To find the unit rate for Elyse's lemonade recipe, we first need to divide the amount of lemonade concentrate by the amount of water to determine how many cups of lemonade concentrate there are per 1 cup of water.
Elyse's lemonade involves:
- Water: \( \frac{1}{5} \frac{4}{1} \) cups of water
- Lemonade concentrate: \( \frac{4}{3} \) cups of lemonade concentrate
First, let's convert \( \frac{1}{5} \frac{4}{1} \) cups of water into an improper fraction:
- Convert \( \frac{4}{1} \) to its improper representation. Since 4 is already a whole number, \( \frac{4}{1} \) is simply 4.
- Hence, \( \frac{1}{5} \frac{4}{1} = \frac{4}{5} \) cups of water.
Now we need to find the unit rate by dividing the amount of lemonade concentrate by the amount of water:
\[ \text{Lemonade concentrate per cup of water} = \frac{\frac{4}{3}}{\frac{4}{5}} \]
To divide fractions, you multiply by the reciprocal of the divisor:
\[ = \frac{4}{3} \times \frac{5}{4} \]
Now we can multiply:
\[ = \frac{4 \times 5}{3 \times 4} = \frac{20}{12} \]
We can simplify \( \frac{20}{12} \) by dividing both the numerator and denominator by their greatest common divisor, which is 4:
\[ = \frac{5}{3} \]
So, Elyse's recipe has \( \frac{5}{3} \) cups of lemonade concentrate for every 1 cup of water.
Next, we can follow a similar process for Jamir's recipe.
Jamir's lemonade involves:
- Water: \( \frac{1}{3} \frac{1}{1} \) cups of water, which is just \( \frac{1}{3} \) cups.
- Lemonade concentrate: \( \frac{9}{8} \) cups of lemonade concentrate.
Now find the unit rate for Jamir's recipe:
\[ \text{Lemonade concentrate per cup of water} = \frac{\frac{9}{8}}{\frac{1}{3}} \]
Multiply by the reciprocal:
\[ = \frac{9}{8} \times \frac{3}{1} = \frac{27}{8} \]
Thus, Jamir's recipe has \( \frac{27}{8} \) cups of lemonade concentrate for every 1 cup of water.
Now we can compare the two unit rates:
- Elyse's unit rate: \( \frac{5}{3} \)
- Jamir's unit rate: \( \frac{27}{8} \)
To easily compare, let's convert both to a decimal or common denominator.
-
For Elyse: \(\frac{5}{3} \approx 1.67\)
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For Jamir: \(\frac{27}{8} = 3.375\)
From the comparison, we see that \(3.375\) (Jamir's unit rate) is greater than \(1.67\) (Elyse's unit rate). Therefore, Jamir's lemonade has a more lemony taste.