To find the unit rate for Elyse's recipe and determine the number of cups of lemonade concentrate per 1 cup of water, we can start by converting Elyse's mixture into an easier format.
Elyse's mixture consists of:
- Water: \( 1 \frac{2}{3} \) cups
- Lemonade concentrate: \( \frac{3}{4} \) cups
First, we need to convert \( 1 \frac{2}{3} \) cups of water into an improper fraction:
\[ 1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} \text{ cups of water} \]
Next, we want to find out how many cups of lemonade concentrate there are per 1 cup of water. We can do this by dividing the amount of lemonade concentrate by the amount of water and then scaling it to 1 cup of water.
To find the unit rate, we take the number of cups of lemonade concentrate and divide it by \( \frac{5}{3} \) (the amount of water), and then we adjust this so that we have it per 1 cup of water.
To do this, we use the formula:
\[ \text{Unit rate} = \frac{\text{Cups of lemonade concentrate}}{\text{Cups of water}} = \frac{\frac{3}{4}}{\frac{5}{3}} \]
Dividing fractions involves multiplying by the reciprocal:
\[ = \frac{3}{4} \times \frac{3}{5} = \frac{9}{20} \]
Thus, Elyse's lemonade has a concentration of \( \frac{9}{20} \) cups of lemonade concentrate per 1 cup of water.
To compare with Jamir's lemonade, we would perform a similar calculation for Jamir's recipe:
Jamir's mixture consists of:
- Water: \( 1 \frac{1}{3} \) cups
- Lemonade concentrate: \( \frac{4}{5} \) cups
Convert \( 1 \frac{1}{3} \) cups of water into an improper fraction:
\[ 1 \frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \text{ cups of water} \]
Now, calculate the unit rate for Jamir:
\[ \text{Unit rate} = \frac{\frac{4}{5}}{\frac{4}{3}} = \frac{4}{5} \times \frac{3}{4} = \frac{3}{5} \]
So Jamir's lemonade has a concentration of \( \frac{3}{5} \) cups of lemonade concentrate per 1 cup of water.
Comparison:
- Elyse's lemonade: \( \frac{9}{20} \)
- Jamir's lemonade: \( \frac{3}{5} = \frac{12}{20} \) when converted for comparison.
Since \( \frac{12}{20} > \frac{9}{20} \), Jamir’s lemonade is more lemony than Elyse's lemonade.