A candy shop owner must mix three types of sweeteners—A, B, and C—in a certain ratio to get the desired...

Understanding the Question

We need to find the ratio of weights \(\mathrm{A:B:C}\) in a sweetener mixture. Let's denote:

• Weight of sweetener A = \(\mathrm{a}\)
• Weight of sweetener B = \(\mathrm{b}\)
• Weight of sweetener C = \(\mathrm{c}\)

What constitutes sufficiency: We need to determine the exact ratio \(\mathrm{a:b:c}\). This could be \(1:1:1\), or \(2:3:5\), or any other specific ratio. If we can uniquely determine this ratio, we have sufficiency.

Analyzing Statement 1

What Statement 1 Tells Us

Statement 1 gives us the individual prices:

• Sweetener A: \(\$40\) per pound
• Sweetener B: \(\$50\) per pound
• Sweetener C: \(\$60\) per pound
• Final mixture average: \(\$50\) per pound

Key Insight: The Balance Point

Notice that \(\$50\) is exactly the price of sweetener B. This is crucial because:

• Sweetener A (\(\$40\)) pulls the average DOWN by \(\$10\) per pound
• Sweetener C (\(\$60\)) pulls the average UP by \(\$10\) per pound
• Sweetener B (\(\$50\)) is neutral - it doesn't affect the average

For the mixture to average exactly \(\$50\):

• Every pound of A (pulling down by \(\$10\)) must be balanced by a pound of C (pulling up by \(\$10\))
• Therefore: \(\mathrm{a = c}\)

The Critical Gap

But what about B? Since B is already at \(\$50\), it doesn't affect the average. We could have:

• 1 lb A, 1 lb C, 0 lb B → ratio \(1:0:1\)
• 1 lb A, 1 lb C, 10 lb B → ratio \(1:10:1\)
• 2 lb A, 2 lb C, 3 lb B → ratio \(2:3:2\)

Different amounts of B give us different ratios, all maintaining the \(\$50\) average.

Conclusion

Statement 1 tells us that \(\mathrm{a = c}\), but gives no information about \(\mathrm{b}\).
Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Important: We now analyze Statement 2 independently, forgetting Statement 1 completely.

What Statement 2 Provides

• Average price of sweeteners A and B together is \(\$45\)
• Average price of sweeteners B and C together is \(\$55\)

The Critical Issue

Wait - Statement 2 talks about "average price" but doesn't tell us the individual prices of A, B, and C. Without knowing what each sweetener costs per pound, we cannot interpret these averages.

For example, if A and B average to \(\$45\):

• Scenario 1: A costs \(\$30\)/lb and B costs \(\$60\)/lb (with ratio \(1:1\))
• Scenario 2: A costs \(\$40\)/lb and B costs \(\$50\)/lb (with ratio \(1:1\))
• Scenario 3: A costs \(\$40\)/lb and B costs \(\$46\)/lb (with ratio \(1:5\))

Each scenario requires different weight ratios to achieve the \(\$45\) average, but we don't know which scenario we're in.

Conclusion

Without individual prices, Statement 2's averages are meaningless for determining weight ratios.
Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Statements

Combined Information

Now we have:

• From Statement 1: A costs \(\$40\)/lb, B costs \(\$50\)/lb, C costs \(\$60\)/lb
• From Statement 2: A and B together average \(\$45\); B and C together average \(\$55\)

The Halfway Principle

Here's the key insight: When two items average to exactly the midpoint between their prices, they must be in equal proportions.

For A (\(\$40\)) and B (\(\$50\)) to average \(\$45\):

• The range from \(\$40\) to \(\$50\) is \(\$10\)
• \(\$45\) is exactly halfway (at the \(\$5\) mark)
• This only happens with equal weights: \(\mathrm{a = b}\)

To visualize: If we had more A than B, the average would be pulled below \(\$45\). If we had more B than A, it would be pulled above \(\$45\). Only equal amounts give exactly \(\$45\).

For B (\(\$50\)) and C (\(\$60\)) to average \(\$55\):

• The range from \(\$50\) to \(\$60\) is \(\$10\)
• \(\$55\) is exactly halfway (at the \(\$5\) mark)
• This only happens with equal weights: \(\mathrm{b = c}\)

The Complete Picture

From our analysis:

• \(\mathrm{a = b}\) (from the A-B average)
• \(\mathrm{b = c}\) (from the B-C average)

Therefore: \(\mathrm{a = b = c}\), giving us the ratio \(1:1:1\)

Why Together They Work

Statement 2's averages only make sense when we know the individual prices from Statement 1. The specific values (\(\$45\) and \(\$55\)) tell us these are "halfway" averages, which uniquely determine equal proportions.

[STOP - Sufficient!] Together, the statements uniquely determine the ratio.

The Answer: C

Both statements together are sufficient to determine the ratio is \(1:1:1\), but neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."