Tüm alıştırma soruları

Soru 1461Soru →

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

(1) $x^3 - y^3 = 26$ and $x^2 + xy + y^2 = 13$
(2) $x + y = 4$

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.

Factoring the target expression gives

x^2 - y^2 = (x + y)(x - y)

. Statement (1) simplifies via the difference of cubes formula

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

to

x - y = 2

and

xy = 3

. However, solving for

x + y

yields

(x + y)^2 = 16

, which gives two solutions (

x + y = 4

or

x + y = -4

) and thus two values for

x^2 - y^2

(

8

or

-8

). Thus, Statement (1) alone is insufficient. Statement (2) gives

x + y = 4

without constraining

x - y

, making it insufficient alone. Combining both statements provides

x + y = 4

and

x - y = 2

, giving a unique product of

8

. Hence, both statements together are sufficient.

Adım Adım Çözüm

1

Rephrase the target expression using algebraic identities

x^2 - y^2 = (x + y)(x - y)

Determining

x^2 - y^2

requires knowing either the product

(x + y)(x - y)

or the individual values of

x + y

and

x - y

.

2

Evaluate Statement (1) independently

Using the difference of cubes identity

x^3 - y^3 = (x - y)(x^2 + xy + y^2)

, substitute the given values:

26 = (x - y)(13) \implies x - y = 2

. Next, expand

(x - y)^2 = x^2 - 2xy + y^2 = 4

. Subtracting this from

x^2 + xy + y^2 = 13

gives

3xy = 9 \implies xy = 3

. Then

(x + y)^2 = (x - y)^2 + 4xy = 4 + 4(3) = 16

, which implies

x + y = 4

or

x + y = -4

. Thus

x^2 - y^2 = (4)(2) = 8

or

(-4)(2) = -8

.

Because Statement (1) produces two distinct possible values (

8

and

-8

), Statement (1) ALONE is NOT sufficient.

3

Evaluate Statement (2) independently

x + y = 4

Without any information about

x - y

,

x^2 - y^2 = 4(x - y)

can take infinitely many values. Statement (2) ALONE is NOT sufficient.

4

Evaluate Statements (1) and (2) combined

From Statement (1),

x - y = 2

. From Statement (2),

x + y = 4

. Multiplying these equations yields

x^2 - y^2 = (x + y)(x - y) = (4)(2) = 8

.

A single, unique numerical value is determined. Therefore, BOTH statements TOGETHER are sufficient.

Anahtar Kavram

Algebraic Factoring and Non-Linear System Ambiguity in Data Sufficiency

Tahmini Süre:2m 0s

Soru 1462Soru →

In 2025, a commercial fruit orchard replaced its traditional synthetic chemical pesticides with an organic biological control system that introduced beneficial predatory insects. During the subsequent harvest season, field inspections revealed that the total population of fruit-damaging pests in the orchard was 30 percent higher than in the previous year. Nevertheless, the proportion of harvested fruit that qualified for premium Grade-A market classification was significantly higher than in any preceding year.

Which of the following, if true, most helps to resolve the apparent discrepancy described above?

The purchasing and release costs of the predatory insects were lower than the annual expense of purchasing and spraying synthetic chemical pesticides.

Consumer demand for organically cultivated fruit has risen sharply, enabling orchards to sell Grade-A produce at a higher retail markup.

The predatory insects specifically suppressed pest larvae that cause internal core rot, leaving pests to cause only light skin blemishes, whereas the chemical pesticides previously used routinely left chemical residue that automatically disqualified fruit from Grade-A status.

Pest populations in neighboring fruit orchards that continued using synthetic chemical pesticides sustained even severe crop damage due to weather conditions.

The total weight of all fruit harvested from the orchard was lower in 2025 than in prior years due to unseasonably late spring frosts.

Cevabı ve açıklamayı göster

Cevap: The predatory insects specifically suppressed pest larvae that cause internal core rot, leaving pests to cause only light skin blemishes, whereas the chemical pesticides previously used routinely left chemical residue that automatically disqualified fruit from Grade-A status.

Cevap

The apparent paradox is resolved by the fact that predatory insects limited pest damage to superficial surface blemishes while eliminating the chemical residues from synthetic sprays that previously triggered automatic Grade-A disqualifications.

The correct answer successfully bridges the gap between increased pest counts and higher Grade-A fruit proportions. It shows that while overall pest numbers rose, the damage caused was restricted to minor surface blemishes that did not fail Grade-A standards. Simultaneously, removing synthetic pesticides eliminated chemical residue disqualifications, boosting the net proportion of fruit achieving Grade-A status.

Adım Adım Çözüm

1

Identify the two contradictory facts in the stimulus.

Fact 1: Total population of fruit-damaging pests increased by 30%. Fact 2: The proportion of harvested fruit qualifying for premium Grade-A classification was significantly higher than in prior years.

Resolving a discrepancy requires finding an underlying cause that allows both statements to be simultaneously true.

2

Evaluate the mechanism provided by each answer choice.

The correct choice explains that predatory insects altered the nature of pest damage (shifting it from internal rot to minor cosmetic marks) while removing synthetic chemical residue, which was a major cause of Grade-A disqualification in previous years.

This dual mechanism accounts for both the survival of more pests and the increased rate of Grade-A qualification.

Anahtar Kavram

Resolving Paradoxes and Discrepancies

Soru 1463Soru →

If $r$ is a real number, is $r^3 < r$ ?

(1) $|r - 0.5| < 0.5$
(2) $|r| < 1$

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

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

r^3 < r

shows that the inequality holds true if and only if

r < -1

or

0 < r < 1

. Statement (1) bounds

r

precisely within the interval

(0, 1)

, ensuring a definitive 'Yes' answer. Statement (2) defines the range

(-1, 1)

, which includes values where the inequality holds (such as

0.5

) and values where it fails (such as

-0.5

), making it insufficient.

Adım Adım Çözüm

1

Rephrase the target inequality algebraically.

The inequality

r^3 < r

rearranges to

r^3 - r < 0

, which factors as

r(r - 1)(r + 1) < 0

.

Simplifying the stem isolates the specific intervals on the number line where the statement holds true.

2

Determine the valid number ranges for the rephrased question.

The expression

r(r - 1)(r + 1)

is strictly negative when

r < -1

or when

0 < r < 1

.

Testing sign changes across key boundary points (

r = -1

,

r = 0

,

r = 1

) identifies the target ranges.

3

Evaluate Statement (1):

|r - 0.5| < 0.5

.

This absolute value inequality expands to

-0.5 < r - 0.5 < 0.5

, which simplifies to

0 < r < 1

.

