To determine how many bottles of mustard were used by David's Snack Shack, we first need to calculate the amount of mustard used based on the amount of ketchup.
We know that they used \( 2 \frac{1}{8} \) bottles of ketchup. To perform calculations, we'll convert this mixed number into an improper fraction.
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Convert \( 2 \frac{1}{8} \) to an improper fraction: \[ 2 \frac{1}{8} = 2 \times 8 + 1 = 16 + 1 = 17 \quad \text{(the numerator)} \quad \text{and the denominator remains 8.} \] Thus, \[ 2 \frac{1}{8} = \frac{17}{8}. \]
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Next, we need to find out how much mustard was used, which is \( 1 \frac{1}{4} \) times the amount of ketchup. First, we convert \( 1 \frac{1}{4} \) to an improper fraction: \[ 1 \frac{1}{4} = 1 \times 4 + 1 = 4 + 1 = 5 \quad \text{(the numerator)} \quad \text{and the denominator remains 4.} \] Thus, \[ 1 \frac{1}{4} = \frac{5}{4}. \]
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Now we multiply the amount of ketchup used by the ratio of mustard to ketchup: \[ \text{Mustard used} = \text{Ketchup used} \times \text{Ratio of mustard to ketchup} = \frac{17}{8} \times \frac{5}{4}. \]
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To multiply the fractions: \[ \frac{17}{8} \times \frac{5}{4} = \frac{17 \times 5}{8 \times 4} = \frac{85}{32}. \]
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Finally, we can convert \( \frac{85}{32} \) to a mixed number if required:
- Divide \( 85 \) by \( 32 \):
- \( 32 \) goes into \( 85 \) 2 times (as \( 32 \times 2 = 64 \)).
- The remainder is \( 85 - 64 = 21 \).
So, we can express \( \frac{85}{32} \) as: \[ 2 \frac{21}{32}. \]
- Divide \( 85 \) by \( 32 \):
Therefore, the answer is:
\[ \boxed{\frac{85}{32}} \quad \text{or} \quad 2 \frac{21}{32} \text{ bottles.} \]