Let's denote the amount of water in each bottle of water as \( w \) ounces, and the amount of lemonade in each bottle of lemonade as \( l \) ounces.
We have two equations based on the information given:
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For Sachi: \[ 22w + 15l = 564 \]
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For Olivia: \[ 28w + 10l = 536 \]
First, let's solve this system of equations. We can multiply each equation to eliminate one variable. Multiply the first equation by 2 and the second by 3:
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From Sachi's equation: \[ 2(22w + 15l) = 2 \times 564 \implies 44w + 30l = 1128 \quad (1) \]
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From Olivia's equation: \[ 3(28w + 10l) = 3 \times 536 \implies 84w + 30l = 1608 \quad (2) \]
Now we can subtract equation (1) from equation (2):
\[ (84w + 30l) - (44w + 30l) = 1608 - 1128 \]
This simplifies to: \[ 40w = 480 \]
Dividing both sides by 40, we find: \[ w = 12 \]
Now that we have the value of \( w \), we can substitute it back into one of the original equations to find \( l \). We will use Sachi's equation:
\[ 22(12) + 15l = 564 \]
Calculating \( 22(12) \): \[ 264 + 15l = 564 \]
Now, isolating \( l \): \[ 15l = 564 - 264 = 300 \]
Dividing both sides by 15 gives: \[ l = 20 \]
Now we know the quantities in each bottle:
- Each bottle of water contains \( 12 \) ounces.
- Each bottle of lemonade contains \( 20 \) ounces.
Next, we calculate how much each player gets:
- Each player receives 2 bottles of water and 1 bottle of lemonade.
Calculating the total amount per player: \[ \text{Total water} = 2 \times 12 = 24 \text{ ounces} \] \[ \text{Total lemonade} = 1 \times 20 = 20 \text{ ounces} \]
Thus, the total fluid ounces per player is: \[ \text{Total per player} = 24 + 20 = 44 \text{ ounces} \]
Therefore, each player gets a total of \(\boxed{44}\) fluid ounces.
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