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Data Sufficiency

263 questions

All types263Multiple choice225True / False38
Question 241Question →

An investment fund holds a portfolio consisting entirely of short-term bonds yielding 4%4\% annually and long-term bonds yielding 8%8\% annually. In 2025, the total dollar amount invested in short-term bonds changed by p%p\%, and the total dollar amount invested in long-term bonds changed by q%q\%. Was the portfolio's overall percentage yield higher in 2025 than in 2024?

(1) In 2024, the amount invested in short-term bonds was twice the amount invested in long-term bonds.
(2) In 2025, the annual interest income earned from long-term bonds was greater than the annual interest income earned from short-term bonds.

Show answer & explanation

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the question stem algebraically demonstrates that the overall portfolio yield increases if and only if the relative proportion of the higher-yielding asset (8%8\%) increases compared to the lower-yielding asset (4%4\%), which simplifies to the condition q>pq > p. Statement (1) alone provides only the initial ratio S=2LS = 2L, leaving pp and qq unconstrained. Statement (2) alone yields 0.08L(1+q/100)>0.04S(1+p/100)0.08L(1 + q/100) > 0.04S(1 + p/100), which cannot be simplified without knowing the ratio S/LS/L. Combining both statements allows substituting S=2LS = 2L into Statement (2)'s inequality, yielding 2L(1+q/100)>2L(1+p/100)  ⟹  q>p2L(1 + q/100) > 2L(1 + p/100) \implies q > p, which definitively answers 'Yes'. Thus, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the question target algebraically.
Overall yield in 2024 is Y2024=0.04S+0.08LS+LY_{2024} = \frac{0.04S + 0.08L}{S + L}. Overall yield in 2025 is Y2025=0.04S(1+p/100)+0.08L(1+q/100)S(1+p/100)+L(1+q/100)Y_{2025} = \frac{0.04S(1 + p/100) + 0.08L(1 + q/100)}{S(1 + p/100) + L(1 + q/100)}. Since YY is a weighted average of two fixed values (4%4\% and 8%8\%), Y2025>Y2024Y_{2025} > Y_{2024} if and only if the weight of the higher-yielding component (8%8\%) increases relative to the lower-yielding component (4%4\%). That requires L(1+q/100)S(1+p/100)>LS\frac{L(1 + q/100)}{S(1 + p/100)} > \frac{L}{S}, which simplifies to 1+q/100>1+p/100  ⟺  q>p1 + q/100 > 1 + p/100 \iff q > p. The target question is rephrased as: 'Is q>pq > p?'
Simplifying the question stem reveals that the initial investment amounts SS and LL drop out when asking about relative yield changes.
2
Evaluate Statement (1) independently.
Statement (1) gives S=2LS = 2L. This provides information about the baseline 2024 portfolio composition, but provides no information regarding pp or qq. Therefore, we cannot determine whether q>pq > p. Statement (1) alone is NOT sufficient.
Knowing initial weights without growth rates does not allow determination of relative growth rate direction.
3
Evaluate Statement (2) independently.
Statement (2) states that interest from long-term bonds in 2025 exceeded interest from short-term bonds: 0.08L(1+q/100)>0.04S(1+p/100)0.08L(1 + q/100) > 0.04S(1 + p/100), which simplifies to 2L(1+q/100)>S(1+p/100)2L(1 + q/100) > S(1 + p/100). Because the ratio S/LS/L is unknown, this inequality could hold even if q≤pq \le p (if SS is very small relative to LL) or only when q>pq > p (if SS is large). Statement (2) alone is NOT sufficient.
Without knowing S/LS/L, the inequality 2L(1+q/100)>S(1+p/100)2L(1+q/100) > S(1+p/100) cannot be reduced to compare qq and pp.
4
Evaluate Statements (1) and (2) together.
From Statement (1), substitute S=2LS = 2L into Statement (2)'s inequality: 2L(1+q/100)>(2L)(1+p/100)2L(1 + q/100) > (2L)(1 + p/100). Dividing both sides by 2L2L (since L>0L > 0) gives 1+q/100>1+p/100  ⟺  q>p1 + q/100 > 1 + p/100 \iff q > p. This provides a definitive 'Yes' to the rephrased question target. BOTH statements together are SUFFICIENT.
Combining the baseline ratio with the interest inequality allows SS and LL to cancel, establishing q>pq > p definitively.

Key Concept

Data Sufficiency Question Stem Rephrasing and Weighted Average Yield Ratios
Question 242Question →

In a GMAT Data Sufficiency question asking for the specific numerical values of two variables xx and yy, if Statement (1) provides the equation 2x+3y=122x + 3y = 12 and Statement (2) provides the equation 4x+6y=244x + 6y = 24, combining both statements is sufficient to determine unique values for xx and yy.

Show answer & explanation

Answer: False

Answer

The statement is False.
The correct answer is False because Statement (2) is a scalar multiple of Statement (1). Linearly dependent equations do not reduce the number of unknown variables or narrow down the solution set to a single unique pair (x,y)(x, y). Therefore, combining both statements does not yield sufficient information to answer the question stem.

Step-by-Step Solution

1
Analyze Statement (1) equation
Statement (1) provides 2x+3y=122x + 3y = 12, which defines a line with infinitely many valid coordinate pairs (x,y)(x, y).
A single linear equation in two variables has infinitely many solutions.
2
Analyze Statement (2) equation and test for linear independence
Dividing Statement (2), 4x+6y=244x + 6y = 24, by 22 yields 2x+3y=122x + 3y = 12, which is identical to Statement (1).
Linearly dependent equations provide redundant data and do not provide a distinct second constraint.
3
Evaluate the combination of Statement (1) and Statement (2)
Combining both statements produces no new information beyond 2x+3y=122x + 3y = 12, leaving infinitely many solution pairs.
Determining unique values for two variables requires two linearly independent equations. Because these statements are dependent, combining them remains insufficient.

Key Concept

Linear Independence in Statement Combination
Question 243Question →

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

(1) x+y=10x + y = 10
(2) 3x−5y=03x - 5y = 0

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Evaluating Statement (2) independently of Statement (1) shows that the equation 3x−5y=03x - 5y = 0 can be rearranged to 3x=5y3x = 5y. Dividing both sides by 3y3y (since y≠0y \neq 0) yields xy=53\frac{x}{y} = \frac{5}{3}. This gives a single, unique value for the target ratio without needing any information from Statement (1). Meanwhile, Statement (1) alone (x+y=10x + y = 10) allows infinitely many pairs of non-zero real numbers (x,y)(x, y), resulting in different values for the ratio xy\frac{x}{y}. Therefore, Statement (2) alone is sufficient, but Statement (1) alone is not sufficient.

