At a craft fair, Ruth's total revenue from selling 2 types of floral arrangements was $832. She sold x arrangements...

Understanding the Question

The question asks us: Did Ruth sell more Type A arrangements than Type B arrangements?

Let's break this down:

• Ruth sold x Type A arrangements and (131 - x) Type B arrangements
• We need to determine if \(\mathrm{x} > (131 - \mathrm{x})\)
• This simplifies to: \(\mathrm{x} > 65.5\)
• Since x must be an integer, we're asking: Is \(\mathrm{x} ≥ 66\)?

This is a yes/no question - we need a definitive "yes" or "no" answer.

Given information:

• Total revenue = $832
• Type A price = y dollars each
• Type B price = (y + 3) dollars each (Type B costs $3 more)
• Both x and y are integers
• Revenue equation: \(\mathrm{xy} + (131 - \mathrm{x})(\mathrm{y} + 3) = 832\)

Expanding this equation:
\(\mathrm{xy} + 131\mathrm{y} + 393 - \mathrm{xy} - 3\mathrm{x} = 832\)
\(131\mathrm{y} - 3\mathrm{x} = 439\)

What makes a statement sufficient? We need to determine whether x is definitely ≥ 66 or definitely < 66. We don't need the exact value of x - just which side of 65.5 it falls on.

Analyzing Statement 1

Statement 1 tells us: If Ruth had sold 2 more Type A arrangements and 5 more Type B arrangements, her total revenue would have been $882.

This gives us a new scenario:

• Type A: (x + 2) arrangements at y dollars each
• Type B: (131 - x + 5) = (136 - x) arrangements at (y + 3) dollars each
• New revenue = $882

Setting up the equation for this new scenario:
\((\mathrm{x} + 2)\mathrm{y} + (136 - \mathrm{x})(\mathrm{y} + 3) = 882\)

Expanding step by step:
\(\mathrm{xy} + 2\mathrm{y} + 136\mathrm{y} + 408 - \mathrm{xy} - 3\mathrm{x} = 882\)
\(138\mathrm{y} - 3\mathrm{x} = 474\)

Now we have two equations:
1. Original scenario: \(131\mathrm{y} - 3\mathrm{x} = 439\)
2. New scenario: \(138\mathrm{y} - 3\mathrm{x} = 474\)

Subtracting equation 1 from equation 2:
\((138\mathrm{y} - 3\mathrm{x}) - (131\mathrm{y} - 3\mathrm{x}) = 474 - 439\)
\(7\mathrm{y} = 35\)
\(\mathrm{y} = 5\)

Substituting y = 5 back into the original equation:
\(131(5) - 3\mathrm{x} = 439\)
\(655 - 3\mathrm{x} = 439\)
\(3\mathrm{x} = 216\)
\(\mathrm{x} = 72\)

Since x = 72, which is greater than 65.5, Ruth sold more Type A arrangements than Type B arrangements.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E. The answer must be A or D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: If Ruth had sold (131 - x) Type A arrangements and x Type B arrangements, her total revenue would have been more than $832.

Here's the elegant insight: Statement 2 is saying that if we swap the quantities between Type A and Type B, the revenue increases.

Let's think about what happens when we swap:

• Original setup: x arrangements at price y, and (131 - x) arrangements at price (y + 3)
• Swapped setup: (131 - x) arrangements at price y, and x arrangements at price (y + 3)

Since Type B costs $3 more than Type A, the revenue change depends on which direction we're moving more arrangements:

• We gain $3 for each of the x arrangements that move from Type A price to Type B price
• We lose $3 for each of the (131 - x) arrangements that move from Type B price to Type A price

Revenue change = Gains - Losses = \(3\mathrm{x} - 3(131 - \mathrm{x})\)
Revenue change = \(3\mathrm{x} - 393 + 3\mathrm{x} = 6\mathrm{x} - 393\)

Statement 2 tells us the new revenue > original revenue, which means:
Revenue change > 0
\(6\mathrm{x} - 393 > 0\)
\(6\mathrm{x} > 393\)
\(\mathrm{x} > 65.5\)

Since x must be an integer, x ≥ 66, which means Ruth sold more Type A arrangements than Type B arrangements.

[STOP - Statement 2 is Sufficient!]

The Answer: D

Both statements independently tell us that Ruth sold more Type A arrangements than Type B arrangements.

Answer Choice D: "Each statement alone is sufficient."