A retailer sold standard boxes at \s each and premium boxes at \p each yesterday, generati… — GMAT

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

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

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Both statements together are sufficient, but neither statement alone is sufficient.

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

(x, y)

such that

x > y

in some cases and

x < y

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

s

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

s = 15

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

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

, which definitively proves

x > y

Step-by-Step Solution

Evaluate Statement (1) independently.

Statement (1) provides

s = 15

and

p = 25

, yielding the equation

15x + 25y = 900

, which simplifies to

3x + 5y = 180

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

(x, y) = (55, 3)

yields

x > y

(Yes), while

(x, y) = (10, 30)

yields

x < y

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

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

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

Statement (2) states that

p = s + 10

and the average price is

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

. Substituting

p = s + 10

gives

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

, which simplifies to

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

. Because the individual box price

s

is unknown,

\frac{y}{x + y}

can vary depending on

s

(e.g., if

s = 10

x < y

; if

s = 15

x > y

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

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

Combine Statements (1) and (2).

From Statement (1),

s = 15

. Substituting

s = 15

into the simplified average equation from Statement (2),

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

, gives

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

. Since

x

and

y

are positive integers,

\frac{x}{y} > 1

, which definitively proves that

x > y

(Yes).

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

x

y

Key Concept

Statement Independence and Statement Combination in Data Sufficiency

Question

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Step-by-Step Solution

Key Concept

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