Step-by-Step Solution

1
Evaluate Statement (1) independently
Statement (1) gives x+y=10x + y = 10. If x=5x = 5 and y=5y = 5, then xy=1\frac{x}{y} = 1. If x=8x = 8 and y=2y = 2, then xy=4\frac{x}{y} = 4.
Since multiple values are possible for the ratio xy\frac{x}{y}, Statement (1) alone is not sufficient.
2
Evaluate Statement (2) independently without carrying over information from Statement (1)
Statement (2) gives 3x−5y=0  ⟹  3x=5y  ⟹  xy=533x - 5y = 0 \implies 3x = 5y \implies \frac{x}{y} = \frac{5}{3}.
Since y≠0y \neq 0, dividing both sides by 3y3y yields a single, unique value of 53\frac{5}{3} for the ratio xy\frac{x}{y}.
3
Conclude Data Sufficiency determination
Statement (2) alone is sufficient, whereas Statement (1) alone is not sufficient.
Statement (2) independently answers the question stem completely.

Key Concept

Evaluating Statement 2 independently of Statement 1 in Data Sufficiency
Question 244Question →

If xx and yy are real numbers such that x≠yx \neq y, what is the value of x+yx−y\frac{x+y}{x-y}?

(1) x2+y2=5xyx^2 + y^2 = 5xy
(2) x2−y2=12x^2 - y^2 = 12

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
Evaluating statement (1) reveals that (x+yx−y)2=73\left(\frac{x+y}{x-y}\right)^2 = \frac{7}{3}, leading to two possible values: 73\sqrt{\frac{7}{3}} and −73-\sqrt{\frac{7}{3}}. Thus, statement (1) alone is insufficient. Statement (2) states that (x+y)(x−y)=12(x+y)(x-y) = 12, which shows that (x+y)(x+y) and (x−y)(x-y) have the same sign, meaning their quotient must be positive. Combining both statements eliminates the negative value, yielding the unique result 73\sqrt{\frac{7}{3}}. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Analyze Statement (1) algebraically.
From x2+y2=5xyx^2 + y^2 = 5xy, add 2xy2xy to both sides to obtain (x+y)2=7xy(x+y)^2 = 7xy, and subtract 2xy2xy from both sides to obtain (x−y)2=3xy(x-y)^2 = 3xy. Taking the ratio gives (x+yx−y)2=7xy3xy=73\left(\frac{x+y}{x-y}\right)^2 = \frac{7xy}{3xy} = \frac{7}{3}. Taking the square root yields x+yx−y=±73\frac{x+y}{x-y} = \pm \sqrt{\frac{7}{3}}.
Since two distinct numerical values are possible, Statement (1) alone is NOT sufficient.
2
Analyze Statement (2) algebraically.
Statement (2) gives x2−y2=12x^2 - y^2 = 12, which factors as (x+y)(x−y)=12(x+y)(x-y) = 12.
Knowing only the product of (x+y)(x+y) and (x−y)(x-y) does not provide enough information to determine the value of their quotient x+yx−y\frac{x+y}{x-y}. Statement (2) alone is NOT sufficient.
3
Combine Statement (1) and Statement (2).
Rewrite the target ratio using Statement (2): x+yx−y=(x+y)(x−y)(x−y)2=12(x−y)2\frac{x+y}{x-y} = \frac{(x+y)(x-y)}{(x-y)^2} = \frac{12}{(x-y)^2}. Because (x−y)2>0(x-y)^2 > 0 for x≠yx \neq y, the ratio must be strictly positive. Combining this with Statement (1), which requires x+yx−y=±73\frac{x+y}{x-y} = \pm \sqrt{\frac{7}{3}}, uniquely isolates the positive solution x+yx−y=73\frac{x+y}{x-y} = \sqrt{\frac{7}{3}}.
Together, the statements determine a single, unique value for the target expression.

Key Concept

Algebraic Rephrasing and Sign Constraints in Data Sufficiency
Estimated Time:2m 0s
Question 245Question →

A bakery sells two types of artisanal bread: Sourdough and Brioche. Last week, the bakery sold a total of 1,2001,200 loaves of bread. Did Sourdough account for more than 60%60\% of the total revenue generated from sales of these two types of bread?

(1) The selling price of a loaf of Brioche was 25%25\% greater than the selling price of a loaf of Sourdough.
(2) The number of Sourdough loaves sold was 70%70\% of the total number of loaves sold.

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
The correct answer identifies that neither statement alone gives both the price ratio and quantity ratio, but taking both statements together provides PSPB=45\frac{P_S}{P_B} = \frac{4}{5} and NSNB=73\frac{N_S}{N_B} = \frac{7}{3}. Multiplying these yields NSPSNBPB=2815≈1.867\frac{N_S P_S}{N_B P_B} = \frac{28}{15} \approx 1.867, which is greater than 1.51.5 (the threshold required for a 60%60\% revenue share). Thus, both statements together provide a definitive 'Yes' answer.

Step-by-Step Solution

1
Rephrase the question stem target into an algebraic condition.
Let NSN_S and NBN_B be the quantities of Sourdough and Brioche sold, and PSP_S and PBP_B be their respective unit prices. Sourdough revenue is RS=NSPSR_S = N_S P_S and total revenue is Rtotal=NSPS+NBPBR_{total} = N_S P_S + N_B P_B. We need to test if NSPSNSPS+NBPB>0.60\frac{N_S P_S}{N_S P_S + N_B P_B} > 0.60, which simplifies to NSPSNBPB>1.5\frac{N_S P_S}{N_B P_B} > 1.5.
Simplifying the target inequality shows that knowing the combined ratio NSPSNBPB=(NSNB)×(PSPB)\frac{N_S P_S}{N_B P_B} = \left(\frac{N_S}{N_B}\right) \times \left(\frac{P_S}{P_B}\right) is necessary and sufficient.
2
Evaluate Statement (1) independently.
Statement (1) states PB=1.25PS=54PSP_B = 1.25 P_S = \frac{5}{4} P_S, so PSPB=45\frac{P_S}{P_B} = \frac{4}{5}.
Since no information is given about the ratio of quantities NSNB\frac{N_S}{N_B}, the overall revenue ratio cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Statement (2) states NS=0.70×1,200=840N_S = 0.70 \times 1,200 = 840, which means NB=360N_B = 360. Thus NSNB=840360=73\frac{N_S}{N_B} = \frac{840}{360} = \frac{7}{3}.
Since no information is given about the relative unit prices PSPB\frac{P_S}{P_B}, the overall revenue ratio cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together.
Combine the price ratio PSPB=45\frac{P_S}{P_B} = \frac{4}{5} and quantity ratio NSNB=73\frac{N_S}{N_B} = \frac{7}{3}. The revenue ratio is NSPSNBPB=(73)×(45)=2815≈1.867\frac{N_S P_S}{N_B P_B} = \left(\frac{7}{3}\right) \times \left(\frac{4}{5}\right) = \frac{28}{15} \approx 1.867.
Since 1.867>1.51.867 > 1.5, Sourdough's revenue share is strictly greater than 60%60\%. This yields a definitive 'Yes', so both statements together are sufficient.

