To find the unit rate of lemonade concentrate per 1 cup of water in Elyse's recipe, we need to divide the amount of lemonade concentrate by the amount of water, scaling the water down to 1 cup.
Elyse’s Recipe:
- Water: \( 1 \frac{7}{3} \) cups
- Lemonade Concentrate: \( 2 \frac{1}{2} \) cups
First, let’s convert the mixed numbers to improper fractions.
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Convert \( 1 \frac{7}{3} \) to an improper fraction: \[ 1 \frac{7}{3} = \frac{3 \times 1 + 7}{3} = \frac{3 + 7}{3} = \frac{10}{3} \]
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Convert \( 2 \frac{1}{2} \) to an improper fraction: \[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]
Now we have:
- Water: \( \frac{10}{3} \) cups
- Lemonade Concentrate: \( \frac{5}{2} \) cups
Next, we want to find how many cups of lemonade concentrate there are per 1 cup of water. To do this, we divide the amount of lemonade concentrate by the amount of water and scale the water to 1 cup:
\[ \text{Lemonade Concentrate Rate} = \frac{\text{Lemonade Concentrate}}{\text{Water}} = \frac{\frac{5}{2}}{\frac{10}{3}} \]
To divide fractions, multiply by the reciprocal of the denominator:
\[ \frac{5}{2} \div \frac{10}{3} = \frac{5}{2} \times \frac{3}{10} \]
Multiply the numerators and the denominators:
\[ = \frac{5 \times 3}{2 \times 10} = \frac{15}{20} = \frac{3}{4} \]
So for Elyse's recipe, there are \( \frac{3}{4} \) cups of lemonade concentrate per 1 cup of water.
Now for Jamir's Recipe:
- Water: \( 1 \frac{3}{1} \) cups (which is just \( 4 \) cups since \( 1 \frac{3}{1} = 4 \))
- Lemonade Concentrate: \( 5 \frac{2}{1} \) cups (which is \( 12 \) cups since \( 5 \frac{2}{1} = 12 \))
Using the same process:
\[ \text{Lemonade Concentrate Rate} = \frac{12}{4} = 3 \]
So, in Jamir's recipe, there are \( 3 \) cups of lemonade concentrate per 1 cup of water.
Conclusion:
- Elyse's lemonade concentrate ratio: \( \frac{3}{4} \)
- Jamir's lemonade concentrate ratio: \( 3 \)
Since \( 3 \) is greater than \( \frac{3}{4} \), Jamir's lemonade has a more lemony taste.