Data Sufficiency Alıştırma Soruları — GMAT — Sayfa 12

Soru 221Soru →

In GMAT Data Sufficiency, when evaluating whether Statement (2) alone is sufficient to answer the question stem, any information or numeric constraints introduced solely within Statement (1) must be completely disregarded.

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Yanlış

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Cevap: True

Cevap

True

The statement is true because standard GMAT Data Sufficiency procedure requires testing Statement (1) alone, then testing Statement (2) alone in complete isolation, and only combining their information if neither statement alone is sufficient.

Adım Adım Çözüm

1

Recall the structural rule for evaluating individual statements in GMAT Data Sufficiency.

Statement (1) and Statement (2) must each be evaluated in complete isolation before any combining phase.

The standard evaluation protocol requires testing whether each statement independently provides sufficient information to answer the question.

2

Analyze whether Statement (1) constraints can be active during Statement (2) standalone analysis.

Statement (1) constraints must be erased from memory when evaluating Statement (2).

Allowing Statement (1) facts to persist during the standalone assessment of Statement (2) causes statement carryover error, leading to incorrect sufficiency conclusions.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 222Soru →

If $x$ and $y$ are positive integers, is $x + y$ divisible by 3?

(1) $x^2 + y^2$ is divisible by 3.
(2) $x^2 - y^2$ is divisible by 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.

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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) alone is sufficient because the square of any integer leaves a remainder of 0 or 1 when divided by 3. The only way for

x^2 + y^2

to be a multiple of 3 is if both

x^2

and

y^2

leave remainders of 0, meaning both

x

and

y

are multiples of 3. Therefore,

x + y

must be a multiple of 3. Statement (2) alone is not sufficient because

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

being divisible by 3 allows

x + y

to be divisible by 3 (e.g.,

x = 2, y = 1

) or not divisible by 3 (e.g.,

x = 4, y = 1

).

Adım Adım Çözüm

1

Analyze remainder properties of perfect squares modulo 3.

For any integer

z

,

z^2 \equiv 0 \pmod 3

if

z

is a multiple of 3, and

z^2 \equiv 1 \pmod 3

if

z

is not a multiple of 3.

Establishing the possible values of perfect squares modulo 3 simplifies the evaluation of both statements.

2

Evaluate Statement (1):

x^2 + y^2

is divisible by 3.

Since

x^2 \pmod 3 \in \{0, 1\}

and

y^2 \pmod 3 \in \{0, 1\}

, the sum

x^2 + y^2 \equiv 0 \pmod 3

can only occur if

x^2 \equiv 0 \pmod 3

and

y^2 \equiv 0 \pmod 3

. This implies both

x

and

y

are divisible by 3, so

x + y

must also be divisible by 3.

Statement (1) yields a definitive 'Yes' answer, making it sufficient.

3

Evaluate Statement (2):

x^2 - y^2

is divisible by 3.

Factor as

(x - y)(x + y)

. If

x = 2

and

y = 1

,

x^2 - y^2 = 3

(divisible by 3) and

x + y = 3

(divisible by 3, answer Yes). If

x = 4

and

y = 1

,

x^2 - y^2 = 15

(divisible by 3) and

x + y = 5

(not divisible by 3, answer No).

Since statement (2) yields both 'Yes' and 'No' outcomes, it is not sufficient.

Anahtar Kavram

Quadratic Remainder Constraints in Data Sufficiency

Soru 223Soru →

If $m$ and $n$ are positive real numbers, is $m$ an integer?

(1) $m^2 n$ is an integer and $m n^2$ is an integer.
(2) $m^3 n$ is an integer and $n$ 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.

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

Cevap

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

Statements (1) and (2) together are not sufficient because setting

m = \frac{1}{2}

and

n = 8

satisfies all given conditions (

m^2 n = 2

,

m n^2 = 32

,

m^3 n = 1

,

n = 8

) while yielding a 'No' to whether

m

is an integer. Meanwhile,

m = 1

and

n = 8

also satisfies all conditions and yields a 'Yes'. Therefore, the two statements combined do not uniquely determine if

m

is an integer.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) is insufficient.

If

m = 1

and

n = 2

, then

m^2 n = 2

and

m n^2 = 4

(both integers), and

m = 1

is an integer. However, if

m = \sqrt[3]{2}

and

n = \sqrt[3]{4}

, then

m^2 n = 2

and

m n^2 = 2

(both integers), but

m = \sqrt[3]{2}

is not an integer.

2

Evaluate Statement (2) independently.

Statement (2) is insufficient.

If

m = 1

and

n = 1

, then

m^3 n = 1

and

n = 1

(both integers), and

m = 1

is an integer. However, if

m = \frac{1}{2}

and

n = 8

, then

n = 8

is an integer and

m^3 n = (\frac{1}{8})(8) = 1

is an integer, but

m = \frac{1}{2}

is not an integer.

3

Evaluate Statements (1) and (2) together.

Statements (1) and (2) together are insufficient.

Test the candidate case

m = \frac{1}{2}

and

n = 8

against both statements simultaneously:
- Under Statement (1):

m^2 n = (\frac{1}{4})(8) = 2

(integer) and

m n^2 = (\frac{1}{2})(64) = 32

(integer).
- Under Statement (2):

n = 8

(integer) and

m^3 n = (\frac{1}{8})(8) = 1

(integer).
Here, all conditions from both statements are satisfied, but

m = \frac{1}{2}

is NOT an integer. Conversely,

m = 1

and

n = 8

also satisfies both statements and

m = 1

IS an integer. Thus, a definitive Yes/No answer cannot be determined.