Key Concept

Data Sufficiency evaluation of revenue proportions using combined quantity and price ratios.
Estimated Time:1m 30s
Question 246Question →

A regional freight company operates two transport fleets: Fleet X and Fleet Y. Last quarter, what was the ratio of the total operating expense of Fleet X to the total operating expense of Fleet Y?

(1) Fleet X's fuel expense was 40%40\% of its total operating expense, and Fleet Y's fuel expense was 25%25\% of its total operating expense.
(2) The combined fuel expense of both fleets was 30%30\% of their combined total operating expenses.

Show answer & explanation

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The statement expressing that both statements together are sufficient while neither alone is sufficient is correct because Statement (1) gives the component percentages (40%40\% and 25%25\%) and Statement (2) gives the overall weighted percentage (30%30\%). Equating the component sum 0.40EX+0.25EY0.40E_X + 0.25E_Y to the weighted total 0.30(EX+EY)0.30(E_X + E_Y) yields a single linear equation that solves directly for the ratio EX/EY=1/2E_X / E_Y = 1/2.

Step-by-Step Solution

1
Define variables and target ratio
Let EXE_X be the total operating expense of Fleet X and EYE_Y be the total operating expense of Fleet Y. The target is to determine the ratio EXEY\frac{E_X}{E_Y}.
Rephrasing the question stem into a formal mathematical expression highlights what relationship needs to be determined.
2
Evaluate Statement (1) alone
Fuel expense of Fleet X =0.40EX= 0.40 E_X and Fuel expense of Fleet Y =0.25EY= 0.25 E_Y.
Statement (1) provides individual percentages, but contains no information relating EXE_X to EYE_Y. Hence, Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone
Total combined fuel expense =0.30(EX+EY)= 0.30 (E_X + E_Y).
Without knowing the individual fuel percentages for Fleet X and Fleet Y, we cannot express the total combined fuel expense in terms of EXE_X and EYE_Y. Hence, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together
Equate the sum of individual fuel expenses to the total combined fuel expense:
0.40EX+0.25EY=0.30(EX+EY)0.40 E_X + 0.25 E_Y = 0.30 (E_X + E_Y)
0.40EX+0.25EY=0.30EX+0.30EY0.40 E_X + 0.25 E_Y = 0.30 E_X + 0.30 E_Y
0.10EX=0.05EY0.10 E_X = 0.05 E_Y
EXEY=0.050.10=12\frac{E_X}{E_Y} = \frac{0.05}{0.10} = \frac{1}{2}
Setting the two expressions for total fuel expense equal creates a linear homogeneous equation that uniquely determines the exact ratio of EXE_X to EYE_Y.

Key Concept

Weighted Average Ratios in Data Sufficiency

Practice More

Practice Data Sufficiency questions involving weighted average mixtures where component percentages lie on opposite sides of the combined percentage.

Alternative Method

Alligation / Weighted Average Formula: The distance between Fleet X's percentage (40%40\%) and the combined percentage (30%30\%) is 10%10\%. The distance between Fleet Y's percentage (25%25\%) and the combined percentage (30%30\%) is 5%5\%. The ratio of Fleet X to Fleet Y is inversely proportional to these distances: EX/EY=5/10=1/2E_X / E_Y = 5 / 10 = 1 / 2.
Estimated Time:1m 30s
Question 247Question →

If aa and bb are non-zero real numbers, what is the value of a2+b2ab\frac{a^2 + b^2}{ab}?

(1) a+b=3aba + b = 3ab
(2) a−b=aba - b = ab

Show answer & explanation

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

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Evaluating each statement independently shows that Statement (1) reduces the expression to 9ab−29ab - 2 and Statement (2) reduces it to ab+2ab + 2, both of which depend on the value of abab. Combining both statements yields a system of linear equations in aa and bb, giving a=1a = 1 and b=1/2b = 1/2. This uniquely determines the value of the target expression as 5/25/2. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the target expression.
a2+b2ab=ab+ba\frac{a^2 + b^2}{ab} = \frac{a}{b} + \frac{b}{a}. Alternatively, squaring identities give a2+b2ab=(a+b)2−2abab=(a+b)2ab−2\frac{a^2+b^2}{ab} = \frac{(a+b)^2 - 2ab}{ab} = \frac{(a+b)^2}{ab} - 2.
Simplifying the target expression shows what combination of variables is required.
2
Evaluate Statement (1) alone: a+b=3aba + b = 3ab.
Substituting a+b=3aba+b = 3ab into the target expression yields (3ab)2−2abab=9ab−2\frac{(3ab)^2 - 2ab}{ab} = 9ab - 2.
Since the value depends on abab, and abab can take multiple non-zero values (e.g., if a=1,b=1/2a=1, b=1/2, ab=1/2ab=1/2 and the expression is 2.52.5; if a=2,b=2/5a=2, b=2/5, ab=4/5ab=4/5 and the expression is 5.25.2), Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) alone: a−b=aba - b = ab.
Expressing a2+b2=(a−b)2+2ab=(ab)2+2aba^2+b^2 = (a-b)^2 + 2ab = (ab)^2 + 2ab, the target expression becomes (ab)2+2abab=ab+2\frac{(ab)^2 + 2ab}{ab} = ab + 2.
Since the value depends on abab, which is not fixed by Statement (2) alone, Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
Adding the two equations (a+b)+(a−b)=3ab+ab(a + b) + (a - b) = 3ab + ab gives 2a=4ab2a = 4ab. Since a≠0a \neq 0, dividing by 2a2a yields b=12b = \frac{1}{2}.
Substituting b=12b = \frac{1}{2} into a−b=aba - b = ab gives a−12=12a  ⟹  a=1a - \frac{1}{2} = \frac{1}{2}a \implies a = 1. With a=1a=1 and b=12b=\frac{1}{2}, ab=12ab = \frac{1}{2}, and a2+b2ab=1+1/41/2=52\frac{a^2+b^2}{ab} = \frac{1 + 1/4}{1/2} = \frac{5}{2}, giving a single unique value.

Key Concept

Algebraic System Reduction and Target Rephrasing in Data Sufficiency
Estimated Time:2m 0s
Question 248Question →

An agricultural supplier sells two grades of organic fertilizer: Premium grade and Standard grade. Last month, all inventory of both grades was sold. What percentage of the supplier's total revenue last month came from sales of Premium grade fertilizer?