Since every value in the range

0 < r < 1

satisfies

r^3 < r

, Statement (1) yields a definitive 'Yes'. Thus, Statement (1) alone is sufficient.

4

Evaluate Statement (2):

|r| < 1

.

This absolute value inequality simplifies to

-1 < r < 1

.

If

r = 0.5

, then

r^3 = 0.125 < 0.5

(Yes). However, if

r = -0.5

, then

r^3 = -0.125 > -0.5

(No). Because Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient.

Anahtar Kavram

Data Sufficiency evaluation of cubic inequalities and absolute value range constraints

Soru 1464Soru →

Each of the $100$ employees at Company K works in Division X, Division Y, or both divisions. The arithmetic mean age of the employees in Division X is $35$ years, and the arithmetic mean age of the employees in Division Y is $45$ years. Is the arithmetic mean age of all $100$ employees at Company K greater than $40$ years?

(1) Exactly $40$ employees work in Division X and exactly $70$ employees work in Division Y.
(2) The arithmetic mean age of the employees who work in both Division X and Division Y is $40$ years.

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 answer the question definitively, but neither statement alone is sufficient.

Combining both statements establishes the exact sizes of all three disjoint subsets (Division X only = 30, Division Y only = 60, both divisions = 10) and the average value of the overlapping subset (40 years). Calculating the total age sum gives 4,150 years, resulting in an exact overall mean of 41.5 years. This provides a definitive 'Yes' answer to whether the mean is greater than 40 years.

Adım Adım Çözüm

1

Formulate the algebraic expressions for the total sum of ages.

Let

x

be the number of employees in Division X only,

y

be the number in Division Y only, and

z

be the number in both divisions.

x + y + z = 100

. Let

S_x, S_y, S_z

be the sum of ages of employees in Division X only, Division Y only, and both divisions, respectively. Then

S_x + S_z = 35(x + z)

and

S_y + S_z = 45(y + z)

. The total sum of ages of all

100

employees is

S_{total} = S_x + S_y + S_z = 35x + 45y + 80z - S_z

.

Because employees in both divisions contribute to the averages of both Division X and Division Y, simply adding

35(x+z)

and

45(y+z)

counts

S_z

twice.

2

Evaluate Statement (1) independently.

Statement (1) states

x + z = 40

and

y + z = 70

. Since

x + y + z = 100

, we find

z = (40 + 70) - 100 = 10

,

x = 30

, and

y = 60

. Thus,

S_{total} = 35(40) + 45(70) - S_z = 4550 - S_z

. The overall mean age is

\frac{4550 - S_z}{100} = 45.5 - \frac{S_z}{100}

. Depending on the value of

S_z

(the sum of ages of the

10

overlap employees), the overall mean can be greater than

40

or less than or equal to

40

.

Without knowing

S_z

or the average age of the overlap group, the overall mean cannot be uniquely bounded. Thus, Statement (1) alone is NOT sufficient.

3

Evaluate Statement (2) independently.

Statement (2) states

\frac{S_z}{z} = 40 \implies S_z = 40z

. Substituting into

S_{total}

gives

S_{total} = 35(x+z) + 45(y+z) - 40z = 35x + 45y + 40z

. The overall mean age is

\frac{35x + 45y + 40z}{x + y + z}

. The condition

\frac{35x + 45y + 40z}{x + y + z} > 40

simplifies to

35x + 45y > 40x + 40y \iff 5y > 5x \iff y > x

.

Statement (2) provides no information about whether

y > x

(whether more employees work exclusively in Division Y than in Division X). Thus, Statement (2) alone is NOT sufficient.

4

Evaluate Statements (1) and (2) together.

From Statement (1),

x = 30

,

y = 60

, and

z = 10

. From Statement (2),

S_z = 40(10) = 400

. Since

y = 60 > x = 30

, the condition

y > x

holds. Substituting these values into

S_{total}

yields

S_{total} = 35(30) + 45(60) + 40(10) = 1050 + 2700 + 400 = 4150

. The arithmetic mean age of all

100

employees is

\frac{4150}{100} = 41.5

years, which is strictly greater than

40

.

Combining both statements yields a unique and definitive 'Yes' answer to the question stem.

Anahtar Kavram

Weighted averages in overlapping sets with double-counted sums

Soru 1465Soru →

A major maritime cargo port recently replaced its conventional diesel cranes with state-of-the-art automated electric gantry cranes, reducing the time required to load and unload containers from a vessel at berth by 40 percent per ship. However, in the six months following the deployment of the new cranes, the average turnaround time for cargo vessels—measured from when a vessel enters the port's harbor to when it departs back to open sea—increased by 15 percent. Which of the following, if true, most helps to resolve the apparent discrepancy described above?

The rapid unloading rate of the new cranes overwhelmed the port's inland customs clearance and truck dispatch facilities, causing severe harbor queuing where ships had to wait at anchor for days before an open berth became available.

The new automated electric cranes require far less scheduled routine maintenance per month than the older diesel cranes required.

The port authority paid a premium price to purchase and install the automated electric cranes compared to traditional equipment.

Shipping lines operating at the port recently expanded their global trade routes to include several new international destinations.

The electricity needed to power the new gantry cranes is generated primarily from clean, renewable energy sources.

Cevabı ve açıklamayı göster

Cevap: The rapid unloading rate of the new cranes overwhelmed the port's inland customs clearance and truck dispatch facilities, causing severe harbor queuing where ships had to wait at anchor for days before an open berth became available.

Cevap

The statement explaining that rapid unloading created a bottleneck at inland customs and truck dispatch, causing ships to wait longer at anchor before reaching a berth, resolves the paradox.

The correct answer resolves the paradox by differentiating between time spent loading at the berth and total time spent in the harbor. If the increased speed at berth causes a downstream bottleneck in customs and truck dispatching, berths cannot clear incoming traffic fast enough, forcing ships to wait at anchor. The prolonged waiting time at anchor outweighs the time saved at berth, causing net turnaround times to rise.

Adım Adım Çözüm

1

Identify the two apparently contradictory facts in the stimulus.

Fact 1: Cargo loading and unloading time at berth decreased by 40% per ship.
Fact 2: Total vessel turnaround time (harbor entry to departure) increased by 15%.

Resolving a paradox requires finding an additional factor that allows both statements to be simultaneously true.

2

Analyze the difference in scope between the two metrics.

Berth loading time is only one component of total harbor turnaround time, which also includes waiting at anchor, maneuvering into berth, and customs processing.

Identifying a scope gap (berth time vs. total harbor time) points to where the hidden delay occurred.

3

Evaluate the choices to find a factor that creates delays elsewhere in the process as a result of or alongside the faster berth speed.

