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Question Stem Simplification and Target Rephrasing

25 soru

Tüm türler25Çoktan seçmeli13Doğru / Yanlış12
Soru 21Soru →

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

(1) x2+y2=2xy+9x^2 + y^2 = 2xy + 9
(2) x+y=5\sqrt{x} + \sqrt{y} = 5

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)(1x+1y)>4(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4 is equivalent to 4+(x−y)2xy>44 + \frac{(x - y)^2}{xy} > 4, which simplifies to x≠yx \neq y for positive real numbers xx and yy. Statement (1) rearranges to (x−y)2=9(x - y)^2 = 9, which proves x≠yx \neq y and provides a definitive 'Yes' answer. Statement (2) allows cases where x=y=6.25x = y = 6.25 as well as x=1,y=16x = 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)(1x+1y)>4(x + y)\left(\frac{1}{x} + \frac{1}{y}\right) > 4 simplifies to asking whether x≠yx \neq y.
Expanding (x+y)(1x+1y)=1+xy+yx+1=2+x2+y2xy=4+(x−y)2xy(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>0x, y > 0, the quantity (x−y)2xy>0\frac{(x - y)^2}{xy} > 0 holds if and only if (x−y)2>0(x - y)^2 > 0, which means x≠yx \neq y.
2
Evaluate Statement (1): x2+y2=2xy+9x^2 + y^2 = 2xy + 9.
Statement (1) is sufficient.
Rearranging yields x2−2xy+y2=9  ⟹  (x−y)2=9x^2 - 2xy + y^2 = 9 \implies (x - y)^2 = 9. Since (x−y)2=9≠0(x - y)^2 = 9 \neq 0, it must be that x≠yx \neq y. This gives a definitive 'Yes' answer to the rephrased question.
3
Evaluate Statement (2): x+y=5\sqrt{x} + \sqrt{y} = 5.
Statement (2) is not sufficient.
If x=6.25x = 6.25 and y=6.25y = 6.25, then 6.25+6.25=2.5+2.5=5\sqrt{6.25} + \sqrt{6.25} = 2.5 + 2.5 = 5. Here x=yx = y, so the answer to 'Is x≠yx \neq y?' is 'No'. If x=1x = 1 and y=16y = 16, then 1+16=1+4=5\sqrt{1} + \sqrt{16} = 1 + 4 = 5. Here x≠yx \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 22Soru →

If aa and bb are non-zero real numbers such that a2≠b2a^2 \neq b^2, is a2b+ab2a3b−ab3>0\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0?

(1) a>ba > b
(2) a+b>0a + b > 0

Cevabı ve açıklamayı göster

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

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The target question asks whether a2b+ab2a3b−ab3>0\frac{a^2 b + a b^2}{a^3 b - a b^3} > 0. Factoring the numerator as ab(a+b)ab(a+b) and the denominator as ab(a+b)(a−b)ab(a+b)(a-b) allows us to cancel common non-zero factors, simplifying the target question to 'Is 1a−b>0\frac{1}{a-b} > 0?', which is equivalent to 'Is a>ba > b?'. Statement (1) directly states that a>ba > b, answering the rephrased question with a definitive 'Yes'. Statement (2) states that a+b>0a+b > 0, which gives no information regarding whether a>ba > b. Therefore, Statement (1) alone is sufficient, while Statement (2) alone is not.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically by factoring numerator and denominator.
Numerator: a2b+ab2=ab(a+b)a^2 b + a b^2 = a b(a + b). Denominator: a3b−ab3=ab(a2−b2)=ab(a+b)(a−b)a^3 b - a b^3 = a b(a^2 - b^2) = a b(a + b)(a - b). The expression becomes ab(a+b)ab(a+b)(a−b)\frac{a b(a + b)}{a b(a + b)(a - b)}.
Simplifying complex rational expressions before evaluating statements prevents misinterpretation of necessary conditions.
2
Cancel non-zero common terms in the simplified fraction.
Since ab≠0a b \neq 0 and a+b≠0a + b \neq 0 (given a2≠b2a^2 \neq b^2), common terms cancel out to yield 1a−b>0\frac{1}{a - b} > 0.
The sign of 1a−b\frac{1}{a - b} is strictly positive if and only if a−b>0a - b > 0, which is equivalent to a>ba > b.
3
Evaluate Statement (1): a>ba > b.
This directly yields a−b>0a - b > 0, so 1a−b>0\frac{1}{a - b} > 0. This provides a definitive 'Yes' answer.
Statement (1) alone provides sufficient information.
4
Evaluate Statement (2): a+b>0a + b > 0.
Knowing a+b>0a + b > 0 gives no information about whether a>ba > b or a<ba < b (e.g., if a=5,b=2a=5, b=2, a>ba > b; if a=2,b=5a=2, b=5, a<ba < b).
Statement (2) alone is not sufficient to determine if a>ba > b.

