient on its own. Statement (2) alone leaves the base price

s

unknown, so the ratio of standard to premium boxes cannot be determined without Statement (1). Combining

s = 15

from Statement (1) with Statement (2) establishes that

\frac{x}{y} = \frac{7}{3}

, which definitively proves

x > y

.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) provides

s = 15

and

p = 25

, yielding the equation

15x + 25y = 900

, which simplifies to

3x + 5y = 180

. Testing positive integer solutions gives multiple valid pairs: for instance,

(x, y) = (55, 3)

yields

x > y

(Yes), while

(x, y) = (10, 30)

yields

x < y

(No). Because both Yes and No are possible, Statement (1) alone is NOT sufficient.

A statement must yield a single definitive Yes or No answer to be sufficient.

2

Evaluate Statement (2) independently without using information from Statement (1).

Statement (2) states that

p = s + 10

and the average price is

\frac{sx + py}{x + y} = 18

. Substituting

p = s + 10

gives

\frac{sx + (s + 10)y}{x + y} = 18

, which simplifies to

s + \frac{10y}{x + y} = 18

. Because the individual box price

s

is unknown,

\frac{y}{x + y}

can vary depending on

s

(e.g., if

s = 10

,

x < y

; if

s = 15

,

x > y

). Thus, Statement (2) alone is NOT sufficient.

Information from Statement (1) must never be carried over when testing Statement (2) independently.

3

Combine Statements (1) and (2).

From Statement (1),

s = 15

. Substituting

s = 15

into the simplified average equation from Statement (2),

15 + \frac{10y}{x + y} = 18

, gives

\frac{10y}{x + y} = 3 \implies 10y = 3x + 3y \implies 7y = 3x \implies \frac{x}{y} = \frac{7}{3}

. Since

x

and

y

are positive integers,

\frac{x}{y} > 1

, which definitively proves that

x > y

(Yes).

Combining the known price from Statement (1) with the weighted average equation from Statement (2) uniquely determines the ratio of

x

to

y

.

Anahtar Kavram

Statement Independence and Statement Combination in Data Sufficiency

Soru 215Soru →

If $a$ and $b$ are real numbers, what is the value of $a^2 - b^2$ ?

(1) $a + b = 5$
(2) $(a - b)^2 = 9$

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

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

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

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

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

Cevap

The statements together are not sufficient to determine a unique value for

a^2 - b^2

, because

a^2 - b^2

can equal either

15

or

-15

.

The correct option states that both statements together are not sufficient. Factoring the target expression yields

a^2 - b^2 = (a+b)(a-b)

. Statement (1) tells us

a+b = 5

, and Statement (2) tells us

a-b = \pm 3

. Combining them yields two possible numerical results (

15

and

-15

). Since Data Sufficiency requires a single, unique numerical value, the information remains insufficient.

Adım Adım Çözüm

1

Rephrase the target expression using algebraic identities.

The target expression

a^2 - b^2

factors into

(a + b)(a - b)

. To find a unique value, we need a unique value for the product

(a + b)(a - b)

.

Factoring highlights the required components: the sum

(a + b)

and the difference

(a - b)

.

2

Evaluate Statement (1) independently.

Statement (1) gives

a + b = 5

. However, the value of

a - b

is completely unknown.

Since

(a - b)

can be any real number,

a^2 - b^2 = 5(a - b)

can take infinitely many values. Statement (1) is not sufficient.

3

Evaluate Statement (2) independently.

Statement (2) gives

(a - b)^2 = 9

, which implies

a - b = 3

or

a - b = -3

.

The sum

a + b

is completely unknown, and

a - b

has two potential values. Statement (2) is not sufficient.

4

Evaluate Statements (1) and (2) together.

From (1),

a + b = 5

. From (2),

a - b = 3

or

a - b = -3

.
If

a - b = 3

, then

a^2 - b^2 = (5)(3) = 15

.
If

a - b = -3

, then

a^2 - b^2 = (5)(-3) = -15

.

Because there are two distinct outcomes (

15

and

-15

), a unique value cannot be determined. Therefore, both statements together are not sufficient.

Anahtar Kavram

Quadratic non-linearity and root ambiguity in Data Sufficiency systems

Soru 216Soru →

If $x$ and $y$ are non-zero real numbers, is $\frac{x}{y}$ an integer?

(1) $x^2 + y^2 = 5xy$
(2) $x$ is a prime number and $xy$ is an integer.

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

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

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

EACH statement ALONE is sufficient.