The option describing inland customs congestion explains why ships spent significantly more time waiting at anchor before berthing, overriding the speed gained at the berth itself.

This bridges the gap without contradicting either premise.

Anahtar Kavram

Resolving Paradoxes: Scope Gaps & System Bottlenecks

Soru 1466Soru →

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

(1) $a^2 + b^2 + 4a - 6b = -13$
(2) $ab = -6$

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) can be rewritten by completing the square as

(a + 2)^2 + (b - 3)^2 = 0

. Since

a

and

b

are constrained to be real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The only way their sum can equal

0

is if

a = -2

and

b = 3

. This uniquely determines

(a + b)^2 = (-2 + 3)^2 = 1

, making Statement (1) alone sufficient. Statement (2) gives

ab = -6

, which allows multiple values for

(a + b)^2

(e.g.,

1

or

25

), making it insufficient. Hence, the option stating that Statement (1) alone is sufficient while Statement (2) alone is not is the correct choice.

Adım Adım Çözüm

1

Analyze Statement (1):

a^2 + b^2 + 4a - 6b = -13

Rearrange and group terms:

(a^2 + 4a) + (b^2 - 6b) = -13

. Complete the square for both variables by adding

4

and

9

to both sides:

(a^2 + 4a + 4) + (b^2 - 6b + 9) = -13 + 4 + 9

, which simplifies to

(a + 2)^2 + (b - 3)^2 = 0

.

Grouping and completing the square reveals the sum-of-squares structure.

2

Evaluate the real number constraint on Statement (1)

Because

a

and

b

are real numbers,

(a + 2)^2 \geq 0

and

(b - 3)^2 \geq 0

. The sum of two non-negative terms can equal zero if and only if both terms are independently zero:

a + 2 = 0 \implies a = -2

and

b - 3 = 0 \implies b = 3

.

The sum of non-negative real squares equaling zero forces each squared term to be zero.

3

Calculate the target expression using values from Statement (1)

(a + b)^2 = (-2 + 3)^2 = (1)^2 = 1

. Since this yields a single unique value, Statement (1) ALONE is sufficient.

A unique value for the target expression establishes sufficiency.

4

Analyze Statement (2):

ab = -6

If

a = 2

and

b = -3

, then

ab = -6

and

(a + b)^2 = (2 - 3)^2 = 1

. If

a = 1

and

b = -6

, then

ab = -6

and

(a + b)^2 = (1 - 6)^2 = 25

. Multiple values are possible.

Testing specific valid cases proves that Statement (2) does not yield a unique result.

5

Conclude Data Sufficiency determination

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

Statement (1) produces a unique answer, whereas Statement (2) does not.

Anahtar Kavram

Completing the Square and Trivial Inequality for Real Squares in Data Sufficiency

Soru 1467Soru →

Among a group of $50$ healthcare professionals, each professional has certified training in either Telemedicine, Robotic Surgery, or both. Exactly $30$ professionals are certified in Telemedicine and exactly $35$ are certified in Robotic Surgery. What is the average (arithmetic mean) years of experience of the professionals who are certified in BOTH Telemedicine and Robotic Surgery?

(1) The average years of experience of all $30$ professionals certified in Telemedicine is $9$ years, and the average years of experience of all $35$ professionals certified in Robotic Surgery is $8$ years.
(2) The average years of experience of the $15$ professionals certified ONLY in Telemedicine is $10$ years, and the average years of experience of the $20$ professionals certified ONLY in Robotic Surgery is $8$ years.

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. Rephrasing the stem shows there are

15

professionals in the 'Telemedicine Only' group,

20

in the 'Robotic Surgery Only' group, and

15

in 'Both'. Statement (1) gives total Telemedicine experience sum

S_T = 270

, but cannot separate

S_{T_{only}}

from

S_B

. Statement (2) gives

S_{T_{only}} = 150

, but gives no total experience bound. Together,

S_B = 270 - 150 = 120

, giving a unique mean of

\frac{120}{15} = 8

years.

Adım Adım Çözüm

1

Rephrase the question stem using overlapping set formulas to find subgroup counts.

Let

N = 50

,

T = 30

, and

R = 35

. Using

N = T + R - B

, we get

50 = 30 + 35 - B \implies B = 15

(professionals in Both). Thus, 'Telemedicine Only' count is

30 - 15 = 15

, and 'Robotic Surgery Only' count is

35 - 15 = 20

. Target: Find the mean experience of the

15

professionals in Both,

\mu_B = \frac{S_B}{15}

.

Simplifying the stem establishes the exact numerical count of professionals in each of the three distinct subgroups: Telemedicine Only (

15

), Robotic Surgery Only (

20

), and Both (

15

).

2

Evaluate Statement (1) independently.

Statement (1) gives total experience

S_T = 30 \times 9 = 270

and

S_R = 35 \times 8 = 280

. Since

S_T = S_{T_{only}} + S_B = 270

,

S_B

depends on

S_{T_{only}}

, which is unknown. Multiple values of

S_B

are possible. NOT sufficient.

Knowing total group averages does not isolate how experience is divided between single-category members and dual-category members.

3

Evaluate Statement (2) independently.

Statement (2) gives

S_{T_{only}} = 15 \times 10 = 150

and

S_{R_{only}} = 20 \times 8 = 160

. Without knowledge of

S_T

,

S_R

, or total group experience,

S_B

can take any real value. NOT sufficient.

Knowing only the single-category subgroup totals provides no boundary or equation for the overlapping group.

4

Evaluate Statement (1) and Statement (2) together.

From Statement (1),

S_T = S_{T_{only}} + S_B = 270

. From Statement (2),

S_{T_{only}} = 150

. Substituting gives

150 + S_B = 270 \implies S_B = 120

. Thus,

\mu_B = \frac{120}{15} = 8

years. SUFFICIENT.

Combining both statements yields a unique value for the total experience sum of the overlapping group.

Anahtar Kavram

Data Sufficiency with Overlapping Sets and Weighted Averages

Tahmini Süre:2m 0s

Soru 1468Soru →

A seminar was attended by $100$ professionals, each of whom speaks at least one of two languages: Spanish or French. Exactly $60$ of the professionals speak Spanish, and exactly $50$ speak French. If all $100$ professionals took a language proficiency examination scored on a scale from $0$ to $100$ , is the average (arithmetic mean) score of all $100$ professionals greater than $75$ ?

(1) The average score of the professionals who speak only Spanish is $80$ , and the average score of the professionals who speak only French is $70$ .
(2) The average score of the professionals who speak both Spanish and French is $85$ .