Anahtar Kavram

Implicit real number constraints versus integer constraints in Data Sufficiency polynomial expressions.

Soru 224Soru →

If $m$ and $n$ are positive integers, what is the value of $m + n$ ?

(1) $m^2 - n^2 = 21$
(2) $m - n = 3$

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.

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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 identifies that neither statement alone provides enough information to determine a single value for

m + n

, but combining both statements allows us to substitute

m - n = 3

directly into the factored form

(m - n)(m + n) = 21

, yielding

m + n = 7

uniquely.

Adım Adım Çözüm

1

Analyze the question target and algebraic constraints

The target is to find a unique value for

m + n

, where

m

and

n

are positive integers.

Establishing positive integer constraints is critical when evaluating factor possibilities.

2

Evaluate Statement (1) independently

Statement (1) states

m^2 - n^2 = 21

, which factors to

(m - n)(m + n) = 21

. Positive integer factor pairs for

21

are

(1, 21)

and

(3, 7)

. If

m - n = 1

and

m + n = 21

, then

m + n = 21

. If

m - n = 3

and

m + n = 7

, then

m + n = 7

. Two values are possible, so Statement (1) alone is INSUFFICIENT.

A single unique value cannot be determined from Statement (1) alone.

3

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

Statement (2) states

m - n = 3

. Infinitely many positive integer pairs

(m, n)

satisfy this equation (e.g.,

(4, 1) \implies m+n=5

;

(5, 2) \implies m+n=7

). Thus, Statement (2) alone is INSUFFICIENT.

Statement (2) must be evaluated strictly isolated from Statement (1).

4

Combine Statement (1) and Statement (2)

Substitute

m - n = 3

into

(m - n)(m + n) = 21

:

3(m + n) = 21 \implies m + n = 7

. A single unique value is determined.

Combining both statements resolves the ambiguity present in each statement individually.

Anahtar Kavram

Statement Independence Evaluation and Algebraic Combination in Data Sufficiency

Tahmini Süre:1m 30s

Soru 225Soru →

If $r$ and $s$ are non-zero real numbers, what is the value of $r^2 + s^2$ ?

(1) $r + s = 6$
(2) $r^3 + s^3 = 126$

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 the unique value of

r^2 + s^2 = 26

, but neither statement alone is sufficient.

The correct answer specifies that both statements together are sufficient, but neither alone is sufficient. Statement (1) provides

r + s = 6

, which leaves

rs

unknown. Statement (2) gives

r^3 + s^3 = 126

, which also leaves

rs

undetermined. When combined, applying the identity

r^3 + s^3 = (r + s)((r + s)^2 - 3rs)

yields

126 = 6(36 - 3rs)

, solving uniquely for

rs = 5

. This allows exact calculation of

r^2 + s^2 = (r+s)^2 - 2rs = 36 - 10 = 26

.

Adım Adım Çözüm

1

Rephrase the target expression using fundamental algebraic identities

Note that

r^2 + s^2 = (r + s)^2 - 2rs

. To find

r^2 + s^2

, we need both

(r + s)

and the product

rs

.

Expressing the target in terms of sum and product simplifies the evaluation of statement sufficiency.

2

Evaluate Statement (1) alone

Statement (1) gives

r + s = 6

. Without knowing

rs

or individual values of

r

and

s

,

r^2 + s^2 = 36 - 2rs

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

A single linear equation with two variables does not fix the value of

rs

.

3

Evaluate Statement (2) alone

Statement (2) gives

r^3 + s^3 = 126

. Multiple pairs satisfy this (e.g.,

r=5, s=1

gives

125+1=126

where

r^2+s^2=26

;

r=\sqrt[3]{126}, s=0

is excluded since non-zero, but other non-integer pairs exist). Statement (2) is NOT sufficient.

A single cubic equation in two variables does not uniquely define

r^2 + s^2

.

4

Evaluate Statement (1) and Statement (2) together

Use the sum of cubes identity

r^3 + s^3 = (r + s)(r^2 - rs + s^2) = (r + s)((r + s)^2 - 3rs)

. Substitute

r + s = 6

and

r^3 + s^3 = 126

:

126 = 6(6^2 - 3rs) \implies 21 = 36 - 3rs \implies 3rs = 15 \implies rs = 5

. Thus,

r^2 + s^2 = (r + s)^2 - 2rs = 6^2 - 2(5) = 26

. Both statements together are SUFFICIENT.

The system yields a unique numerical value for

r^2 + s^2

.

Anahtar Kavram

Using polynomial identities (sum of cubes and square of a binomial) to evaluate sufficiency of non-linear symmetric systems.

Soru 226Soru →

An agricultural facility blends two fertilizer solutions, Solution $X$ (which is $20\%$ nitrogen by volume) and Solution $Y$ (which contains nitrogen and inert liquid). When Solution $X$ and Solution $Y$ are mixed, the resulting blend has an overall nitrogen concentration of $32\%$ by volume. What is the percentage of nitrogen by volume in Solution $Y$ ?

(1) The volume of Solution $Y$ in the blend is $50\%$ greater than the volume of Solution $X$ .
(2) The blend contains a total of $160$ liters of Solution $X$ .

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.

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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 question asks for a relative percentage concentration, which depends strictly on the volumetric ratio of the two component solutions. Statement (1) specifies that Solution Y has a volume 50% greater than Solution X, fixing the ratio of Solution Y to Solution X at 3 to 2. Substituting this ratio into the weighted average concentration formula yields a single unique solution for the nitrogen concentration of Solution Y. Thus, Statement (1) alone is sufficient. Statement (2) provides an absolute volume for Solution X without establishing the volume or ratio of Solution Y, leaving the mixture concentration under-determined and thus insufficient.