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

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) allows us to set up a quadratic equation for the ratio

\frac{x}{y}

, yielding

\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 1 = 0

. The roots of this quadratic equation are

\frac{5 \pm \sqrt{21}}{2}

, which are irrational numbers. Therefore,

\frac{x}{y}

cannot be an integer under any circumstance, providing a definitive 'No' answer to the question. Statement (1) is therefore sufficient. Statement (2) allows

\frac{x}{y}

to be an integer (e.g.,

x=3, y=1

) or a non-integer (e.g.,

x=3, y=2

), so it is not sufficient. Thus, the option stating that Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient is the correct choice.

Adım Adım Çözüm

1

Analyze the target question stem

The target asks if the quotient

\frac{x}{y}

is an integer, where

x

and

y

are real numbers (not restricted to integers).

Establishing that

x

and

y

are real numbers prevents making unjustified integer assumptions.

2

Evaluate Statement (1):

x^2 + y^2 = 5xy

Divide both sides by

y^2

(since

y \neq 0

):

\left(\frac{x}{y}\right)^2 + 1 = 5\left(\frac{x}{y}\right)

, which rearranges to

\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 1 = 0

. Setting

k = \frac{x}{y}

, we get

k^2 - 5k + 1 = 0

. Solving for

k

using the quadratic formula gives

k = \frac{5 \pm \sqrt{21}}{2}

.

Since

\sqrt{21}

is irrational,

k = \frac{x}{y}

is an irrational number and can NEVER be an integer.

3

Determine sufficiency for Statement (1)

Statement (1) yields a definitive 'No' to the question 'Is

\frac{x}{y}

an integer?'. Thus, Statement (1) alone is SUFFICIENT.

In Data Sufficiency Yes/No questions, a definitive 'No' answer is a sufficient result.

4

Evaluate Statement (2):

x

is a prime number and

xy

is an integer

Case A: Let

x = 3

and

y = 1

. Then

xy = 3

(an integer), and

\frac{x}{y} = 3

(an integer) -> YES.
Case B: Let

x = 3

and

y = 2

. Then

xy = 6

(an integer), and

\frac{x}{y} = \frac{3}{2}

(not an integer) -> NO.

Because

\frac{x}{y}

can be an integer or not an integer, Statement (2) alone is NOT sufficient.

Anahtar Kavram

Data Sufficiency Yes/No decision logic combined with irrational root analysis and real number constraints.

Tahmini Süre:2m 0s

Soru 217Soru →

A logistics center operates two automated sorting divisions: Division $P$ and Division $Q$ . Yesterday, Division $P$ had an error rate of $x\%$ of the packages it processed, and Division $Q$ had an error rate of $y\%$ of the packages it processed. What was the overall package error rate for the two divisions combined yesterday?

(1) Division $P$ processed $50\%$ more packages yesterday than Division $Q$ processed yesterday, and Division $P$ 's error rate was $2.0\%$ .
(2) Yesterday, Division $Q$ processed $40\%$ of the total packages processed by both divisions combined, and Division $Q$ 's error rate was $5.0\%$ .

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

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

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

EACH statement ALONE is sufficient.

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

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 the option stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) provides the relative volume weights and Division

P

's error rate, but lacks Division

Q

's error rate. Statement (2) provides the relative volume weights and Division

Q

's error rate, but lacks Division

P

's error rate. Combining both statements supplies all necessary values (

x = 2.0\%

,

y = 5.0\%

, and volume ratio

60:40

) to uniquely calculate the combined weighted error rate of

3.2\%

.

Adım Adım Çözüm

1

Formulate the algebraic target for the overall combined error rate.

Let

N_P

and

N_Q

represent the package volumes processed by Division

P

and Division

Q

, respectively. The overall error rate

E

is given by the weighted average equation:

E = \frac{x\cdot N_P + y\cdot N_Q}{N_P + N_Q} = x\left(\frac{N_P}{N_P + N_Q}\right) + y\left(\frac{N_Q}{N_P + N_Q}\right)

. To find

E

, we need

x

,

y

, and the relative weight ratio

\frac{N_P}{N_Q}

(or the fraction of total volume contributed by each division).

Rephrasing the Data Sufficiency question stem shows that absolute counts for

N_P

and

N_Q

are unnecessary; only the error rates

x

and

y

and their relative proportions are required.

2

Evaluate Statement (1) independently.

Statement (1) states that

N_P = 1.5 N_Q

, which means

\frac{N_P}{N_Q} = \frac{3}{2}

and the relative weights are

\frac{N_P}{N_P + N_Q} = 0.60

and

\frac{N_Q}{N_P + N_Q} = 0.40

. It also gives

x = 2.0\%

. However, no information is provided about Division

Q

's error rate (

y

). Thus,

E = 0.60(2.0\%) + 0.40(y\%)

, which varies depending on

y

. Statement (1) alone is INSUFFICIENT.

