To determine if the ratios in the table are equivalent, we first need to analyze the provided values. The amounts given seem to be associated with bottles (2, 6, 10, 12) and cases (10, 30, 50, 60).
Let's convert each set of values into a ratio.
- Bottles to Cases Ratios:
- The first part appears to suggest that for every number of bottles, we compare it to certain cases.
Let's take the calculations step by step based on available bottle and case numbers. Assuming the ratio format remains consistent (bottles:cases):
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For the first ratio (2 bottles to 10 cases): \[ Ratio = \frac{2}{10} = \frac{1}{5} \]
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For the second ratio (6 bottles to 30 cases): \[ Ratio = \frac{6}{30} = \frac{1}{5} \]
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For the third ratio (10 bottles to 50 cases): \[ Ratio = \frac{10}{50} = \frac{1}{5} \]
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For the fourth ratio (12 bottles to 60 cases): \[ Ratio = \frac{12}{60} = \frac{1}{5} \]
Now we can observe that all simplified ratios of bottles to cases equal \(\frac{1}{5}\).
Next, we'll express the ratios in terms of \(x\) (for "bottles") and relevant case numbers.
- Expressing these ratios in terms of equivalence: \[ \text{Let } x = 1 \]
We can express all bottle counts based on the simplest ratio we derived:
- Based on the expression \(x\):
- For \(2 bottles: 10 cases\) ==> \( 1:5 \)
- For \(6 bottles: 30 cases\) ==> \(1:5 \)
- For \(10 bottles: 50 cases\) ==> \(1:5 \)
- For \(12 bottles: 60 cases\) ==> \(1:5 \)
In conclusion, every ratio simplifies to the same equivalence, proving they are indeed equal: \[ \text{All ratios are } \frac{1}{5} \]
If you want to find the specific equivalent formats in this context:
- \(210 = 42 \times 5\) (using equivalence from any part of the ratios, when actual numbers are provided)
- \(630 = 126 \times 5\)
- \(1050 = 210 \times 5\)
- \(1260 = 252 \times 5\)
Hence you can conclude: \[ 210 = 42, 630 = 126, 1050 = 210, 1260 = 252 \]
Feel free to ask if you need any further clarifications or details!