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 response identifies that both statements together are sufficient while neither statement alone is sufficient. By using the principle of inclusion-exclusion on the overlapping set sizes, we find that there are 50 professionals who speak only Spanish, 40 who speak only French, and 10 who speak both. Statement (1) alone is insufficient because the average score of the 10 dual-language professionals remains unknown, allowing the overall average to fall either above or below 75. Statement (2) alone is insufficient because it provides no score data for 90 of the 100 professionals. When both statements are combined, all three disjoint subgroup averages are known, giving an exact overall average of 76.5, which is definitively greater than 75.

Adım Adım Çözüm

1

Rephrase the question stem using overlapping set principles

Let

S

be Spanish speakers (

60

) and

F

be French speakers (

50

). Since every professional speaks at least one language,

|S \cup F| = 100

. By the inclusion-exclusion principle,

|S \cup F| = |S| + |F| - |S \cap F| \implies 100 = 60 + 50 - |S \cap F|

, so

|S \cap F| = 10

. The group partitions into: Spanish only =

60 - 10 = 50

, French only =

50 - 10 = 40

, and Both =

10

.

Deconstructing the total population into three mutually exclusive subgroups establishes the exact weights for computing the overall weighted average score.

2

Evaluate Statement (1) alone

Statement (1) provides average score for Spanish-only (

80

) and French-only (

70

). The total score sum is

50(80) + 40(70) + 10(A_{both}) = 6800 + 10(A_{both})

. The overall average is

68 + 0.1(A_{both})

. Depending on

A_{both}

(

0 \le A_{both} \le 100

), the overall average can range from

68

to

78

. For instance, if

A_{both} = 70

, overall average is

75

(not

>75

); if

A_{both} = 100

, overall average is

78

(

>75

). Thus, Statement (1) alone is NOT sufficient.

Since the score of the overlap group is unknown, the overall average cannot be uniquely tested against the threshold of 75.

3

Evaluate Statement (2) alone

Statement (2) provides

A_{both} = 85

, but gives no information about the average scores of the Spanish-only (

50

people) or French-only (

40

people) groups. Thus, the overall average score could be very low or very high. Statement (2) alone is NOT sufficient.

Without data on 90% of the population, statement (2) alone leaves the overall mean undetermined.

4

Evaluate Statements (1) and (2) together

Combining both statements gives:

50

people with average

80

,

40

people with average

70

, and

10

people with average

85

. Overall total score sum

= 50(80) + 40(70) + 10(85) = 4000 + 2800 + 850 = 7650

. Overall average

= 7650 / 100 = 76.5

. Since

76.5 > 75

, we get a definitive YES answer.

Having full weighted average data for all three disjoint components of the set yields a single, precise overall mean.

Anahtar Kavram

Weighted Average across Mutually Exclusive Partitions of Overlapping Sets

Tahmini Süre:2m 0s

Soru 1469Soru →

Consider the following passage:

'A 12-month clinical study evaluated the efficacy of Drug Y in 500 patients diagnosed with Condition Z. Every patient who was administered Drug Y experienced a statistically significant reduction in Condition Z symptoms. Additionally, exactly 15 percent of the patients receiving Drug Y experienced mild side effect W. None of the patients in the control group, who received a placebo, experienced side effect W or any reduction in Condition Z symptoms.'

Match each assertion below to its correct logical classification based on the passage.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The statement regarding trial participants experiencing both side effect W and symptom reduction matches Valid Inference. The statement attributing symptom reduction to pharmacological action rather than external variables matches Unstated Assumption. The statement predicting identical results in general clinical practice matches Speculative Extrapolation.

The item correctly categorizes each claim according to GMAT Critical Reasoning criteria: absolute mathematical overlap derived from premises forms a Valid Inference; presupposed control of background variables forms an Unstated Assumption; and extending trial results to general real-world populations forms a Speculative Extrapolation.

Adım Adım Çözüm

1

Analyze the logical scope of the statement asserting that participants experiencing side effect W also experienced symptom reduction.

Identify that 100% of Drug Y recipients had symptom reduction. Since side effect W occurred exclusively within the Drug Y group (15% of recipients), 100% of those with side effect W necessarily had symptom reduction.

Deductions that must be true strictly based on given set boundaries are valid inferences.

2

Evaluate the statement attributing the trial results to Drug Y rather than external variables.

Recognize that for trial observations to prove drug efficacy, unmentioned confounding variables must be assumed not to have influenced the outcome.

An unstated premise necessary to bridge observed data to a valid conclusion constitutes an assumption.

3

Evaluate the statement predicting general clinical outcomes outside the study.

Determine that broader real-world performance cannot be guaranteed from a single 500-patient trial.

Claims extending beyond the temporal, demographic, or environmental boundaries of given data are speculative extrapolations.

Anahtar Kavram

Distinguishing Inferences from Assumptions and Speculations

Soru 1470Soru →

In 2024, a coastal municipality enacted a strict ordinance prohibiting food-service establishments from using single-use plastic takeaway containers, aiming to curb plastic waste. Municipal sanitation records show that in the year following the ordinance, the total number of single-use plastic packaging items collected from commercial areas dropped by 25 percent. However, data recorded at the city's primary landfill revealed a 15 percent increase in the total weight of discarded plastic material over the exact same period. Which of the following, if true, most helps to resolve the apparent paradox described above?

The ordinance applied exclusively to consumer food packaging, while heavy industrial manufacturing in the municipality expanded rapidly during the same period, generating large amounts of dense plastic scrap.

Many local residents continued to purchase single-use plastic goods from e-commerce retailers based outside the municipality's regulatory jurisdiction.

Most restaurants complied with the ordinance by switching to lightweight aluminum containers that are easier for municipal services to collect.

The municipal landfill recently upgraded its automated sorting equipment to separate recyclable paper products from general waste.

Municipal officials raised tipping fees for commercial garbage trucks entering the landfill to encourage private waste diversion programs.

Cevabı ve açıklamayı göster

Cevap: The ordinance applied exclusively to consumer food packaging, while heavy industrial manufacturing in the municipality expanded rapidly during the same period, generating large amounts of dense plastic scrap.

Cevap

The correct option is the one stating that the ordinance applied exclusively to consumer food packaging while heavy industrial manufacturing expanded rapidly, generating large amounts of dense plastic scrap.

The correct choice reconciles both facts by distinguishing between the scope of the ordinance (consumer food packaging) and another major source of plastic waste (industrial manufacturing). If industrial manufacturing expanded and generated heavy plastic scrap, the total weight of plastic entering the landfill could easily rise by 15% even as consumer packaging items fell by 25%.

Adım Adım Çözüm

1

Identify the two contradictory facts in the stimulus.