Adım Adım Çözüm

1

Formulate the weighted average equation from the question stem.

Let

V_X

and

V_Y

be the volumes of Solution

X

and Solution

Y

, respectively, and let

y\%

be the nitrogen concentration of Solution

Y

. The total nitrogen content equation is

0.20 V_X + \frac{y}{100} V_Y = 0.32 (V_X + V_Y)

.

The total volume of nitrogen in the final mixture is the sum of the nitrogen contributed by each individual solution.

2

Simplify the target equation to isolate

y

in terms of the volume ratio

\frac{V_Y}{V_X}

.

Dividing the equation by

V_X

yields

0.20 + \frac{y}{100} \left(\frac{V_Y}{V_X}\right) = 0.32 \left(1 + \frac{V_Y}{V_X}\right)

.

Knowing the ratio of

V_Y

to

V_X

is both necessary and sufficient to determine

y

, making absolute volume values unnecessary.

3

Evaluate Statement (1): The volume of Solution

Y

is

50\%

greater than the volume of Solution

X

.

V_Y = 1.5 V_X \implies \frac{V_Y}{V_X} = 1.5 = \frac{3}{2}

. Substituting

\frac{3}{2}

into the simplified equation gives

0.20 + \frac{y}{100}(1.5) = 0.32(2.5) \implies 0.20 + 0.015 y = 0.80 \implies 0.015 y = 0.60 \implies y = 40\%

.

Statement (1) provides a unique numerical value for

y

, so Statement (1) alone is SUFFICIENT.

4

Evaluate Statement (2): The blend contains a total of

160

liters of Solution

X

.

V_X = 160

liters, but

V_Y

and total volume

V_X + V_Y

remain unknown.

Without knowing

V_Y

or the ratio

\frac{V_Y}{V_X}

,

y

cannot be uniquely determined. Statement (2) alone is NOT SUFFICIENT.

Anahtar Kavram

Weighted Averages and Volumetric Ratios in Data Sufficiency

Tahmini Süre:2m 0s

Soru 227Soru →

In a GMAT Data Sufficiency question asking for the unique value of a variable $x$ , if Statement (1) alone limits the possible values of $x$ to the set $\{2, 5\}$ and Statement (2) alone limits the possible values of $x$ to the set $\{5, 8\}$ , then the two statements combined are sufficient to answer the question.

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Yanlış

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Cevap: True

Cevap

True. Combining two consistent Data Sufficiency statements yields a solution set equal to the intersection of their individual solution sets. The intersection of

\{2, 5\}

and

\{5, 8\}

is

\{5\}

, which uniquely identifies

x

and is therefore sufficient.

The statement is true because taking the intersection of the two solution sets

\{2, 5\}

and

\{5, 8\}

leaves only the single value

x = 5

, satisfying the GMAT Data Sufficiency requirement for a unique value.

Adım Adım Çözüm

1

Evaluate Statement (1) alone

Statement (1) yields

x \in \{2, 5\}

, which contains two possible values and is insufficient on its own.

A Value Data Sufficiency question requires a single unique numerical value for sufficiency.

2

Evaluate Statement (2) alone independently

Statement (2) yields

x \in \{5, 8\}

, which contains two possible values and is insufficient on its own.

Statement (2) must be evaluated without carrying over any constraints from Statement (1).

3

Combine Statement (1) and Statement (2)

The combined solution set is

\{2, 5\} \cap \{5, 8\} = \{5\}

.

Combining statements requires identifying values that satisfy both statements simultaneously.

4

Determine sufficiency of the combined solution set

Since

x = 5

is the only value in the combined set, the statements together provide a single unique answer.

A single unique result satisfies the criterion for Data Sufficiency.

Anahtar Kavram

Statement Combination and Solution Set Intersection

Soru 228Soru →

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

(1) $x^3 + y^3 = 28$
(2) $x^2 - xy + y^2 = 7$

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 sum of cubes factors as

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

. Neither statement alone provides enough information to determine

x + y

, but combining both statements gives

28 = (x + y)(7)

, which uniquely determines

x + y = 4

.

Adım Adım Çözüm

1

Rephrase the question target using the sum of cubes identity.

Recall

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

. Thus, if

x^2 - xy + y^2 \neq 0

, then

x + y = \frac{x^3 + y^3}{x^2 - xy + y^2}

.

Factoring allows connecting the expression in Statement (1) directly to the expression in Statement (2).

2

Evaluate Statement (1) alone.

Statement (1) gives

x^3 + y^3 = 28

. For example, if

x=3

and

y=-1

,

x^3+y^3 = 27 - 1 = 26 \neq 28

; if

x=\sqrt[3]{28}

and

y=0

, then

x+y=\sqrt[3]{28}

. If

x=3, y=1

,

x^3+y^3=28

and

x+y=4

. Multiple values are possible.

A single cubic equation in two variables does not fix the sum

x+y

.

3

Evaluate Statement (2) alone.

Statement (2) gives

x^2 - xy + y^2 = 7

. Multiple pairs such as

(x,y) = (3,2)

give

9-6+4=7

(where

x+y=5

) and

(x,y) = (3,1)

give

9-3+1=7

(where

x+y=4

).

A quadratic equation in two variables permits multiple sums for

x+y

.

4

Evaluate both statements together.

Substitute Statement (1) and Statement (2) into the identity:

28 = (x + y)(7) \implies x + y = 4

.

Dividing

x^3 + y^3

by

x^2 - xy + y^2

yields a single unique value of

4

for

x + y

.

