To find out how much cake was left after Aliyah’s siblings and friends ate, we need to first convert the mixed numbers for the pieces of cake eaten into improper fractions and then subtract them from 1 (representing the whole cake).
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Convert mixed numbers to improper fractions:
- Siblings ate \( 14 \frac{1}{4} = 14 + \frac{1}{4} = \frac{56}{4} + \frac{1}{4} = \frac{57}{4} \)
- Friends ate \( 23 \frac{2}{3} = 23 + \frac{2}{3} = \frac{69}{3} + \frac{2}{3} = \frac{71}{3} \)
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Find a common denominator to add the fractions:
- The least common multiple (LCM) of 4 and 3 is 12.
- Convert \( \frac{57}{4} \) to a denominator of 12: \[ \frac{57}{4} = \frac{57 \times 3}{4 \times 3} = \frac{171}{12} \]
- Convert \( \frac{71}{3} \) to a denominator of 12: \[ \frac{71}{3} = \frac{71 \times 4}{3 \times 4} = \frac{284}{12} \]
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Add the fractions: \[ \frac{171}{12} + \frac{284}{12} = \frac{455}{12} \]
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Find the amount of cake left: Since the whole cake is represented as \( \frac{12}{12} \): \[ \text{Amount left} = 1 - \frac{455}{12} = \frac{12}{12} - \frac{455}{12} = \frac{12 - 455}{12} = \frac{-443}{12} \]
Thus, it appears that more cake has been eaten than was made, leading to a negative actual value, which indicates there is no cake left, and we are short by \( \frac{443}{12} \) (or 36⅔) of a cake.
So the correct conclusion is that there is no cake left. If we consider options or look further into the context, the representation shows the cake was entirely consumed and then some.
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