(1) The price per bag of Premium grade fertilizer was 50%50\% higher than the price per bag of Standard grade fertilizer.
(2) The number of bags of Standard grade fertilizer sold last month was 20%20\% less than the number of bags of Premium grade fertilizer sold.

Show answer & explanation

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

Answer

Both statements together are sufficient to determine the exact percentage of total revenue coming from Premium grade fertilizer, but neither statement alone is sufficient.
The correct choice identifies that neither statement alone gives both the price ratio and the quantity ratio required to calculate the percentage contribution to total revenue. Statement (1) provides only the relative prices (pPpS=1.5\frac{p_P}{p_S} = 1.5), while Statement (2) provides only the relative quantities sold (S=0.8PS = 0.8P). Combining both gives the combined revenue ratio S⋅pSP⋅pP=45×23=815\frac{S \cdot p_S}{P \cdot p_P} = \frac{4}{5} \times \frac{2}{3} = \frac{8}{15}, which allows calculating the exact percentage 1523\frac{15}{23}. Thus, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the question target algebraically.
Let PP and SS be the quantities of Premium and Standard bags sold, and let pPp_P and pSp_S be their respective unit prices. Total revenue is Rtotal=P⋅pP+S⋅pSR_{total} = P \cdot p_P + S \cdot p_S. The target percentage is P⋅pPP⋅pP+S⋅pS=11+S⋅pSP⋅pP\frac{P \cdot p_P}{P \cdot p_P + S \cdot p_S} = \frac{1}{1 + \frac{S \cdot p_S}{P \cdot p_P}}. Finding this percentage requires knowing the ratio of total revenues S⋅pSP⋅pP\frac{S \cdot p_S}{P \cdot p_P}, which simplifies to (SP)×(pSpP)\left(\frac{S}{P}\right) \times \left(\frac{p_S}{p_P}\right).
Simplifying the target expression reduces the problem from requiring four individual variable values to requiring only one combined ratio.
2
Evaluate Statement (1) independently.
Statement (1) states pP=1.5⋅pSp_P = 1.5 \cdot p_S, which gives pSpP=11.5=23\frac{p_S}{p_P} = \frac{1}{1.5} = \frac{2}{3}. However, no information is given regarding the ratio of quantities sold SP\frac{S}{P}. Statement (1) alone is NOT sufficient.
Knowing only the price ratio leaves the quantity ratio unknown.
3
Evaluate Statement (2) independently.
Statement (2) states S=0.8⋅PS = 0.8 \cdot P, which gives SP=0.8=45\frac{S}{P} = 0.8 = \frac{4}{5}. However, no information is given regarding the price ratio pSpP\frac{p_S}{p_P}. Statement (2) alone is NOT sufficient.
Knowing only the quantity ratio leaves the price ratio unknown.
4
Evaluate Statements (1) and (2) together.
Combining both statements yields S⋅pSP⋅pP=(45)×(23)=815\frac{S \cdot p_S}{P \cdot p_P} = \left(\frac{4}{5}\right) \times \left(\frac{2}{3}\right) = \frac{8}{15}. Substituting this into the target expression gives 11+815=1523≈65.2%\frac{1}{1 + \frac{8}{15}} = \frac{15}{23} \approx 65.2\%. Therefore, both statements together are sufficient.
Combining the relative prices and relative quantities provides the exact ratio of revenues.

Key Concept

Data Sufficiency Target Rephrasing for Percentage of Total Revenue
Estimated Time:2m 0s
Question 249Question →

If uu and vv are real numbers, what is the value of u+2vu + 2v?

(1) 3u+6v=153u + 6v = 15
(2) u2−4v2=0u^2 - 4v^2 = 0

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Dividing Statement (1) by 3 directly gives u+2v=5u + 2v = 5, providing a unique numerical answer. Statement (2) permits multiple values for u+2vu + 2v because (u−2v)(u+2v)=0(u - 2v)(u + 2v) = 0 only guarantees that at least one factor is zero, not necessarily u+2v=0u + 2v = 0. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Analyze Statement (1) algebraically.
The equation is 3u+6v=153u + 6v = 15. Factoring out 3 gives 3(u+2v)=153(u + 2v) = 15. Dividing both sides by 3 yields u+2v=5u + 2v = 5.
This determines a unique, definitive numerical value for the target expression u+2vu + 2v. Therefore, Statement (1) alone is sufficient.
2
Analyze Statement (2) independently.
The equation u2−4v2=0u^2 - 4v^2 = 0 factors as (u−2v)(u+2v)=0(u - 2v)(u + 2v) = 0.
This statement implies that either u+2v=0u + 2v = 0 or u−2v=0u - 2v = 0. If u=4u = 4 and v=2v = 2, then u−2v=0u - 2v = 0 and u+2v=8u + 2v = 8. If u=0u = 0 and v=0v = 0, then u+2v=0u + 2v = 0. Since u+2vu + 2v can take multiple values, Statement (2) alone is not sufficient.

Key Concept

Algebraic Expression Simplification and Unique Determination in Data Sufficiency
Question 250Question →

If pp and qq are real numbers, what is the value of p2+4q2p^2 + 4q^2?

(1) p+2q=8p + 2q = 8
(2) pq=6pq = 6

Show answer & explanation

Answer: BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The target expression p2+4q2p^2 + 4q^2 can be related to the binomial square (p+2q)2=p2+4pq+4q2(p + 2q)^2 = p^2 + 4pq + 4q^2. Rearranging gives p2+4q2=(p+2q)2−4pqp^2 + 4q^2 = (p + 2q)^2 - 4pq. Neither statement alone provides both p+2qp + 2q and pqpq. However, combining Statement (1) (p+2q=8p + 2q = 8) and Statement (2) (pq=6pq = 6) allows direct substitution: p2+4q2=82−4(6)=40p^2 + 4q^2 = 8^2 - 4(6) = 40. Because this yields a single unique value, both statements together are sufficient.

Step-by-Step Solution

1
Evaluate Statement (1) independently
Statement (1) states p+2q=8p + 2q = 8. Squaring both sides yields (p+2q)2=p2+4pq+4q2=64(p + 2q)^2 = p^2 + 4pq + 4q^2 = 64, so p2+4q2=64−4pqp^2 + 4q^2 = 64 - 4pq. Without the value of pqpq, a unique numerical value for p2+4q2p^2 + 4q^2 cannot be determined.
Statement (1) alone leaves one degree of freedom (the product pqpq is unknown).
2
Evaluate Statement (2) independently
Statement (2) states pq=6pq = 6. Knowing only the product of pp and qq allows infinitely many pairs (p,q)(p, q), resulting in infinitely many values for p2+4q2p^2 + 4q^2.
Statement (2) alone does not constrain the linear sum p+2qp + 2q.
3
Combine Statements (1) and (2)
From Statement (1), p2+4q2=64−4pqp^2 + 4q^2 = 64 - 4pq. Substituting pq=6pq = 6 from Statement (2) yields p2+4q2=64−4(6)=40p^2 + 4q^2 = 64 - 4(6) = 40. This provides a unique, definitive numerical answer.
Combining both statements eliminates all unknown parameters from the target expression p2+4q2p^2 + 4q^2.

