Let's denote the number of pounds of peanuts as \( p \) and the number of pounds of almonds as \( a \).
We know the following from the problem statement:
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The total weight of the mixture is 20 pounds: \[ p + a = 20 \]
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The cost of peanuts is $2.95 per pound and the cost of almonds is $5.95 per pound. The total cost of the mixture is $4.00 per pound for 20 pounds, which means the total cost of the mixture is: \[ 20 \times 4.00 = 80.00 \]
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The total cost for the peanuts and the almonds can also be expressed as: \[ 2.95p + 5.95a = 80.00 \]
Now we have a system of equations:
- \( p + a = 20 \)
- \( 2.95p + 5.95a = 80.00 \)
From the first equation, we can express \( a \) in terms of \( p \): \[ a = 20 - p \]
Now we substitute this expression for \( a \) into the second equation: \[ 2.95p + 5.95(20 - p) = 80.00 \]
Expanding this gives: \[ 2.95p + 119 - 5.95p = 80.00 \]
Combining like terms: \[ -3p + 119 = 80.00 \]
Now, isolate \( p \): \[ -3p = 80.00 - 119 \] \[ -3p = -39 \] \[ p = \frac{-39}{-3} = 13 \]
So, the number of pounds of peanuts in the mixture is \( \boxed{13} \).
To find the number of pounds of almonds, we can substitute \( p \) back into the equation for \( a \): \[ a = 20 - p = 20 - 13 = 7 \]
Thus, the final answer is \( p = 13 \) pounds of peanuts and \( a = 7 \) pounds of almonds.