Anahtar Kavram

Question Stem Simplification and Target Rephrasing
Tahmini Süre:1m 40s
Soru 23Soru →

For all positive real numbers aa and bb, the Data Sufficiency Yes/No target question "Is a2+3abb2+3ab>1\frac{a^2 + 3ab}{b^2 + 3ab} > 1?" is algebraically equivalent to asking "Is a>ba > b?"

Cevabı ve açıklamayı göster

Cevap: True

Cevap

True. The target question simplifies directly to "Is a>ba > b?"
The statement is true because cross-multiplying the strictly positive denominator b2+3abb^2 + 3ab and subtracting the shared term 3ab3ab reduces a2+3abb2+3ab>1\frac{a^2 + 3ab}{b^2 + 3ab} > 1 directly to a2>b2a^2 > b^2. Since aa and bb are both positive, a2>b2a^2 > b^2 is logically equivalent to a>ba > b.

Adım Adım Çözüm

1
Determine the sign of the denominator.
Since a>0a > 0 and b>0b > 0, b2+3ab>0b^2 + 3ab > 0.
Knowing the denominator is strictly positive ensures we can cross-multiply without reversing the inequality sign.
2
Cross-multiply the inequality by the denominator.
a2+3ab>b2+3aba^2 + 3ab > b^2 + 3ab
Multiplying both sides of a2+3abb2+3ab>1\frac{a^2 + 3ab}{b^2 + 3ab} > 1 by (b2+3ab)(b^2 + 3ab) clears the fraction.
3
Subtract 3ab3ab from both sides.
a2>b2a^2 > b^2
Eliminating the common linear term isolates the quadratic terms.
4
Take the square root of both sides.
a>ba > b
Because aa and bb are positive, a2>b2a^2 > b^2 holds if and only if a>ba > b.

Anahtar Kavram

Data Sufficiency Question Stem Simplification
Soru 24Soru →

If kk is a positive integer, is k3−4kk+2\frac{k^3 - 4k}{k + 2} an odd integer?

(1) k2+3kk^2 + 3k is an even integer.
(2) 3k+13k + 1 is an odd integer.

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.
Rephrasing the target question k3−4kk+2\frac{k^3 - 4k}{k + 2} by factoring the numerator gives k(k−2)(k+2)k+2=k(k−2)\frac{k(k-2)(k+2)}{k+2} = k(k-2). Because kk and k−2k-2 share the same parity, their product k(k−2)k(k-2) is odd if and only if kk is odd. Thus, the question simplifies to: 'Is kk an odd integer?' Statement (1) tells us k2+3k=k(k+3)k^2 + 3k = k(k+3) is even, which is true for all integers kk, so it provides no information about whether kk is odd or even (NOT sufficient). Statement (2) tells us 3k+13k + 1 is odd, which implies 3k3k is even, and therefore kk must be even. Since kk is definitively even, the answer to 'Is kk an odd integer?' is a definitive 'No'. Therefore, Statement (2) alone is sufficient.