Key Concept

Algebraic Identity Expansion and System Combination in Data Sufficiency
Estimated Time:1m 30s
Question 251Question →

If xx and yy are real numbers, what is the value of x−yx - y?

(1) (x−y)2=16(x - y)^2 = 16
(2) x2−y2=24x^2 - y^2 = 24

Show answer & explanation

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The option stating that statements (1) and (2) together are not sufficient is correct because combining both equations results in two valid candidate solution sets for (x,y)(x, y), specifically (5,1)(5, 1) and (−5,−1)(-5, -1). These produce two different values for the target expression x−yx - y (44 and −4-4), preventing a single unique determination.

Step-by-Step Solution

1
Evaluate Statement (1) independently.
(x−y)2=16  ⟹  x−y=4(x - y)^2 = 16 \implies x - y = 4 or x−y=−4x - y = -4.
Taking the square root of both sides yields two possible values for x−yx - y, so Statement (1) alone is not sufficient.
2
Evaluate Statement (2) independently.
x2−y2=(x−y)(x+y)=24x^2 - y^2 = (x - y)(x + y) = 24.
Without knowing x+yx + y, the expression x−yx - y can take infinitely many numerical values, so Statement (2) alone is not sufficient.
3
Evaluate Statements (1) and (2) together.
Case 1: If x−y=4x - y = 4, then 4(x+y)=24  ⟹  x+y=64(x + y) = 24 \implies x + y = 6, giving (x,y)=(5,1)(x, y) = (5, 1) where x−y=4x - y = 4. Case 2: If x−y=−4x - y = -4, then −4(x+y)=24  ⟹  x+y=−6-4(x + y) = 24 \implies x + y = -6, giving (x,y)=(−5,−1)(x, y) = (-5, -1) where x−y=−4x - y = -4.
Both coordinate pairs (5,1)(5, 1) and (−5,−1)(-5, -1) satisfy both given statements, but yield two distinct values (44 and −4-4) for x−yx - y. Therefore, both statements combined remain insufficient.

Key Concept

Degree Ambiguity in Systems of Non-Linear Algebraic Equations
Question 252Question →

A cloud infrastructure provider operates two categories of servers: High-Memory servers and High-CPU servers. Last month, High-Memory servers accounted for 40%40\% of the total server-hours logged by the provider, and High-CPU servers accounted for the remaining 60%60\%. What was the average operational cost per server-hour across all servers operated by the provider last month?

(1) The operational cost per server-hour for a High-Memory server was $3.00\$3.00 greater than the operational cost per server-hour for a High-CPU server.

(2) The total operational cost for a combination of 200200 server-hours of High-Memory servers and 300300 server-hours of High-CPU servers was $1,400\$1,400.

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The correct response is that statement (2) alone is sufficient while statement (1) alone is not. Rephrasing the question stem shows that the overall average cost per hour is 0.40m+0.60c0.40m + 0.60c, where mm and cc represent the hourly rates of High-Memory and High-CPU servers, respectively. Statement (1) gives only the difference m−c=3m - c = 3, which leaves the overall average dependent on an unknown base rate. Statement (2) provides the equation 200m+300c=1,400200m + 300c = 1,400. Dividing this equation by 500500 yields 0.40m+0.60c=2.800.40m + 0.60c = 2.80, directly matching the target expression without requiring individual values for mm or cc.

Step-by-Step Solution

1
Rephrase the question stem algebraically.
Let mm be the hourly cost of High-Memory servers and cc be the hourly cost of High-CPU servers. The target average cost per server-hour is A=0.40m+0.60cA = 0.40m + 0.60c.
Rephrasing identifies the exact linear combination of variables required to solve the question.
2
Evaluate Statement (1) independently.
Statement (1) states m=c+3m = c + 3. Substituting into the target gives A=0.40(c+3)+0.60c=c+1.20A = 0.40(c + 3) + 0.60c = c + 1.20.
Since the value of cc remains unknown, a single numerical value for AA cannot be determined. Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) gives 200m+300c=1,400200m + 300c = 1,400. Dividing the entire equation by 500500 yields 200500m+300500c=1,400500\frac{200}{500}m + \frac{300}{500}c = \frac{1,400}{500}, which simplifies to 0.40m+0.60c=2.800.40m + 0.60c = 2.80.
The coefficients of the given equation match the target proportion (40%40\% and 60%60\%), directly providing the exact value A=$2.80A = \$2.80. Statement (2) alone IS sufficient.

Key Concept

Weighted Averages and Expression Rephrasing in Data Sufficiency
Estimated Time:2m 0s
Question 253Question →

A specialty coffee roaster produces a house blend composed entirely of Arabica beans, which cost 12perpound,andRobustabeans,whichcost12 per pound, and Robusta beans, which cost 8 per pound. What was the average cost per pound of the house blend produced last week?

(1) Arabica beans accounted for 60% of the total weight of the house blend produced last week.
(2) The total weight of the house blend produced last week was 500 pounds.

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The average cost per pound of a blend depends only on the relative percentage weighting of its components. Since Statement (1) provides that Arabica beans constitute 60% of the blend, the remaining 40% must be Robusta. The weighted average cost per pound can be calculated directly as 0.60($12)+0.40($8)=$10.400.60(\$12) + 0.40(\$8) = \$10.40. Statement (2) gives total weight but no ratio information, leaving the average cost indeterminate. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not.

Step-by-Step Solution

1
Formulate the algebraic target for weighted average cost per pound.
Let wAw_A be the proportion of total weight consisting of Arabica beans and wR=1−wAw_R = 1 - w_A be the proportion consisting of Robusta beans. Average cost per pound =12wA+8(1−wA)=4wA+8= 12 w_A + 8 (1 - w_A) = 4 w_A + 8.
Rephrasing the question stem shows that knowing the weight fraction wAw_A alone is sufficient to calculate a unique average cost per pound.
2
Evaluate Statement (1) independently.
Statement (1) specifies wA=0.60w_A = 0.60. Average cost per pound =4(0.60)+8=$10.40= 4(0.60) + 8 = \$10.40.
A single unique numerical value is obtained, making Statement (1) ALONE sufficient.
3
Evaluate Statement (2) independently.
Statement (2) specifies that total weight is 500500 pounds, but the proportion wAw_A can be any value between 00 and 11.
Without knowing the proportion of each bean type, the average cost per pound cannot be uniquely determined, so Statement (2) ALONE is not sufficient.

