A certain cake has two layers with pieces of fruit on top of each layer. Of all the pieces of...

Understanding the Question

We have a two-layer cake with fruit on top of each layer, where \(25%\) of ALL the fruit pieces are strawberries. We need to find: How many pieces of fruit are on the first layer?

Given Information

• The cake has exactly two layers
• Each layer has fruit pieces on top
• \(25%\) of all fruit pieces (across both layers) are strawberries

What We Need to Determine

To answer "sufficient," we need to be able to find a unique value for the number of fruit pieces on the first layer. If multiple values are possible, the statement(s) are not sufficient.

Key Insight

Since strawberries make up exactly \(25%\) of all fruit, this means for every 4 pieces of fruit total, exactly 1 must be a strawberry. This fixed ratio will be crucial in determining whether we can find the answer.

Analyzing Statement 1

Statement 1 tells us: Of the pieces of fruit on the first layer, 6 are strawberries.

What We Know and Don't Know

• ✓ We know: First layer has exactly 6 strawberries
• ✗ We don't know: Total number of fruit pieces on the first layer
• ✗ We don't know: Any information about the second layer

Testing Different Scenarios

Let's see if different values for the first layer are possible:

Scenario 1: First layer has 24 pieces total (6 strawberries, 18 other fruit)

• If the second layer has 16 pieces with 4 strawberries:
  • Total fruit = \(24 + 16 = 40\) pieces
  • Total strawberries = \(6 + 4 = 10\)
  • Check: \(10 ÷ 40 = 0.25 = 25%\) ✓

Scenario 2: First layer has 12 pieces total (6 strawberries, 6 other fruit)

• If the second layer has 28 pieces with 4 strawberries:
  • Total fruit = \(12 + 28 = 40\) pieces
  • Total strawberries = \(6 + 4 = 10\)
  • Check: \(10 ÷ 40 = 0.25 = 25%\) ✓

Since the first layer could have 24 pieces OR 12 pieces (among other possibilities), we cannot determine a unique answer.

Conclusion

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2 tells us: 20 percent of the 20 pieces of fruit on the second layer are strawberries.

This means:

• Second layer has exactly 20 pieces total
• Second layer has \(20% × 20 = 4\) strawberries

What We Know and Don't Know

• ✓ We know: Second layer has 20 total pieces with 4 strawberries
• ✗ We don't know: Any information about the first layer

Testing Different Scenarios

Let's see if different values for the first layer are possible:

Scenario 1: First layer has 20 pieces with 6 strawberries

• Total fruit = \(20 + 20 = 40\) pieces
• Total strawberries = \(6 + 4 = 10\)
• Check: \(10 ÷ 40 = 0.25 = 25%\) ✓

Scenario 2: First layer has 60 pieces with 16 strawberries

• Total fruit = \(60 + 20 = 80\) pieces
• Total strawberries = \(16 + 4 = 20\)
• Check: \(20 ÷ 80 = 0.25 = 25%\) ✓

Since the first layer could have 20 pieces OR 60 pieces (among other possibilities), we cannot determine a unique answer.

Conclusion

Statement 2 alone is NOT sufficient.

This eliminates choice B.

Combining Statements

Let's use BOTH statements together:

• From Statement 1: The first layer has 6 strawberries
• From Statement 2: The second layer has 20 pieces total with 4 strawberries

Combined Information

Total strawberries across both layers = \(6 + 4 = 10\) strawberries

Since strawberries are \(25%\) of all fruit:

• If 10 strawberries = \(25%\) of all fruit
• Then total fruit = \(10 ÷ 0.25 = 40\) pieces

We know the second layer has 20 pieces, so:

• First layer must have \(40 - 20 = 20\) pieces

Verification

Let's verify our answer:

• First layer: 20 pieces (with 6 strawberries)
• Second layer: 20 pieces (with 4 strawberries)
• Total: 40 pieces with 10 strawberries
• Strawberry percentage: \(10 ÷ 40 = 25%\) ✓

[STOP - Sufficient!] We found a unique answer: the first layer has exactly 20 pieces of fruit.

Why Together They Are Sufficient

The combination gives us the exact count of all strawberries (10), which with the \(25%\) constraint determines the total fruit count (40). Since we know the second layer size (20), we can uniquely determine the first layer size.

This eliminates choice E.

The Answer: C

Both statements together provide enough information to determine that the first layer has exactly 20 pieces of fruit, but neither statement alone is sufficient.

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