Fact 1: Plastic takeaway packaging items collected dropped by 25%. Fact 2: Total weight of plastic material at the landfill increased by 15%.

Resolving a paradox requires clearly defining the two statements that seem mutually inconsistent.

2

Evaluate the choices to find a factor that allows both facts to be true simultaneously.

Industrial plastic scrap (not covered by the ordinance) increased total plastic weight, even though consumer packaging item count fell.

A valid resolution introduces a third factor that reconciles count versus weight across different waste categories.

Anahtar Kavram

Resolving Paradoxes through Scope and Category Differentiation

Tahmini Süre:1m 30s

Soru 1471Soru →

If $x$ is a real number, is $|x^3 - x| > x^2 - 1$ ?

(1) $|x - 1| > 0$
(2) $|x + 2| < 1$

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.

The correct option is the one stating that Statement (2) alone is sufficient, but Statement (1) alone is not sufficient. Simplifying the stem target

|x^3 - x| > x^2 - 1

into

|x||x^2 - 1| > x^2 - 1

reveals that the inequality holds for every real number except

x = 1

and

x = -1

(where both sides equal zero). Statement (1) states

|x - 1| > 0

, meaning

x \neq 1

, but it permits

x = -1

, which yields a 'No' answer, making Statement (1) insufficient. Statement (2) states

|x + 2| < 1

, which expands to

-3 < x < -1

. Because this interval strictly excludes both

x = -1

and

x = 1

, every value of

x

in this range satisfies the stem inequality, yielding a definitive 'Yes'.

Adım Adım Çözüm

1

Rephrase the question stem target algebraically.

The target inequality

|x^3 - x| > x^2 - 1

simplifies to

|x(x^2 - 1)| > x^2 - 1

, which is equivalent to

|x| \cdot |x^2 - 1| > x^2 - 1

.

Factoring allows analysis of the critical values where

x^2 - 1 = 0

.

2

Analyze the conditions under which

|x| \cdot |x^2 - 1| > x^2 - 1

holds true.

Case 1: If

x^2 - 1 = 0

(meaning

x = 1

or

x = -1

), LHS =

0

and RHS =

0

, giving

0 > 0

, which is FALSE. Case 2: If

x^2 - 1 < 0

(meaning

-1 < x < 1

), LHS is positive and RHS is negative, so non-negative > negative is TRUE. Case 3: If

x^2 - 1 > 0

(meaning

x > 1

or

x < -1

), dividing both sides by

x^2 - 1 > 0

yields

|x| > 1

, which is TRUE. Thus, the inequality holds for ALL real numbers EXCEPT

x = 1

and

x = -1

. The question simplifies to: Is

x \neq 1

and

x \neq -1

?

Simplifying the target reveals that the inequality is true everywhere except at the two boundary roots

x = 1

and

x = -1

.

3

Evaluate Statement (1):

|x - 1| > 0

.

This implies

x \neq 1

. However,

x

could equal

-1

. If

x = -1

,

|-1 - 1| = 2 > 0

is satisfied, but

|(-1)^3 - (-1)| = 0

and

(-1)^2 - 1 = 0

, giving

0 > 0

(NO). If

x = 2

,

|2 - 1| = 1 > 0

is satisfied, and

|8 - 2| = 6 > 3

(YES). Since both 'Yes' and 'No' are possible, Statement (1) is NOT sufficient.

Statement (1) rules out

x = 1

but permits

x = -1

.

4

Evaluate Statement (2):

|x + 2| < 1

.

Solving the absolute value inequality gives

-1 < x + 2 < 1

, which simplifies to

-3 < x < -1

. In this range,

x

cannot be

1

or

-1

because the upper boundary is strictly less than

-1

. Thus, for all

x \in (-3, -1)

,

x \neq 1

and

x \neq -1

is guaranteed, yielding a definitive 'YES'. Statement (2) is SUFFICIENT.

The open interval

(-3, -1)

excludes both

x = -1

and

x = 1

.

Anahtar Kavram

Question Stem Simplification with Absolute Value Inequalities

Tahmini Süre:2m 0s

Soru 1472Soru →

The table below displays the number of employees and the average salary per employee across three departments at Company Z:

Department	Number of Employees	Average Salary ($)
Marketing	20	60,000
Engineering	$E$	80,000
Sales	30	50,000

What was the overall average salary per employee across all three departments combined?

(1) The total number of employees across all three departments combined is 100.
(2) The total annual payroll for the Engineering department is $4,000,000.

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

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

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

EACH statement ALONE is sufficient.

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

Cevabı ve açıklamayı göster

Cevap: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

The question target requires finding the weighted average salary across all departments, which reduces to finding the value of the unknown employee count

E

. Statement (1) yields

20 + E + 30 = 100

, so

E = 50

. Statement (2) yields

E \times 80,000 = 4,000,000

, so

E = 50

. Since each statement independently determines

E

, each statement alone is sufficient.

Adım Adım Çözüm

1

Rephrase the question stem target

Overall average salary =

\frac{20(60,000) + E(80,000) + 30(50,000)}{20 + E + 30} = \frac{2,700,000 + 80,000E}{50 + E}

. Finding the value of

E

is sufficient to answer the question.

The overall average salary is a weighted average that depends solely on the single unknown parameter

E

.

2

Evaluate Statement (1) independently

Total employees =

20 + E + 30 = 100 \implies E = 50

. Since

E

is uniquely determined, the overall average salary can be calculated.

Statement (1) provides a direct linear equation in

E

.

3

Evaluate Statement (2) independently

Engineering total payroll =

E \times 80,000 = 4,000,000 \implies E = 50

. Since

E

is uniquely determined, the overall average salary can be calculated.

Statement (2) gives the total department cost, which divided by the known average salary gives

E

directly.

4

Conclude sufficiency classification

Each statement alone gives

E = 50

, making each statement individually sufficient.

Both Statement (1) and Statement (2) yield a unique value for

E

independently.

Anahtar Kavram

Weighted average determination from tabular data using Data Sufficiency logic

Tahmini Süre:1m 0s

Soru 1473Soru →

To protect 14th-century sandstone monuments from winter freeze-thaw weathering, municipal heritage conservators plan to apply a hydrophobic silicone nanofilm across the exterior surfaces of all downtown stone structures. Laboratory tests confirm that the nanofilm prevents external rainwater from penetrating sandstone pores by up to 98 percent. The conservators conclude that applying this coating will significantly reduce the rate of structural cracking and surface spalling caused by freeze-thaw cycles over the next decade.

Which of the following, if true, most seriously weakens the conservators' argument?

Groundwater drawn up through the porous foundations of sandstone monuments evaporates primarily through exterior stone surfaces, and the nanofilm prevents trapped internal moisture from escaping.