Key Concept

Weighted Average Ratios in Data Sufficiency
Estimated Time:1m 30s
Question 254Question →

If xx and yy are real numbers, what is the value of (x+y)2(x + y)^2?

(1) x2+y2=25x^2 + y^2 = 25
(2) x2−y2=7x^2 - y^2 = 7

Show answer & explanation

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

Answer

Statements (1) and (2) TOGETHER are NOT sufficient.
The correct response states that Statements (1) and (2) together are not sufficient because solving the system yields x2=16x^2 = 16 and y2=9y^2 = 9. This gives x=±4x = \pm 4 and y=±3y = \pm 3, allowing xyxy to be either 1212 or −12-12. Consequently, (x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy can equal either 4949 or 11, which prevents finding a single unique value.

Step-by-Step Solution

1
Rephrase the target expression using algebraic identities
(x+y)2=x2+y2+2xy(x + y)^2 = x^2 + y^2 + 2xy
Expanding the square shows that we need the sum of squares x2+y2x^2 + y^2 and the product term 2xy2xy (or the individual values of xx and yy) to determine a unique value.
2
Evaluate Statement (1) alone
x2+y2=25x^2 + y^2 = 25, but xyxy can take multiple values.
For example, if x=5x = 5 and y=0y = 0, then (x+y)2=25(x + y)^2 = 25. If x=4x = 4 and y=3y = 3, then (x+y)2=49(x + y)^2 = 49. Multiple values are possible, so Statement (1) alone is insufficient.
3
Evaluate Statement (2) alone
x2−y2=7x^2 - y^2 = 7, but (x+y)2(x + y)^2 is not uniquely determined.
If x=4x = 4 and y=3y = 3, then x2−y2=16−9=7x^2 - y^2 = 16 - 9 = 7 and (x+y)2=49(x + y)^2 = 49. If x=7x = \sqrt{7} and y=0y = 0, then x2−y2=7x^2 - y^2 = 7 and (x+y)2=7(x + y)^2 = 7. Multiple values are possible, so Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined
x2=16x^2 = 16 and y2=9y^2 = 9, which leads to x=±4x = \pm 4 and y=±3y = \pm 3.
Adding the two equations gives 2x2=32  ⟹  x2=162x^2 = 32 \implies x^2 = 16. Subtracting gives 2y2=18  ⟹  y2=92y^2 = 18 \implies y^2 = 9. Thus x=4x = 4 or −4-4, and y=3y = 3 or −3-3.
5
Check for sign ambiguity in the cross-term xyxy
If x=4,y=3x = 4, y = 3, then (x+y)2=72=49(x + y)^2 = 7^2 = 49. If x=4,y=−3x = 4, y = -3, then (x+y)2=12=1(x + y)^2 = 1^2 = 1.
Because the system of non-linear equations determines x2x^2 and y2y^2 but not the relative signs of xx and yy, there are two distinct valid values for (x+y)2(x + y)^2. Therefore, Statements (1) and (2) together are NOT sufficient.

Key Concept

Non-linear systems of equations and degree ambiguity in Data Sufficiency
Estimated Time:2m 0s
Question 255Question →

If aa and bb are real numbers, what is the value of a3−b3a^3 - b^3?

(1) a−b=4a - b = 4
(2) a2+b2=26a^2 + b^2 = 26

Show answer & explanation

Answer: BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Answer

BOTH statements (1) and (2) TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
The correct answer choice states that both statements together are sufficient, but neither alone is sufficient. Statement (1) gives the linear difference between the variables, and Statement (2) gives the sum of their squares. Neither alone yields a single value for a3−b3a^3 - b^3. However, squaring Statement (1) allows us to isolate the product ab=5ab = 5. Using the standard factoring identity a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2), we can substitute the known components to find the unique value 124.

Step-by-Step Solution

1
Rephrase the target expression using algebraic factoring identities.
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Expanding or factoring the expression reveals that we need the values of (a−b)(a - b), (a2+b2)(a^2 + b^2), and abab.
2
Evaluate Statement (1) independently.
a−b=4a - b = 4
Knowing a−b=4a - b = 4 leaves a2+ab+b2a^2 + ab + b^2 unknown. For example, if a=4,b=0a=4, b=0, then a3−b3=64a^3-b^3=64. If a=5,b=1a=5, b=1, then a3−b3=124a^3-b^3=124. Multiple values exist, so Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
a2+b2=26a^2 + b^2 = 26
If a=5,b=1a=5, b=1, then a3−b3=124a^3-b^3=124. If a=26,b=0a=\sqrt{26}, b=0, then a3−b3=2626a^3-b^3=26\sqrt{26}. Multiple values exist, so Statement (2) alone is insufficient.
4
Evaluate both statements combined.
ab=5ab = 5 and a3−b3=124a^3 - b^3 = 124
From Statement (1), (a−b)2=a2−2ab+b2=42=16(a - b)^2 = a^2 - 2ab + b^2 = 4^2 = 16. Substituting Statement (2) into this equation gives 26−2ab=16  ⟹  2ab=10  ⟹  ab=526 - 2ab = 16 \implies 2ab = 10 \implies ab = 5. Now substituting all values into the factored identity gives a3−b3=4×(26+5)=4×31=124a^3 - b^3 = 4 \times (26 + 5) = 4 \times 31 = 124. This gives a single, unique value.

Key Concept

Algebraic Factoring of Difference of Cubes and System Solvability
Question 256Question →

An eco-friendly power facility generates electricity exclusively using solar panels and wind turbines. Last month, energy from solar panels was produced at a constant cost of 0.04perkilowatt−hour(kWh),andenergyfromwindturbineswasproducedataconstantcostof0.04 per kilowatt-hour (kWh), and energy from wind turbines was produced at a constant cost of 0.06 per kWh. What was the average cost per kWh of the total energy generated by the facility last month?

(1) Last month, the facility generated 300,000 kWh of energy using solar panels.
(2) Last month, the ratio of the energy generated by wind turbines to the energy generated by solar panels was 2 to 3.