Without

y

, a unique numerical value for

E

cannot be calculated.

3

Evaluate Statement (2) independently.

Statement (2) states that Division

Q

processed

40\%

of the total packages, meaning

\frac{N_Q}{N_P + N_Q} = 0.40

and

\frac{N_P}{N_P + N_Q} = 0.60

. It also provides

y = 5.0\%

. However, no information is provided about Division

P

's error rate (

x

). Thus,

E = 0.60(x\%) + 0.40(5.0\%)

, which varies depending on

x

. Statement (2) alone is INSUFFICIENT.

Without

x

, a unique numerical value for

E

cannot be calculated.

4

Evaluate Statements (1) and (2) together.

Combining both statements gives

x = 2.0\%

,

y = 5.0\%

, and consistent relative weights (

\frac{N_P}{N_P + N_Q} = 0.60

and

\frac{N_Q}{N_P + N_Q} = 0.40

). The overall combined error rate can be computed directly:

E = 0.60(2.0\%) + 0.40(5.0\%) = 1.2\% + 2.0\% = 3.2\%

. A unique value is obtained. Both statements together are SUFFICIENT.

All required variables (

x

,

y

, and relative volume weighting) are known when combining both statements.

Anahtar Kavram

Weighted Averages and Ratio Sufficiency in Data Sufficiency

Tahmini Süre:2m 0s

Soru 218Soru →

A private equity firm allocated its initial capital between two portfolio ventures: Venture X and Venture Y. In 2025, Venture X yielded a profit equal to $20\%$ of its initial investment, while Venture Y incurred a loss equal to $10\%$ of its initial investment. What was the firm's overall percentage profit or loss across both ventures combined?

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

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

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

EACH statement ALONE is sufficient.

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

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 target shows that the overall percentage return depends strictly on the ratio of the investment amounts,

\frac{X}{Y}

. Statement (1) directly gives

X = 1.5Y

, establishing a fixed ratio

\frac{X}{Y} = \frac{3}{2}

, which uniquely yields an overall profit of

8\%

. Statement (2) reveals the dollar investment in Venture X (

200,000

) but leaves the investment in Venture Y completely unconstrained, making it impossible to compute a unique overall percentage.

Adım Adım Çözüm

1

Rephrase the target question algebraically.

Let

X

be the initial investment in Venture X and

Y

be the initial investment in Venture Y. The total net profit is

0.20X - 0.10Y

, and the total investment is

X + Y

. The overall percentage return is given by

\frac{0.20X - 0.10Y}{X + Y} \times 100\% = \left(0.20 \left(\frac{X}{X+Y}\right) - 0.10 \left(\frac{Y}{X+Y}\right)\right) \times 100\%

. Thus, finding the ratio

\frac{X}{Y}

is sufficient to determine the overall return.

Data Sufficiency stems asking for a combined percentage rate depend only on the ratio of the component base values, not their absolute amounts.

2

Evaluate Statement (1): The amount invested in Venture X was

50\%

greater than the amount invested in Venture Y.

This implies

X = 1.5Y = \frac{3}{2}Y

, so

\frac{X}{Y} = \frac{3}{2}

. Substituting

X = 1.5Y

into the overall return formula yields

\frac{0.20(1.5Y) - 0.10Y}{1.5Y + Y} = \frac{0.30Y - 0.10Y}{2.5Y} = \frac{0.20Y}{2.5Y} = \frac{0.20}{2.5} = 0.08 = 8\%

profit.

Since a unique overall percentage (8% profit) is determined, Statement (1) alone is sufficient.

3

Evaluate Statement (2): The dollar amount of profit generated by Venture X was

\$40,000

.

This gives

0.20X = 40,000 \implies X = 200,000

. However, no information is provided about the value of

Y

. If

Y = 100,000

, overall profit is

\frac{40,000 - 10,000}{300,000} = 10\%

. If

Y = 400,000

, overall profit is

\frac{40,000 - 40,000}{600,000} = 0\%

.

Multiple overall percentage returns are possible depending on

Y

, so Statement (2) alone is not sufficient.

Anahtar Kavram

Weighted Average Percentage Yields and Stem Simplification in Data Sufficiency

Tahmini Süre:2m 0s

Soru 219Soru →

If $m$ and $n$ are real numbers, what is the value of $m + n$ ?