Anahtar Kavram

Algebraic Expression Rephrasing and Factoring Identities in Data Sufficiency

Soru 229Soru →

If $x$ is a positive integer, is $x^2 + 5x$ divisible by 6?

(1) $x$ is a prime number.
(2) $x + 1$ is divisible by 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: 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.

Factoring the stem expression gives

x(x+5)

. For any positive integer

x

, one of

x

or

x+5

is even, making

x(x+5)

divisible by 2. Thus, the question reduces to whether

x(x+5)

is divisible by 3. Statement (1) allows

x=2

(

x^2+5x=14

, not divisible by 6) and

x=3

(

x^2+5x=24

, divisible by 6), yielding both 'No' and 'Yes' responses, so it is insufficient. Statement (2) states that

x+1

is a multiple of 3, meaning

x \equiv 2 \pmod 3

. Evaluating

x(x+5) \equiv 2(2+5) = 14 \equiv 2 \pmod 3

shows

x^2+5x

is never divisible by 3, and therefore never divisible by 6. This consistent 'No' answer makes Statement (2) alone sufficient.

Adım Adım Çözüm

1

Analyze and rephrase the question stem target expression.

Factor

x^2 + 5x = x(x + 5)

. Since

x

and

x + 5

have opposite parity, their product is always even (divisible by 2). Thus,

x^2 + 5x

is divisible by 6 if and only if

x(x + 5)

is divisible by 3.

Determining divisibility by 6 requires checking divisibility by both 2 and 3.

2

Evaluate Statement (1):

x

is a prime number.

If

x = 2

,

x^2 + 5x = 14

, which is not divisible by 6 (Answer: NO). If

x = 3

,

x^2 + 5x = 24

, which is divisible by 6 (Answer: YES). Since both 'Yes' and 'No' are possible, Statement (1) alone is INSUFFICIENT.

A statement must yield a single consistent answer ('Yes' or 'No') to be sufficient.

3

Evaluate Statement (2):

x + 1

is divisible by 3.

If

x + 1

is divisible by 3, then

x = 3k - 1

for some integer

k \ge 1

. Substituting

x = 3k - 1

into

x(x + 5)

yields

(3k - 1)(3k + 4) = 9k^2 + 9k - 4

. Dividing this expression by 3 gives a remainder of 2. Because

x(x + 5)

is never divisible by 3,

x^2 + 5x

is NEVER divisible by 6.

A definitive 'No' answer for all permitted values of

x

establishes sufficiency.

Anahtar Kavram

Number Properties and Integer Constraints in Data Sufficiency

Alternatif Yöntem

Test explicit values for Statement (2):

x \in \{2, 5, 8, 11\}

. Evaluating

x^2 + 5x

yields

14, 50, 104, 176

, none of which are divisible by 6. Recognizing this uniform 'No' outcome confirms Statement (2) is sufficient.

Tahmini Süre:2m 0s

Soru 230Soru →

If $x$ and $y$ are positive real numbers, what is the value of the ratio $\frac{x}{y}$ ?

(1) $\frac{x + 2y}{y} = 5$

(2) $x^2 - 9y^2 = 0$

Which of the following 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: EACH statement ALONE is sufficient.

Cevap

EACH statement ALONE is sufficient.

Evaluating each statement independently demonstrates sufficiency for both. Statement (1) simplifies directly to

\frac{x}{y} = 3

. Statement (2) factors to

(x-3y)(x+3y)=0

, and since

x

and

y

are positive real numbers as stated in the question stem,

x+3y

cannot equal zero, leaving

x-3y=0

, or

\frac{x}{y} = 3

. Because each statement independently yields a unique value for the ratio, each statement alone is sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently

Simplifying

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

gives

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

, so

\frac{x}{y} = 3

.

This yields a single unique value for the target ratio, so Statement (1) alone is sufficient.

2

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

Factoring

x^2 - 9y^2 = 0

yields

(x - 3y)(x + 3y) = 0

, meaning

x = 3y

or

x = -3y

. Since the stem specifies

x

and

y

are positive real numbers,

x + 3y > 0

, so

x = -3y

is impossible. Thus,

x = 3y

, which gives

\frac{x}{y} = 3

.

Because of the domain constraint in the stem, Statement (2) independently provides a single unique ratio of 3, making Statement (2) alone sufficient.

3

Conclude overall Data Sufficiency decision

Since Statement (1) alone is sufficient and Statement (2) alone is sufficient, each statement alone is sufficient.

Matching standard GMAT Data Sufficiency evaluation rules.

Anahtar Kavram

Statement Independence & Stem Constraint Application

Soru 231Soru →

A biopharmaceutical facility operates two automated synthesis lines, Line X and Line Y, to produce a therapeutic compound. On a specific shift, Line X operated for $h_X$ hours at a constant production rate of $r_X$ liters per hour, and Line Y operated for $h_Y$ hours at a constant production rate of $r_Y$ liters per hour. Was the overall average hourly production rate for both lines combined across the total operating hours greater than 45 liters per hour?

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 determine whether the combined average hourly rate

R_{avg} = \frac{h_X r_X + h_Y r_Y}{h_X + h_Y}

is greater than 45 liters per hour, we evaluate each statement independently. Statement (1) provides

r_X = 40

,

r_Y = 60

, and

h_X = 2 h_Y

. Substituting

h_X = 2 h_Y

gives

R_{avg} = \frac{2 h_Y (40) + h_Y (60)}{3 h_Y} = \frac{140}{3} = 46\frac{2}{3}

liters per hour. Since

46\frac{2}{3} > 45

, Statement (1) alone is sufficient. Statement (2) provides

r_X = 50

and

r_Y = 55

. Because any weighted average of two positive values must lie strictly between the smaller and larger value (

50 < R_{avg} < 55

), the combined rate must be strictly greater than 50, which guarantees it is greater than 45. Thus, Statement (2) alone is also sufficient. Since each statement independently yields a definitive 'Yes', the correct choice is that each statement alone is sufficient.