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The question asks for the weighted average cost per kWh of electricity produced by two sources with given unit costs (0.04and0.04 and 0.06). A weighted average of two component rates depends solely on the ratio of the quantities involved, not on the total absolute quantities. Statement (1) gives only the quantity of solar energy without providing any information about wind energy, making it insufficient. Statement (2) gives the exact ratio of wind energy to solar energy (2:32:3), which allows us to find the exact weighted average cost ($0.048 per kWh) without knowing the total energy produced. Therefore, Statement (2) alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target using weighted averages.
Let SS be the kWh generated from solar panels and WW be the kWh generated from wind turbines. The average cost per kWh is given by 0.04S+0.06WS+W=0.04(SS+W)+0.06(WS+W)\frac{0.04S + 0.06W}{S + W} = 0.04\left(\frac{S}{S+W}\right) + 0.06\left(\frac{W}{S+W}\right).
Simplifying the formula shows that the average cost depends strictly on the relative ratio of WW to SS (or the proportion of each source to the total), rather than absolute kWh quantities.
2
Evaluate Statement (1): Last month, the facility generated 300,000 kWh of energy using solar panels.
S=300,000S = 300,000, but WW is unknown.
Since the amount of wind energy WW is completely unknown, we cannot calculate the weighted average cost. Statement (1) is NOT sufficient.
3
Evaluate Statement (2): The ratio of energy generated by wind turbines to solar panels was 2 to 3.
WS=23⇒W=23S\frac{W}{S} = \frac{2}{3} \Rightarrow W = \frac{2}{3}S. Substituting into the average cost expression yields 0.04S+0.06(23S)S+23S=0.04S+0.04S53S=0.085/3=$0.048\frac{0.04S + 0.06(\frac{2}{3}S)}{S + \frac{2}{3}S} = \frac{0.04S + 0.04S}{\frac{5}{3}S} = \frac{0.08}{5/3} = \$0.048 per kWh.
Because SS cancels out completely, knowing only the ratio gives a unique numerical value for the average cost per kWh. Statement (2) is ALONE sufficient.

Key Concept

Weighted Average Ratios in Data Sufficiency
Question 257Question →

A logistics facility processes two types of parcels: Express packages and Standard packages. Last week, what percentage of the total parcels processed were Express packages?

(1) The average weight of an Express package was 4.54.5 kilograms, and the average weight of a Standard package was 2.52.5 kilograms.
(2) The overall average weight of all parcels processed last week was 3.13.1 kilograms.

Show answer & explanation

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

Answer

Both statements together are sufficient, but neither statement alone is sufficient.
The correct response indicates that both statements together are sufficient, but neither statement alone is sufficient. Statement (1) provides only individual package weights, and Statement (2) provides only the overall average package weight. Neither statement alone reveals the mixture ratio. Combining both statements yields the weighted average equation 4.5E+2.5S=3.1(E+S)4.5E + 2.5S = 3.1(E + S), which simplifies to 1.4E=0.6S1.4E = 0.6S, establishing the unique ratio ES=37\frac{E}{S} = \frac{3}{7}. This uniquely determines that Express packages comprise 30%30\% of the total.

Step-by-Step Solution

1
Define target variable and analyze the question stem
Let EE be the number of Express packages and SS be the number of Standard packages. The target question asks for the percentage EE+S×100%\frac{E}{E + S} \times 100\%. Finding the ratio ES\frac{E}{S} is sufficient to determine this percentage.
Rephrasing a percentage target into a ratio target simplifies statement evaluation.
2
Evaluate Statement (1) independently
Statement (1) states that the average weight of an Express package is 4.54.5 kg and a Standard package is 2.52.5 kg. Without knowing the overall average weight or relative proportion, ES\frac{E}{S} cannot be determined.
Knowing individual component averages alone leaves the weighted ratio undetermined.
3
Evaluate Statement (2) independently
Statement (2) states that the overall average weight is 3.13.1 kg. Without the individual component weights, the relative breakdown cannot be determined.
Knowing the combined average weight alone provides no boundary values for individual component rates.
4
Evaluate Statements (1) and (2) together
Using weighted average principles: 4.5E+2.5S=3.1(E+S)⇒4.5E+2.5S=3.1E+3.1S⇒1.4E=0.6S⇒ES=0.61.4=374.5E + 2.5S = 3.1(E + S) \Rightarrow 4.5E + 2.5S = 3.1E + 3.1S \Rightarrow 1.4E = 0.6S \Rightarrow \frac{E}{S} = \frac{0.6}{1.4} = \frac{3}{7}. The percentage of Express packages is 33+7×100%=30%\frac{3}{3 + 7} \times 100\% = 30\%.
Combining individual component averages with the overall weighted average determines a unique ratio of components.

Key Concept

Weighted Average Ratios in Data Sufficiency
Question 258Question →

An agricultural processing facility produces a liquid concentrate by blending organic apple juice, which contains 12%12\% sugar by volume, and organic grape juice, which contains 18%18\% sugar by volume. If a batch of the concentrate is made by mixing only these two types of juice, is the total volume of grape juice used in the batch greater than the total volume of apple juice used in the batch?

(1) The concentration of sugar in the batch of concentrate is 16%16\% by volume.
(2) The total volume of the batch of concentrate is 150150 liters.

Show answer & explanation

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Evaluating Statement (1) alone shows that a batch with a 16%16\% overall sugar concentration made from 12%12\% and 18%18\% components must contain twice as much grape juice as apple juice (G=2AG = 2A). Because juice volumes are strictly positive, GG must be greater than AA, providing a definitive 'Yes' answer. Statement (2) alone provides only the total volume A+G=150A + G = 150, which allows for infinitely many combinations of AA and GG where G>AG > A could be either true or false. Thus, Statement (1) alone is sufficient and Statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question target using variables
Let AA be the volume of apple juice and GG be the volume of grape juice in liters. The question asks: Is G>AG > A?
Establishing clear algebraic expressions simplifies evaluating sufficiency.
2
Evaluate Statement (1) algebraically
Total sugar volume = 0.12A+0.18G0.12A + 0.18G. Total batch volume = A+GA + G.
Given sugar concentration = 16%16\%, set up equation: 0.12A+0.18GA+G=0.16\frac{0.12A + 0.18G}{A + G} = 0.16.
Multiply out: 0.12A+0.18G=0.16A+0.16G  ⟹  0.02G=0.04A  ⟹  G=2A0.12A + 0.18G = 0.16A + 0.16G \implies 0.02G = 0.04A \implies G = 2A.
Since volumes are positive (A>0A > 0), G=2A>AG = 2A > A is ALWAYS true (Answer is a definitive 'Yes').
Statement (1) uniquely fixes the ratio of grape juice to apple juice at 2:12:1, which fully answers the Yes/No question.
3
Evaluate Statement (2) independently
Given A+G=150A + G = 150.
Case 1: If A=100A = 100 and G=50G = 50, then G>AG > A is False ('No').
Case 2: If A=50A = 50 and G=100G = 100, then G>AG > A is True ('Yes').
Since both 'Yes' and 'No' are possible, Statement (2) is NOT sufficient.
Without concentration or proportion data, knowing total volume alone does not determine which component has the larger volume.