Synthetic sealants previously tested on modern concrete structures required reapplication every three years, whereas the new nanofilm maintains its water repellency for fifteen years.

The total financial cost of coating all downtown sandstone monuments exceeds the annual emergency repairs budget of the heritage conservation department.

Nearby granite architectural structures suffer significantly lower rates of spalling and surface cracking during severe winters than untreated sandstone monuments do.

The chemical application procedure temporarily alters the natural surface hue of historic sandstone for several weeks before curing into an invisible layer.

Cevabı ve açıklamayı göster

Cevap: Groundwater drawn up through the porous foundations of sandstone monuments evaporates primarily through exterior stone surfaces, and the nanofilm prevents trapped internal moisture from escaping.

Cevap

The argument is most seriously weakened by the statement that groundwater drawn up through porous foundations evaporates through exterior surfaces and that the nanofilm prevents this internal moisture from escaping.

The conservators assume that blocking external rainwater will prevent water from collecting and freezing inside the sandstone. The correct option reveals that moisture also enters the monuments from below via groundwater. Because the nanofilm creates an impermeable seal, this rising groundwater becomes trapped beneath the stone surface, where it will freeze during winter and cause severe internal cracking and spalling. This directly negates the expected benefit of the plan.

Adım Adım Çözüm

1

Deconstruct the argument structure into premise and conclusion

Premise: Hydrophobic nanofilm stops 98% of external rainwater from penetrating sandstone. Conclusion: Applying the nanofilm will significantly reduce freeze-thaw cracking and spalling over the next decade.

Identifying the explicit logical bridge allows us to uncover the unstated assumption.

2

Identify the central unstated assumption of the author

The argument assumes that external rainwater is the sole or primary source of water responsible for freeze-thaw damage inside the sandstone.

Weakening questions require finding evidence that demonstrates why the premise might fail to guarantee the conclusion.

3

Evaluate the impact of new information on the underlying assumption

If groundwater enters through foundations and cannot escape due to the barrier, trapped internal moisture will freeze and expand, exacerbating structural damage despite keeping rainwater out.

Demonstrating an alternative cause of severe moisture damage directly breaks the connection between keeping rainwater out and preventing freeze-thaw cracking.

Anahtar Kavram

Weakening Causal Arguments by Identifying Trapped Alternative Factors

Soru 1474Soru →

For a real number $p$ , is $p$ an integer?

(1) $p^3 - p$ is a positive prime number.
(2) $p^2$ 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) is sufficient because factoring

p^3 - p

gives

(p-1)p(p+1)

, which represents the product of three consecutive integers whenever

p

is an integer. The product of any three consecutive integers must be divisible by 6. Since no prime number is a multiple of 6,

p

cannot be an integer if

p^3 - p

is prime. This provides a definitive 'No' answer to the question 'Is

p

an integer?', establishing sufficiency. Statement (2) is insufficient because

p

could be an integer like 3 (yielding 'Yes') or an irrational number like

\sqrt{3}

(yielding 'No').

Adım Adım Çözüm

1

Analyze Statement (1):

p^3 - p

is a positive prime number.

p

cannot be an integer, yielding a definitive 'No' to the question stem.

Factor

p^3 - p

as

(p-1)p(p+1)

. If

p

were an integer, this product would represent three consecutive integers. Any three consecutive integers contain at least one even factor and exactly one multiple of 3, making their product divisible by 6. A prime number is a positive integer greater than 1 with no positive divisors other than 1 and itself, so no prime number can be a multiple of 6. Thus, no integer

p

can make

p^3 - p

a prime number. Since

p^3 - p

is given to be a positive prime,

p

must be a non-integer real number. A definitive 'No' answer establishes that Statement (1) alone is sufficient.

2

Analyze Statement (2):

p^2

is an integer.

p

could be an integer or a non-integer, so the question cannot be answered uniquely.

If

p = 3

,

p^2 = 9

(an integer), and

p

is an integer (Yes). If

p = \sqrt{3}

,

p^2 = 3

(an integer), but

p

is not an integer (No). Because both 'Yes' and 'No' outcomes are possible, Statement (2) alone is not sufficient.

3

Determine the overall Data Sufficiency choice.

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

Statement (1) provides a conclusive answer on its own, whereas Statement (2) remains ambiguous.

Anahtar Kavram

Number Properties and Integer Constraints in Data Sufficiency

Soru 1475Soru →

If $x$ and $y$ are real numbers, what is the value of $x + y$ ?

(1) $x = 5$
(2) $y = 3$

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.

Neither statement alone provides values for both

x

and

y

. Evaluating each statement independently shows that one variable remains unknown in each case. When combined, Statement (1) gives

x = 5

and Statement (2) gives

y = 3

, allowing us to uniquely calculate

x + y = 8

. Thus, both statements together are sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

x = 5

, but

y

is unknown, so

x + y

cannot be uniquely determined.

Statement (1) alone is not sufficient.

2

Evaluate Statement (2) independently

y = 3

, but

x

is unknown, so

x + y

cannot be uniquely determined.

Statement (2) alone is not sufficient. No information from Statement (1) can be carried over during independent evaluation.

3

Combine Statement (1) and Statement (2)

x = 5

and

y = 3

, yielding

x + y = 5 + 3 = 8

.

Combining both statements provides a single unique numerical answer.

Anahtar Kavram

Statement Independence Evaluation and Statement Combination

Soru 1476Soru →

If $x$ and $y$ are positive real numbers, is $(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4$ ?

(1) $x^2 + y^2 = 2xy + 9$
(2) $\sqrt{x} + \sqrt{y} = 5$

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 target question stem shows that

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

is equivalent to

4 + \frac{(x - y)^2}{xy} > 4

, which simplifies to

x \neq y

for positive real numbers

x

and

y

. Statement (1) rearranges to

(x - y)^2 = 9

, which proves

x \neq y

and provides a definitive 'Yes' answer. Statement (2) allows cases where

x = y = 6.25

as well as

x = 1, y = 16

, leaving the answer uncertain. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Adım Adım Çözüm

1

Rephrase the target question stem algebraically.

The target question asking whether

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4

simplifies to asking whether

x \neq y

.

Expanding

(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) = 1 + \frac{x}{y} + \frac{y}{x} + 1 = 2 + \frac{x^2 + y^2}{xy} = 4 + \frac{(x - y)^2}{xy}

. Since

x, y > 0

, the quantity

\frac{(x - y)^2}{xy} > 0

holds if and only if

(x - y)^2 > 0

, which means

x \neq y

.

2

Evaluate Statement (1):

x^2 + y^2 = 2xy + 9

.