Adım Adım Çözüm

1

Formulate the algebraic expression for the target condition.

The combined average hourly rate is

R_{avg} = \frac{h_X r_X + h_Y r_Y}{h_X + h_Y}

. We need to determine if

R_{avg} > 45

.

Establishing the target formula is necessary to analyze statement inputs.

2

Evaluate Statement (1):

r_X = 40

,

r_Y = 60

, and

h_X = 2 h_Y

.

Substitute

h_X = 2 h_Y

into

R_{avg}

:

R_{avg} = \frac{2 h_Y (40) + h_Y (60)}{2 h_Y + h_Y} = \frac{140 h_Y}{3 h_Y} = \frac{140}{3} = 46.67

liters per hour. Since

46.67 > 45

, Statement (1) yields a definitive 'Yes'.

Knowing the ratio of hours allows the variable

h_Y

to cancel out, providing an exact rate.

3

Evaluate Statement (2):

r_X = 50

and

r_Y = 55

.

Since both individual rates

r_X

and

r_Y

are strictly greater than 45 liters per hour, any weighted average of these two positive quantities must lie between 50 and 55 liters per hour (

50 < R_{avg} < 55

). Thus,

R_{avg}

is guaranteed to be greater than 45, yielding a definitive 'Yes'.

A weighted average of two values is strictly bounded by the minimum and maximum of those values.

4

Conclude Data Sufficiency decision.

Since Statement (1) alone yields a definitive 'Yes' and Statement (2) alone yields a definitive 'Yes', each statement alone is sufficient.

Each statement independently answers the question.

Anahtar Kavram

Weighted Average Bounds in Data Sufficiency

Soru 232Soru →

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

(1) $k^2$ is an integer.
(2) $k^3$ 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, but NEITHER statement ALONE is sufficient.

The correct option states that both statements together are sufficient, but neither alone is sufficient. Statement (1) allows irrational square roots like

\sqrt{2}

, making it insufficient. Statement (2) allows irrational cube roots like

\sqrt[3]{2}

, making it insufficient. Combined, taking the ratio

\frac{k^3}{k^2} = k

shows that

k

is the quotient of two integers and therefore rational. A rational number whose square is an integer must be an integer itself, giving a definitive 'Yes'.

Adım Adım Çözüm

1

Evaluate Statement (1):

k^2

is an integer.

If

k = 2

,

k^2 = 4

(an integer) and

k

is an integer (Yes). If

k = \sqrt{2}

,

k^2 = 2

(an integer), but

k

is not an integer (No). Statement (1) is NOT sufficient.

The question specifies

k

is a real number, so non-integer square roots must be tested.

2

Evaluate Statement (2):

k^3

is an integer.

If

k = 2

,

k^3 = 8

(an integer) and

k

is an integer (Yes). If

k = \sqrt[3]{2}

,

k^3 = 2

(an integer), but

k

is not an integer (No). Statement (2) is NOT sufficient.

Cube roots of non-cube integers yield irrational real numbers.

3

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

If

k = 0

,

k

is an integer (Yes). If

k \neq 0

, then

k = \frac{k^3}{k^2}

. Since

k^3

and

k^2

are both integers,

k

must be a rational number. Let

k = \frac{p}{q}

in simplest form where

p, q \in \mathbb{Z}

and

\gcd(p, q) = 1

. Since

k^2 = \frac{p^2}{q^2}

is an integer and

\gcd(p^2, q^2) = 1

,

q^2

must equal

1

, so

q = \pm 1

. Thus,

k

is an integer. Combined, the statements are SUFFICIENT.

Combining algebraic operations reveals that

k

is rational, and a rational number with an integer square must itself be an integer.

Anahtar Kavram

Data Sufficiency evaluation of number properties with real number constraints and rational number properties

Soru 233Soru →

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

(1) $p^2 + q^2 = 25$
(2) $pq = 12$

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.

Expanding the target expression gives

(p + q)^2 = p^2 + 2pq + q^2

. Statement (1) provides

p^2 + q^2 = 25

but leaves

pq

unknown, making it insufficient alone. Statement (2) provides

pq = 12

but leaves

p^2 + q^2

unknown, making it insufficient alone. Combining both statements allows direct substitution into the identity:

(p + q)^2 = 25 + 2(12) = 49

, which yields a single unique answer.

Adım Adım Çözüm

1

Rephrase the question stem algebraically.

(p + q)^2 = p^2 + 2pq + q^2

Expanding the target expression shows that calculating

(p + q)^2

requires knowing the sum of squares

(p^2 + q^2)

and the product

pq

.

2

Evaluate Statement (1) independently.

Insufficient

Given

p^2 + q^2 = 25

, the term

2pq

remains unknown. For example, if

p = 5

and

q = 0

, then

(p+q)^2 = 25

; if

p = 3

and

q = 4

, then

(p+q)^2 = 49

. Since multiple outcomes exist, Statement (1) alone is insufficient.

3

Evaluate Statement (2) independently.

Insufficient

Given

pq = 12

, the term

p^2 + q^2

remains unknown. For example, if

p = 3

and

q = 4

, then

(p+q)^2 = 49

; if

p = 1

and

q = 12

, then

(p+q)^2 = 169

. Since multiple outcomes exist, Statement (2) alone is insufficient.