Key Concept

Weighted Average Mixture Ratios in Data Sufficiency
Question 259Question →

A technology firm sells only two products: Model Alpha and Model Beta. Last month, what percentage of the firm's total revenue was generated from sales of Model Alpha?

(1) Last month, the unit price of Model Alpha was 20%20\% higher than the unit price of Model Beta.
(2) Last month, the firm sold 50%50\% more units of Model Alpha than units of Model Beta.

Show answer & explanation

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

Answer

Both statements together are sufficient to answer the question, but neither statement alone is sufficient.
The question asks for the percentage of total revenue coming from Model Alpha, which simplifies algebraically to finding the ratio of Model Alpha's revenue (pAqAp_A q_A) to Model Beta's revenue (pBqBp_B q_B). Neither statement alone provides both the price ratio and quantity ratio needed. Statement (1) gives only the price ratio (pA/pB=1.2p_A/p_B = 1.2), and Statement (2) gives only the quantity ratio (qA/qB=1.5q_A/q_B = 1.5). Together, multiplying these ratios yields pAqA/(pBqB)=1.8p_A q_A / (p_B q_B) = 1.8, which uniquely determines Model Alpha's percentage share of total revenue as 9/149/14 or approximately 64.29%64.29\%. Thus, both statements together are sufficient.

Step-by-Step Solution

1
Define the target variable and rephrase the question algebraically.
Let pAp_A and pBp_B be the unit prices of Model Alpha and Model Beta, respectively. Let qAq_A and qBq_B be the quantities sold. The total revenue generated by Model Alpha is RA=pAqAR_A = p_A q_A, and total revenue generated by Model Beta is RB=pBqBR_B = p_B q_B. Total firm revenue is Rtotal=pAqA+pBqBR_{total} = p_A q_A + p_B q_B. We need to find pAqApAqA+pBqB×100%\frac{p_A q_A}{p_A q_A + p_B q_B} \times 100\%, which simplifies to 11+pBqBpAqA×100%\frac{1}{1 + \frac{p_B q_B}{p_A q_A}} \times 100\%. Thus, we only need the value of the ratio pAqApBqB\frac{p_A q_A}{p_B q_B}.
Rephrasing the target from finding individual revenue components to finding the single ratio of Model Alpha's revenue to Model Beta's revenue simplifies statement evaluation.
2
Evaluate Statement (1) independently.
Statement (1) states pA=1.2pBp_A = 1.2 p_B, which gives pApB=1.2\frac{p_A}{p_B} = 1.2. However, we have no information about the ratio of quantities qAqB\frac{q_A}{q_B}.
Without quantity data, the revenue ratio pAqApBqB\frac{p_A q_A}{p_B q_B} cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Statement (2) states qA=1.5qBq_A = 1.5 q_B, which gives qAqB=1.5\frac{q_A}{q_B} = 1.5. However, we have no information about the ratio of unit prices pApB\frac{p_A}{p_B}.
Without price data, the revenue ratio pAqApBqB\frac{p_A q_A}{p_B q_B} cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined.
Combining both statements gives pAqApBqB=(pApB)×(qAqB)=1.2×1.5=1.8\frac{p_A q_A}{p_B q_B} = \left(\frac{p_A}{p_B}\right) \times \left(\frac{q_A}{q_B}\right) = 1.2 \times 1.5 = 1.8. Therefore, Model Alpha's revenue share is 1.8pBqB1.8pBqB+1.0pBqB=1.82.8=914≈64.29%\frac{1.8 p_B q_B}{1.8 p_B q_B + 1.0 p_B q_B} = \frac{1.8}{2.8} = \frac{9}{14} \approx 64.29\%.
Since a unique numerical percentage can be calculated, both statements together are sufficient.

Key Concept

Data Sufficiency Evaluation for Revenue Ratios and Percentages
Question 260Question →

Two automated cargo vessels, Vessel X and Vessel Y, transported grain along a straight route of 12001{}200 nautical miles from Port A to Port B at constant speeds of vxv_x and vyv_y knots, respectively. What was the average speed vxv_x, in knots, of Vessel X for the trip?

(1) Vessel X completed the trip in 20%20\% less time than Vessel Y.
(2) If Vessel X had traveled at a speed 1010 knots faster, it would have completed the trip in 1010 hours less time.

Show answer & explanation

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The choice stating that Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient is correct. Statement (1) provides only a relative ratio between the speeds of Vessel X and Vessel Y (vy=0.8vxv_y = 0.8v_x), leaving infinitely many possible values for vxv_x. Statement (2) sets up the equation 1200vx−1200vx+10=10\frac{1{}200}{v_x} - \frac{1{}200}{v_x + 10} = 10, which reduces to the quadratic equation vx2+10vx−1200=0v_x^2 + 10v_x - 1{}200 = 0. Because speed must be positive, vx=30v_x = 30 is the unique solution.

Step-by-Step Solution

1
Rephrase the question stem target
The target is to find the exact numerical value of vx=1200Txv_x = \frac{1{}200}{T_x}, where TxT_x is the time in hours taken by Vessel X.
Establishing the relationship between distance, rate, and time (D=v⋅tD = v \cdot t) clarifies what data is necessary.
2
Evaluate Statement (1) independently
Statement (1) states Tx=0.80TyT_x = 0.80 T_y, which implies 1200vx=0.80(1200vy)\frac{1{}200}{v_x} = 0.80 \left(\frac{1{}200}{v_y}\right), or vy=0.80vxv_y = 0.80 v_x.
This relationship gives the ratio of the two speeds but provides no specific numerical value for vxv_x or TxT_x. Thus, statement (1) ALONE is not sufficient.
3
Evaluate Statement (2) independently
Statement (2) gives 1200vx−1200vx+10=10\frac{1{}200}{v_x} - \frac{1{}200}{v_x + 10} = 10. Dividing by 1010 yields 120vx−120vx+10=1\frac{120}{v_x} - \frac{120}{v_x + 10} = 1, which simplifies to 120(10)=vx(vx+10)120(10) = v_x(v_x + 10), or vx2+10vx−1200=0v_x^2 + 10 v_x - 1{}200 = 0. Factoring gives (vx+40)(vx−30)=0(v_x + 40)(v_x - 30) = 0. Since speed must be positive, vx=30v_x = 30 knots.
Statement (2) yields a single valid positive value for vxv_x. Thus, statement (2) ALONE is sufficient.

Key Concept

Data Sufficiency evaluation for rate problems involving algebraic equations of degree two where physical constraints eliminate non-positive roots.

Alternative Method

For statement (2), test clean factors of 12001{}200 for vx(vx+10)=1200v_x(v_x + 10) = 1{}200. Since 30×40=120030 \times 40 = 1{}200, vx=30v_x = 30 can be quickly verified without full quadratic expansion.
Estimated Time:1m 45s
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