Statement (1) is sufficient.

Rearranging yields

x^2 - 2xy + y^2 = 9 \implies (x - y)^2 = 9

. Since

(x - y)^2 = 9 \neq 0

, it must be that

x \neq y

. This gives a definitive 'Yes' answer to the rephrased question.

3

Evaluate Statement (2):

\sqrt{x} + \sqrt{y} = 5

.

Statement (2) is not sufficient.

If

x = 6.25

and

y = 6.25

, then

\sqrt{6.25} + \sqrt{6.25} = 2.5 + 2.5 = 5

. Here

x = y

, so the answer to 'Is

x \neq y

?' is 'No'. If

x = 1

and

y = 16

, then

\sqrt{1} + \sqrt{16} = 1 + 4 = 5

. Here

x \neq y

, so the answer is 'Yes'. Because both 'Yes' and 'No' are possible, Statement (2) is insufficient.

Anahtar Kavram

Data Sufficiency Question Stem Simplification

Tahmini Süre:1m 30s

Soru 1477Soru →

To combat urban air pollution, the municipal transit authority of a major city plans to replace its entire fleet of diesel buses with electric buses powered by lithium-iron-phosphate (LFP) batteries. Opponents of the plan argue that the frequent fast-charging required during peak operational hours will cause rapid battery degradation, forcing costly battery replacements within three years and rendering the transition economically unviable. However, transit officials maintain that the transition will substantially lower overall fleet maintenance costs over a ten-year period. Which of the following, if true, provides the strongest support for the transit officials' contention?

Lithium-iron-phosphate batteries subjected to frequent fast-charging in transit conditions retain over 85 percent of their initial operational capacity after eight years of service.

Specialized high-voltage fast-charging stations require annual equipment inspections that cost significantly more than routine maintenance for diesel fueling stations.

Passengers surveyed on existing electric transit routes in neighboring cities reported higher satisfaction levels regarding cabin noise than passengers on diesel routes.

The municipal transit authority currently allocates thirty percent of its annual operating budget to purchasing diesel fuel.

Modern diesel engine technology has significantly reduced particulate emissions compared to models manufactured a decade ago.

Cevabı ve açıklamayı göster

Cevap: Lithium-iron-phosphate batteries subjected to frequent fast-charging in transit conditions retain over 85 percent of their initial operational capacity after eight years of service.

Cevap

The statement showing that lithium-iron-phosphate batteries retain over 85 percent of their capacity after eight years of frequent fast-charging provides the strongest support for the transit officials' claim.

The transit officials claim that transitioning to electric buses will reduce maintenance costs over a 10-year period, while opponents argue that battery degradation within 3 years will destroy any financial benefit. The option stating that LFP batteries maintain over 85 percent capacity after 8 years of fast-charging directly disproves the opponents' premise of early failure and provides factual backing for the officials' 10-year savings projection.

Adım Adım Çözüm

1

Analyze the argument structure

Premise: Switching to electric buses reduces pollution, but opponents claim frequent fast-charging degrades batteries within 3 years, causing high replacement costs. Conclusion: Transit officials contend the switch will substantially lower overall fleet maintenance costs over 10 years.

Identifying the conclusion and the opponents' counter-argument is essential to determine what assumption needs support.

2

Identify the weakness or gap in the argument

The officials' conclusion depends on overcoming the opponents' premise that rapid battery degradation within 3 years will negate long-term savings.

A strong supporting premise must rule out the primary financial obstacle raised by the opponents.

3

Evaluate the choices to find the statement that solidifies the officials' conclusion

Showing empirically that LFP batteries endure eight years of fast-charging with minimal capacity loss directly removes the opponents' objection and supports the 10-year savings claim.

Demonstrating data reliability and battery durability bridges the gap between battery life expectancy and 10-year cost reductions.

Anahtar Kavram

Strengthening Arguments by Ruling Out Alternative Vulnerabilities and Invalidating Opponents' Premises

Soru 1478Soru →

If $x$ is a real number, is $x$ an integer?

(1) $x^2 + 2x$ is an integer.
(2) $x^3 + 2x^2$ 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: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Cevap

Both statements together are sufficient to determine that

x

is an integer, but neither statement alone is sufficient.

The correct response is that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) permits irrational values like

x = -1 + \sqrt{2}

, making it insufficient on its own. Statement (2) permits irrational values like

x = \frac{-1 + \sqrt{5}}{2}

, making it insufficient on its own. Combining both statements allows us to express

x

as the quotient of two integers

b/a

when

x^2 + 2x \neq 0

, which mathematically forces

x

to be an integer.

Adım Adım Çözüm

1

Evaluate Statement (1) independently:

x^2 + 2x

is an integer.

Statement (1) is NOT sufficient.

Let

x^2 + 2x = 1

. Solving

x^2 + 2x - 1 = 0

gives

x = -1 + \sqrt{2}

, which is a real non-integer. However, if

x = 1

,

x^2 + 2x = 3

is also an integer. Since

x

could be an integer or a non-integer, Statement (1) alone is insufficient.

2

Evaluate Statement (2) independently:

x^3 + 2x^2

is an integer.

Statement (2) is NOT sufficient.

Let

x^3 + 2x^2 = 1

. The equation

x^3 + 2x^2 - 1 = 0

factors as

(x + 1)(x^2 + x - 1) = 0

. Setting

x^2 + x - 1 = 0

yields non-integer root

x = \frac{-1 + \sqrt{5}}{2}

. For this non-integer value,

x^3 + 2x^2 = 1

, which is an integer. Since

x

can also be the integer

-1

, Statement (2) alone is insufficient.

3

Evaluate Statements (1) and (2) together.

Both statements together are SUFFICIENT.

From Statement (1), let

x^2 + 2x = a

, where

a

is an integer. From Statement (2), let

x^3 + 2x^2 = b

, where

b

is an integer. Notice that

x^3 + 2x^2 = x(x^2 + 2x) = x \cdot a = b

. Case 1: If

a = 0

, then

x^2 + 2x = 0 \implies x(x + 2) = 0 \implies x = 0

or

x = -2

, both of which are integers. Case 2: If

a \neq 0

, then

x = \frac{b}{a}

, meaning

x

is a rational number. Let

x = \frac{m}{n}

in lowest terms where

\gcd(m, n) = 1

and

n > 0

. Substituting

x = \frac{m}{n}

into

x^2 + 2x = a

gives

\frac{m^2}{n^2} + \frac{2m}{n} = a \implies m^2 + 2mn = a n^2 \implies m(m + 2n) = a n^2

. If

n > 1

, any prime factor

p

of

n

must divide

m(m + 2n)

, which implies

p

divides

m^2

and thus

p

divides

m

. This contradicts

\gcd(m, n) = 1

. Thus,

n

must equal

1

, proving

x

is an integer.