(1) $m^2 - n^2 = 0$
(2) $m^2 + n^2 = 50$

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

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

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

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

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

Cevap

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

Evaluating both statements together yields four possible coordinate pairs:

(5, 5)

,

(-5, -5)

,

(5, -5)

, and

(-5, 5)

. The sum

m + n

can equal

10

,

-10

, or

0

. Because a Data Sufficiency question requires a single unique value to be deemed sufficient, having three distinct possible values means the statements together are insufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

m^2 - n^2 = 0 \implies (m - n)(m + n) = 0 \implies m = n

or

m = -n

.

If

m = -n

, then

m + n = 0

. However, if

m = n

, then

m + n = 2m

, which varies. Since we cannot determine a single numerical value for

m + n

, Statement (1) alone is NOT sufficient.

2

Evaluate Statement (2) independently.

m^2 + n^2 = 50

.

Different pairs

(m, n)

satisfy

m^2 + n^2 = 50

. For example, if

(m, n) = (5, 5)

, then

m + n = 10

. If

(m, n) = (7, 1)

, then

m + n = 8

. Since

m + n

is not uniquely determined, Statement (2) alone is NOT sufficient.

3

Evaluate Statements (1) and (2) combined.

From (1),

m^2 = n^2

. Substitute into (2):

n^2 + n^2 = 50 \implies 2n^2 = 50 \implies n^2 = 25 \implies n = 5

or

n = -5

.

Since

m^2 = 25

,

m

can also be

5

or

-5

. The possible ordered pairs

(m, n)

are

(5, 5)

,

(-5, -5)

,

(5, -5)

, and

(-5, 5)

. Calculating

m + n

for these pairs gives

10

,

-10

, and

0

. Because multiple outcomes are possible, the combined statements are NOT sufficient.

Anahtar Kavram

Non-Linear Systems and Multiple Solution Ambiguity in Data Sufficiency

Soru 220Soru →

A manufacturing plant produces two types of electronic components: Component A and Component B. In Year 1, the unit production cost of Component A was $20\%$ greater than the unit production cost of Component B. In Year 2, the unit production cost of Component A increased by $10\%$ , while the unit production cost of Component B increased by $25\%$ . Was the overall average unit production cost across all components produced by the plant higher in Year 2 than in Year 1?

(1) In Year 2, the plant produced $50\%$ more units of Component B than it did in Year 1.
(2) In Year 1, Component A accounted for $60\%$ of the total number of components produced by the plant.

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

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

BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient.

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient to answer the question with a definitive 'Yes'.

The correct answer is the option stating that EACH statement ALONE is sufficient. Analyzing the range of possible weighted averages directly from the stem reveals that the minimum possible average unit cost in Year 2 (

1.25c

) is strictly greater than the maximum possible average unit cost in Year 1 (

1.20c

). Thus, the average unit cost in Year 2 must be greater than in Year 1 regardless of the proportion of Component A and Component B produced in either year. Since the stem alone proves a definitive 'Yes', each statement alone is sufficient.

Adım Adım Çözüm

1

Define variables for Year 1 unit costs and establish the range of Year 1 overall average cost.

Let the Year 1 unit cost of Component B be

c

, where

c > 0

. Then the Year 1 unit cost of Component A is

1.20c

. Any weighted average of these two costs in Year 1,

\text{Avg}_1

, must satisfy

c \le \text{Avg}_1 \le 1.20c

. In particular,

\text{Avg}_1 \le 1.20c

.

The overall weighted average of a set of values cannot exceed the maximum individual value in that set.

2

Calculate Year 2 unit costs and establish the range of Year 2 overall average cost.

In Year 2, the unit cost of Component A becomes

1.20c \times 1.10 = 1.32c

. The unit cost of Component B becomes

c \times 1.25 = 1.25c

. Any weighted average in Year 2,

\text{Avg}_2

, must satisfy

1.25c \le \text{Avg}_2 \le 1.32c

. In particular,

\text{Avg}_2 \ge 1.25c

.

The overall weighted average of a set of values cannot be less than the minimum individual value in that set.

3

Compare the upper bound of Year 1 average cost with the lower bound of Year 2 average cost.

Since

\text{Avg}_2 \ge 1.25c > 1.20c \ge \text{Avg}_1

, it follows that

\text{Avg}_2 > \text{Avg}_1

holds unconditionally for all possible production volumes and ratios in both years.

The minimum possible average cost in Year 2 (

1.25c

) is strictly greater than the maximum possible average cost in Year 1 (

1.20c

).

4

Evaluate Data Sufficiency statements based on the stem analysis.

Because the stem alone yields a definitive 'Yes' answer, Statement (1) alone is sufficient and Statement (2) alone is sufficient.

When the question stem itself contains sufficient information to answer the target question definitively, each statement independently yields a definitive answer.

Anahtar Kavram

Weighted Average Extreme Value Bounds in Data Sufficiency

Data Sufficiency

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