4

Evaluate Statements (1) and (2) together.

Sufficient

Substituting

p^2 + q^2 = 25

and

pq = 12

into

(p + q)^2 = (p^2 + q^2) + 2(pq)

yields

25 + 2(12) = 49

. This determines a unique value.

Anahtar Kavram

Algebraic Identity Expansion and Substitution in Data Sufficiency

Soru 234Soru →

In GMAT Data Sufficiency, when evaluating whether Statement (2) alone is sufficient to answer the question stem, any algebraic conditions or variable constraints introduced exclusively within Statement (1) must be completely disregarded.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. Statement (2) must be evaluated in strict isolation from Statement (1).

Each statement in a Data Sufficiency question must be evaluated independently first. Statement (1) constraints cannot be referenced or assumed during the standalone evaluation of Statement (2).

Adım Adım Çözüm

1

Recall the fundamental procedural rules of GMAT Data Sufficiency.

Statement (1) and Statement (2) must be evaluated independently before any combining step is attempted.

Independent evaluation determines whether either statement alone answers the question stem.

2

Apply the rule of independence specifically to the evaluation phase of Statement (2).

When evaluating Statement (2), Statement (1) is treated as non-existent.

Carrying over constraints from Statement (1) creates a false sense of sufficiency for Statement (2) alone.

Anahtar Kavram

Statement Independence in Data Sufficiency

Soru 235Soru →

If $x$ and $y$ are positive real numbers, is the product $xy$ an odd integer?

(1) $x^2y$ is an odd integer.
(2) $xy^2$ is an odd 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: Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

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

The correct option is the choice stating that Statements (1) and (2) together are NOT sufficient. Since

x

and

y

are positive real numbers, setting

x = y = \sqrt[3]{3}

satisfies both

x^2y = 3

and

xy^2 = 3

(both odd integers), but gives

xy = \sqrt[3]{9}

, which is not an integer. Conversely, setting

x = 1/\sqrt[3]{3}

and

y = \sqrt[3]{9}

yields

x^2y = 3

and

xy^2 = 9

(both odd integers), giving

xy = 3

, which is an odd integer. Because both a 'No' and a 'Yes' answer are possible, the information remains insufficient even when both statements are combined.

Adım Adım Çözüm

1

Analyze the stem constraints and target question.

The target asks whether

xy

is an odd integer, given that

x

and

y

are positive real numbers (not necessarily integers).

Recognizing that

x

and

y

are not restricted to integers is essential to avoid invalid integer assumption traps.

2

Evaluate Statement (1) independently.

Statement (1) gives

x^2y = k

for some odd integer

k

. If

x = \sqrt{3}

and

y = 1

, then

x^2y = 3

(an odd integer), but

xy = \sqrt{3}

(not an integer). If

x = 3

and

y = 1

, then

x^2y = 9

(an odd integer) and

xy = 3

(an odd integer). Thus, Statement (1) alone is NOT sufficient.

Testing non-integer values demonstrates that

xy

may or may not be an integer.

3

Evaluate Statement (2) independently.

Statement (2) gives

xy^2 = m

for some odd integer

m

. If

x = 1

and

y = \sqrt{3}

, then

xy^2 = 3

(an odd integer), but

xy = \sqrt{3}

(not an integer). If

x = 1

and

y = 3

, then

xy^2 = 9

(an odd integer) and

xy = 3

(an odd integer). Thus, Statement (2) alone is NOT sufficient.

Symmetry with Statement (1) confirms that Statement (2) alone produces variable outcomes.

4

Evaluate Statements (1) and (2) combined.

Multiplying the two expressions gives

(x^2y)(xy^2) = x^3y^3 = (xy)^3 = km

. Since

k

and

m

are odd integers, their product

km

is also an odd integer, say

P

. Thus,

(xy)^3 = P \implies xy = \sqrt[3]{P}

. If

k = 3

and

m = 3

, then

P = 9

, so

xy = \sqrt[3]{9}

(not an integer). If

k = 3

and

m = 9

, then

P = 27

, so

xy = \sqrt[3]{27} = 3

(an odd integer). Because

xy

can be either an odd integer or a non-integer, the combined statements are NOT sufficient.

Combining the statements yields

(xy)^3

equal to an odd integer, which does not guarantee that

xy

itself is an integer.

Anahtar Kavram

Implicit Real vs. Integer Constraints in Data Sufficiency

Tahmini Süre:2m 0s

Soru 236Soru →

A municipal water treatment plant fills a main reservoir by pumping water from Inlet Pipe $A$ and Inlet Pipe $B$ simultaneously for 6 hours. The water entering through Pipe $A$ has a mineral concentration of $a$ milligrams per liter ( $\text{mg/L}$ ), and the water entering through Pipe $B$ has a mineral concentration of $b\text{ mg/L}$ , where $a < b$ . Is the final mineral concentration of the water in the reservoir less than $\frac{a + b}{2}\text{ mg/L}$ ?

(1) The rate at which Pipe $A$ pumps water is 20 percent greater than the rate at which Pipe $B$ pumps water.
(2) The total volume of water pumped by Pipe $B$ during the 6 hours is 4,500 liters.

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 stem shows that the overall mineral concentration is less than the midpoint

\frac{a+b}{2}

if and only if a larger volume of water is drawn from the pipe with lower concentration (Pipe A), i.e.,

V_A > V_B

. Statement (1) establishes that Pipe A's pumping rate is

1.2

times Pipe B's rate, so over any identical time frame,

V_A > V_B

. This yields a definitive 'Yes' answer, making Statement (1) alone sufficient. Statement (2) provides only the total volume from Pipe B without any information about Pipe A, which is insufficient.