Anahtar Kavram

Using algebraic combination of Data Sufficiency statements and integer polynomial constraints to establish sufficiency without assuming variables are integers.

Soru 1479Soru →

If $x$ and $y$ are non-zero real numbers, what is the value of $\frac{x}{y}$ ?

(1) $x^2 y - x y^2 = 2xy$
(2) $x^2 + y^2 = 5xy$

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.

The statement that both statements together are not sufficient is correct because combining Statement (1) (

x = y + 2

) and Statement (2) (

x^2 + y^2 = 5xy

) leads to a quadratic equation

3y^2 + 6y - 4 = 0

with two distinct real roots for

y

. Evaluating

\frac{x}{y} = 1 + \frac{2}{y}

for both roots yields two distinct possible values,

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

and

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

. Because a unique numerical value cannot be determined, the information provided is not sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Simplifies to

x - y = 2 \implies x = y + 2

.

Since

x, y \neq 0

,

xy \neq 0

. Dividing

x^2 y - x y^2 = 2xy

by

xy

yields

x - y = 2

. Thus

\frac{x}{y} = \frac{y+2}{y} = 1 + \frac{2}{y}

, which varies with

y

. Not sufficient.

2

Evaluate Statement (2) independently.

Yields

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

.

Dividing

x^2 + y^2 = 5xy

by

y^2

gives a quadratic in

k = \frac{x}{y}

:

k^2 - 5k + 1 = 0

. Solving for

k

gives

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

. Because there are two distinct real solutions for the ratio, Statement (2) alone is not sufficient.

3

Evaluate Statements (1) and (2) combined.

Two distinct pairs of real numbers satisfy both conditions, giving two distinct values for

\frac{x}{y}

.

Substitute

x = y + 2

into

x^2 + y^2 = 5xy

:

(y+2)^2 + y^2 = 5y(y+2) \implies 3y^2 + 6y - 4 = 0

. This gives two real roots for

y

:

y = \frac{-3 \pm \sqrt{21}}{3}

. Substituting each

y

back into

\frac{x}{y} = 1 + \frac{2}{y}

yields

\frac{x}{y} = \frac{5 + \sqrt{21}}{2}

and

\frac{x}{y} = \frac{5 - \sqrt{21}}{2}

. Since two distinct ratios remain possible, combined statements are not sufficient.

Anahtar Kavram

Non-linear systems in Data Sufficiency often result in multiple valid solutions, requiring explicit verification of solution uniqueness rather than assuming two equations with two variables yield a single solution.

Tahmini Süre:2m 15s

Soru 1480Soru →

An agricultural research station evaluated a sample of $90$ fruit trees. Each tree was treated with Fertilizer X, Fertilizer Y, or both. Exactly $60$ trees were treated with Fertilizer X, and exactly $50$ trees were treated with Fertilizer Y. What was the average (arithmetic mean) yield, in kilograms, of all $90$ trees?

(1) The average yield of the trees treated with Fertilizer X was $45$ kg, and the average yield of the trees treated only with Fertilizer Y was $35$ kg.
(2) The average yield of the trees treated with Fertilizer Y was $42$ kg, and the average yield of the trees treated only with Fertilizer X was $48$ kg.

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.

Using the principal formula for overlapping sets

N(\text{Total}) = N(\text{X}) + N(\text{Y}) - N(\text{X } \cap \text{ Y})

, we find that

90 = 60 + 50 - N(\text{X } \cap \text{ Y})

, meaning exactly

20

trees received both fertilizers. This partitions the

90

trees into three mutually exclusive groups:

40

trees receiving Only X,

20

trees receiving Both, and

30

trees receiving Only Y.

Statement (1) provides the average for all trees receiving X (which combines 'Only X' and 'Both', totaling

60

trees) as

45

kg, giving a subgroup total yield of

60 \times 45 = 2,700

kg. It also gives the average for the remaining

30

trees ('Only Y') as

35

kg, giving

30 \times 35 = 1,050

kg. Summing these gives the exact total yield of all

90

trees (

3,750

kg), which allows computing a unique overall mean. Hence Statement (1) alone is sufficient.

Statement (2) provides the average for all trees receiving Y (combining 'Only Y' and 'Both', totaling

50

trees) as

42

kg, giving a subgroup total yield of

50 \times 42 = 2,100

kg. It also gives the average for the remaining

40

trees ('Only X') as

48

kg, giving

40 \times 48 = 1,920

kg. Summing these gives the exact total yield of all

90

trees (

4,020

kg), which allows computing a unique overall mean. Hence Statement (2) alone is sufficient.

Since each statement alone is sufficient, the correct option is the one stating that each statement alone is sufficient.

Adım Adım Çözüm

1

Determine the number of trees in each disjoint subset using overlapping set principles.

Number of trees receiving both fertilizers is

20

; 'Only X' is

40

; 'Only Y' is

30

.

By the inclusion-exclusion principle:

N(\text{Total}) = N(\text{X}) + N(\text{Y}) - N(\text{Both})

. Thus,

90 = 60 + 50 - N(\text{Both})

, which yields

N(\text{Both}) = 20

. Consequently,

N(\text{Only X}) = 60 - 20 = 40

and

N(\text{Only Y}) = 50 - 20 = 30

.

2

Evaluate Statement (1) independently.

Total yield

= 3,750

kg, yielding a unique overall average of

\frac{3,750}{90} = \frac{125}{3}

kg.

Statement (1) gives the average yield for all

60

trees treated with Fertilizer X (

45

kg) and for the

30

trees treated only with Fertilizer Y (

35

kg). The total yield of all

90

trees is

(60 \times 45) + (30 \times 35) = 2,700 + 1,050 = 3,750

kg. Dividing by

90

gives a single, deterministic value.

3

Evaluate Statement (2) independently.

Total yield

= 4,020

kg, yielding a unique overall average of

\frac{4,020}{90} = \frac{134}{3}

kg.

Statement (2) gives the average yield for all

50

trees treated with Fertilizer Y (

42

kg) and for the

40

trees treated only with Fertilizer X (

48

kg). The total yield of all

90

trees is

(50 \times 42) + (40 \times 48) = 2,100 + 1,920 = 4,020

kg. Dividing by

90

gives a single, deterministic value.

4

Synthesize data sufficiency evaluation.

Since each statement independently allows us to calculate the exact average yield, each statement alone is sufficient.

Data Sufficiency requires identifying whether each statement alone yields a unique solution to the question asked.

Anahtar Kavram

Overlapping Sets and Weighted Averages

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