Adım Adım Çözüm

1
Rephrase and simplify the question stem algebraically.
Since kk is a positive integer, k+2>0k + 2 > 0. Factoring the numerator gives k3−4k=k(k2−4)=k(k−2)(k+2)k^3 - 4k = k(k^2 - 4) = k(k - 2)(k + 2). Dividing by k+2k + 2 yields k(k−2)(k+2)k+2=k(k−2)\frac{k(k - 2)(k + 2)}{k + 2} = k(k - 2).
Simplifying the target expression reduces cognitive complexity before evaluating statements.
2
Determine the parity condition for k(k−2)k(k - 2) to be an odd integer.
The factors kk and k−2k - 2 differ by 2, so they have the same parity. If kk is even, k(k−2)k(k - 2) is (even)(even) = even. If kk is odd, k(k−2)k(k - 2) is (odd)(odd) = odd. Thus, the target question 'Is k(k−2)k(k - 2) an odd integer?' simplifies to 'Is kk an odd integer?'
Rephrasing the target question to 'Is kk odd?' establishes a clear, minimal sufficiency criterion.
3
Evaluate Statement (1): k2+3kk^2 + 3k is an even integer.
Factor k2+3kk^2 + 3k as k(k+3)k(k + 3). The terms kk and k+3k + 3 differ by 3, meaning one is always even and the other is odd. Their product is ALWAYS even regardless of whether kk is odd or even. Therefore, Statement (1) provides no information about whether kk is odd. NOT sufficient.
A statement that holds true for all integers provides no distinguishing information about kk.
4
Evaluate Statement (2): 3k+13k + 1 is an odd integer.
If 3k+13k + 1 is odd, then 3k3k must be even. Since 3 is odd, 3k3k is even if and only if kk is even. Hence, kk is definitely even. This gives a definitive 'No' to the rephrased question 'Is kk an odd integer?'. A definitive 'No' means the statement IS sufficient.
In Data Sufficiency Yes/No questions, a statement that yields a conclusive 'No' is sufficient.

Anahtar Kavram

Data Sufficiency Question Stem Simplification and Parity Analysis
Tahmini Süre:1m 30s
Soru 25Soru →

If mm and nn are positive integers such that m≠nm \neq n, is m2+mn−2n2m−n\frac{m^2 + mn - 2n^2}{m - n} an even integer?

(1) m+3m + 3 is an odd integer.
(2) nn is an odd integer.

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.
By factoring the numerator of the expression in the question stem, m2+mn−2n2m−n=(m+2n)(m−n)m−n=m+2n\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n. Since 2n2n is always even, m+2nm + 2n is even if and only if mm is even. Rephrasing the question target yields: "Is mm an even integer?" Statement (1) tells us m+3m + 3 is odd, which means mm must be even (giving a definitive "Yes" answer). Statement (2) provides information about nn, which is irrelevant to whether mm is even. Thus, Statement (1) alone is sufficient, but Statement (2) alone is not.

Adım Adım Çözüm

1
Simplify the target expression in the question stem
m2+mn−2n2m−n=(m+2n)(m−n)m−n=m+2n\frac{m^2 + mn - 2n^2}{m - n} = \frac{(m + 2n)(m - n)}{m - n} = m + 2n
Factoring the numerator simplifies the target expression for m≠nm \neq n.
2
Rephrase the target question using parity properties
Since 2n2n is an even integer for any integer nn, m+2nm + 2n is even if and only if mm is an even integer. The rephrased question target is: "Is mm an even integer?"
Simplifying the target question eliminates the variable nn from the requirement.
3
Evaluate Statement (1): m+3m + 3 is an odd integer
If m+3m + 3 is odd, then m=odd−3=evenm = \text{odd} - 3 = \text{even}. Thus, mm is definitively even. Statement (1) alone is SUFFICIENT.
Subtracting an odd integer from an odd integer yields an even integer.
4
Evaluate Statement (2): nn is an odd integer
Statement (2) gives information about nn, but gives no information about mm. Since the rephrased question depends solely on mm, Statement (2) alone is INSUFFICIENT.
Knowing nn does not determine whether mm is even.

Anahtar Kavram

Algebraic factoring and target rephrasing in Data Sufficiency questions
ÖncekiSayfa 2 / 2
Question Stem Simplification and Target Rephrasing Alıştırma Soruları — GMAT — Sayfa 2 | Examkin