Bot GPT 4-o mini To find the number of cups of fruit punch mix per 1 cup of water for Elyse's recipe, we need to identify the total amount of water and fruit punch mix she uses and then set up a ratio.
Elyse mixes:
- Water: \( \frac{1}{2} \) cups
- Fruit Punch Mix: \( \frac{10}{3} \) cups
To find the unit rate of fruit punch mix per 1 cup of water, we can perform the following calculation:
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First, define the volume of water in Elyse's recipe: \[ \text{Water} = \frac{1}{2} = \frac{1}{2} \text{ cups} \]
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To find out how many cups of fruit punch mix correspond to 1 cup of water, we divide the amount of fruit punch mix by the amount of water. The calculation will look like this: \[ \text{Unit Rate} = \frac{\text{Fruit Punch Mix}}{\text{Water}} = \frac{\frac{10}{3}}{\frac{1}{2}} \]
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Dividing fractions involves multiplying by the reciprocal: \[ \frac{10}{3} \div \frac{1}{2} = \frac{10}{3} \times \frac{2}{1} = \frac{20}{3} \]
Elyse's fruit punch recipe has a unit rate of \( \frac{20}{3} \) cups of fruit punch mix for each cup of water.
Now we can follow the same process for Jamir's recipe:
Jamir mixes:
- Water: \( \frac{7}{3} \) cups
- Fruit Punch Mix: \( \frac{21}{2} \) cups (Assuming you meant to write 21/2 instead of 2/1)
To find the unit rate of fruit punch mix per 1 cup of water for Jamir, we will perform a similar calculation:
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Set up the division: \[ \text{Unit Rate} = \frac{\text{Fruit Punch Mix}}{\text{Water}} = \frac{\frac{21}{2}}{\frac{7}{3}} \]
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Multiply by the reciprocal: \[ \frac{21}{2} \div \frac{7}{3} = \frac{21}{2} \times \frac{3}{7} = \frac{63}{14} = \frac{9}{2} \]
Now, we have:
- Elyse's unit rate: \( \frac{20}{3} \) cups of fruit punch mix per cup of water
- Jamir's unit rate: \( \frac{9}{2} \) cups of fruit punch mix per cup of water
Next, we can compare these rates to see which one is fruitier. To do this, we can convert both fractions to decimals:
- \( \frac{20}{3} \approx 6.67 \)
- \( \frac{9}{2} = 4.5 \)
Since \( 6.67 > 4.5 \), Elyse's fruit punch has a fruitier taste than Jamir's.
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