Adım Adım Çözüm

1

Simplify and rephrase the question target mathematically.

The final concentration is

C = \frac{V_A a + V_B b}{V_A + V_B}

. The question asks if

C < \frac{a + b}{2}

, which expands to

2(V_A a + V_B b) < (V_A + V_B)(a + b) \iff 2V_A a + 2V_B b < V_A a + V_A b + V_B a + V_B b \iff V_A(a - b) < V_B(a - b)

. Since

a < b

, we know that

(a - b) < 0

. Dividing both sides of the inequality by

(a - b)

reverses the inequality sign, yielding

V_A > V_B

. Thus, the question asks: 'Is the volume of water supplied by Pipe A greater than the volume supplied by Pipe B?'

Rephrasing the question stem reveals the underlying relationship required to determine sufficiency.

2

Evaluate Statement (1) independently.

Statement (1) states that Pipe A's pumping rate

R_A

is 20 percent greater than Pipe B's rate

R_B

, so

R_A = 1.2 R_B

. Since both pipes operate for the exact same duration of 6 hours,

V_A = 6 R_A = 7.2 R_B

and

V_B = 6 R_B

. Since

R_B > 0

,

V_A = 1.2 V_B > V_B

is guaranteed to be true. Statement (1) gives a definitive 'Yes' to the rephrased question.

A statement that yields a definitive 'Yes' answer to a Yes/No question is sufficient.

3

Evaluate Statement (2) independently.

Statement (2) gives

V_B = 4,500

liters, but provides no information regarding Pipe A's pumping rate or total volume

V_A

. If

V_A = 5,000

liters, then

V_A > V_B

(Yes); if

V_A = 3,000

liters, then

V_A < V_B

(No). Because multiple outcomes are possible, Statement (2) is not sufficient.

An inability to determine a unique 'Yes' or 'No' answer makes a statement insufficient.

Anahtar Kavram

Question Stem Simplification and Weighted Average Inequalities in Data Sufficiency

Tahmini Süre:2m 0s

Soru 237Soru →

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

(1) $(m + n)^2 = 36$
(2) $m^2 + n^2 = 20$

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 correct response identifies that statements (1) and (2) together leave sign ambiguity in the difference of squares because

m

and

n

can swap values while maintaining the same sum and sum of squares.

Adım Adım Çözüm

1

Rephrase the target question

The target expression is

m^2 - n^2 = (m + n)(m - n)

. To find a unique value, we need a unique numerical value for

(m + n)(m - n)

.

Factoring the difference of squares clarifies what information is necessary.

2

Evaluate Statement (1) alone

(m + n)^2 = 36 \implies m + n = 6

or

m + n = -6

. We have no information about

m - n

.

Since

m - n

can take infinitely many values,

m^2 - n^2

is not uniquely determined. Statement (1) is insufficient.

3

Evaluate Statement (2) alone

m^2 + n^2 = 20

.

Different pairs of numbers like

(m, n) = (\sqrt{20}, 0)

give

m^2 - n^2 = 20

, while

(m, n) = (0, \sqrt{20})

give

m^2 - n^2 = -20

. Statement (2) is insufficient.

4

Evaluate Statements (1) and (2) combined

Expand Statement (1):

m^2 + 2mn + n^2 = 36

. Substitute Statement (2):

20 + 2mn = 36 \implies 2mn = 16 \implies mn = 8

.
Now test values satisfying

m + n = 6

and

mn = 8

:
Case 1:

m = 4, n = 2 \implies m^2 - n^2 = 4^2 - 2^2 = 16 - 4 = 12

.
Case 2:

m = 2, n = 4 \implies m^2 - n^2 = 2^2 - 4^2 = 4 - 16 = -12

.
Since

m^2 - n^2

yields two distinct values (

12

and

-12

), the statements together are insufficient.

Symmetry between variables in a non-linear system creates multiple possible values for non-symmetric target expressions.

Anahtar Kavram

Symmetry and Degree Ambiguity in Algebraic Data Sufficiency

Soru 238Soru →

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

(1) $x^2 - y^2 = 15$
(2) $x^2 y - x y^2 = 12$

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.

Evaluating each statement individually reveals that neither provides enough constraints on its own to solve for the sum of the variables. When both statements are combined, substituting the expression for the sum into the product equation yields a quartic polynomial in terms of the difference of the variables. Because the function is strictly increasing for positive values, it possesses exactly one positive real root. This single valid root uniquely determines the sum of the variables, making both statements together sufficient.

Adım Adım Çözüm

1

Evaluate Statement (1) independently.

Statement (1) gives

x^2 - y^2 = (x-y)(x+y) = 15

. Multiple pairs of positive real numbers satisfy this (e.g.,

x=4, y=1

gives

x+y=5

;

x=8, y=7

gives

x+y=15

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

A single non-linear equation in two variables does not restrict

x+y

to a unique value.

2

Evaluate Statement (2) independently.

Statement (2) gives

x^2 y - x y^2 = xy(x-y) = 12

. Infinitely many positive real pairs satisfy this condition (e.g.,

x=4, y=1

gives

x+y=5

;

x=3, y=1

does not yield 12, but continuous choices of

x, y > 0

allow multiple sums). Thus, Statement (2) alone is NOT sufficient.

Knowing the product

xy(x-y) = 12

leaves the value of

x+y

undetermined.

