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A certain cake has two layers with pieces of fruit on top of each layer. Of all the pieces of fruit on the cake, \(25\) percent are strawberries. How many pieces of fruit are on the first layer of the cake?
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?
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.
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.
Statement 1 tells us: Of the pieces of fruit on the first layer, 6 are strawberries.
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)
Scenario 2: First layer has 12 pieces total (6 strawberries, 6 other fruit)
Since the first layer could have 24 pieces OR 12 pieces (among other possibilities), we cannot determine a unique answer.
Statement 1 alone is NOT sufficient.
This eliminates choices A and D.
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:
Let's see if different values for the first layer are possible:
Scenario 1: First layer has 20 pieces with 6 strawberries
Scenario 2: First layer has 60 pieces with 16 strawberries
Since the first layer could have 20 pieces OR 60 pieces (among other possibilities), we cannot determine a unique answer.
Statement 2 alone is NOT sufficient.
This eliminates choice B.
Let's use BOTH statements together:
Total strawberries across both layers = \(6 + 4 = 10\) strawberries
Since strawberries are \(25%\) of all fruit:
We know the second layer has 20 pieces, so:
Let's verify our answer:
[STOP - Sufficient!] We found a unique answer: the first layer has exactly 20 pieces of fruit.
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.
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."