3

Combine Statement (1) and Statement (2).

Let

u = x-y

and

v = x+y

. Note that

u > 0

because

x^2 - y^2 = 15 > 0

and

x, y > 0

.
From (1),

v = \frac{15}{u}

.
Also,

xy = \frac{(x+y)^2 - (x-y)^2}{4} = \frac{v^2 - u^2}{4}

.
Substituting into (2):

\left(\frac{v^2 - u^2}{4}\right) u = 12 \implies u(v^2 - u^2) = 48

.
Substitute

v = \frac{15}{u}

into the equation:

u\left(\frac{225}{u^2} - u^2\right) = 48 \implies \frac{225}{u} - u^3 = 48 \implies u^4 + 48u - 225 = 0

.

Express the system in terms of the difference

u = x-y

and sum

v = x+y

to analyze uniqueness.

4

Determine the number of positive real roots for

u^4 + 48u - 225 = 0

.

Let

f(u) = u^4 + 48u - 225

. For

u > 0

, the derivative

f'(u) = 4u^3 + 48 > 0

, meaning

f(u)

is strictly increasing for all positive

u

. Since

f(0) = -225 < 0

and

\lim_{u \to \infty} f(u) = +\infty

, there is exactly ONE positive real root for

u

. Testing integer values shows

f(3) = 3^4 + 48(3) - 225 = 81 + 144 - 225 = 0

, so

u = 3

uniquely. Substituting

u = 3

into

v = \frac{15}{u}

yields

v = x+y = 5

uniquely (with positive numbers

x = 4, y = 1

).

A strictly increasing continuous function transitioning from negative to positive values has exactly one root.

Anahtar Kavram

Non-Linear Algebraic Systems and Root Uniqueness in Data Sufficiency

Soru 239Soru →

A manufacturing facility produced two types of products, Type X and Type Y. In July, the total cost to produce $N$ units of Type X and $M$ units of Type Y was $4,200. What was the unit manufacturing cost of a Type X product?

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 provides sufficient data because the total cost for Type X products is given as

2,800 for

N = 70

units. Dividing

2,800 by 70 gives a unique unit manufacturing cost of $40 for Type X products, completely independent of Statement (1) and without needing information regarding Type Y.

Adım Adım Çözüm

1

Rephrase the target question

Let

C_x

be the unit cost of Type X and

C_y

be the unit cost of Type Y. Total cost is

N \cdot C_x + M \cdot C_y = 4200

. The target question asks for the unique value of

C_x

.

Clarify what specific variable needs to be solved for.

2

Evaluate Statement (1) independently

Statement (1) tells us

C_y = 30

and

N = 2M

. Substituting into the total cost equation gives

N \cdot C_x + (N/2) \cdot 30 = 4200

, which simplifies to

N(C_x + 15) = 4200

. Since

N

is unknown,

C_x

cannot be uniquely determined. Statement (1) alone is NOT sufficient.

Determine if Statement (1) yields a single unique numerical value for

C_x

.

3

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

Statement (2) states that the total cost for Type X products is

N \cdot C_x = 2800

, and

N = 70

. Solving for

C_x

gives

C_x = 2800 / 70 = 40

. This uniquely answers the question stem directly. Statement (2) alone IS sufficient.

Test Statement (2) strictly in isolation.

Anahtar Kavram

Evaluating Statement Independence in Data Sufficiency

Soru 240Soru →

A tech company sells two types of service plans: Basic and Enterprise. Last month, what percentage of the company's total revenue from these two service plans was generated by sales of the Enterprise plan?

(1) Last month, the price of an Enterprise plan was $250\%$ of the price of a Basic plan, and the number of Basic plans sold was $60\%$ greater than the number of Enterprise plans sold.
(2) Last month, total revenue generated from sales of the Basic plan was $\$120,000$ .

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: 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) establishes exact multiplier relationships for both price and quantity (

P_E = 2.5 P_B

and

N_B = 1.6 N_E

). When computing the proportion of revenue generated by the Enterprise plan, all unknown variables (

P_B

and

N_E

) cancel out cleanly, producing a single, deterministic percentage of

\frac{25}{41} \times 100\%

. Statement (2) only specifies the dollar revenue for the Basic plan, leaving the Enterprise plan's revenue completely unconstrained. Thus, Statement (1) alone is sufficient, while Statement (2) alone is not.

Adım Adım Çözüm

1

Formulate the target ratio algebraically

Target percentage

= \frac{R_E}{R_E + R_B} \times 100\%

, where

R_E = P_E \times N_E

and

R_B = P_B \times N_B

.

Total revenue is the sum of Enterprise revenue (

R_E

) and Basic revenue (

R_B

).

2

Evaluate Statement (1)

P_E = 2.5 P_B

and

N_B = 1.6 N_E

. Therefore,

R_E = 2.5 P_B N_E

and

R_B = 1.6 P_B N_E

. The revenue ratio is

\frac{R_E}{R_B} = \frac{2.5}{1.6} = \frac{25}{16}

.

Substituting these expressions into the target formula yields

\frac{25 P_B N_E}{25 P_B N_E + 16 P_B N_E} = \frac{25}{41} \approx 60.98\%

. Because a single, unique percentage is obtained, Statement (1) alone is sufficient.

3

Evaluate Statement (2)

R_B = \$120,000

, but

R_E

remains unknown.

Knowing only the total revenue from Basic plans provides no information about the Enterprise revenue or its share of total revenue. Thus, Statement (2) alone is insufficient.

Anahtar Kavram

Data Sufficiency Evaluation of Percentages via Relative Variable Relationships

Data Sufficiency

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