Question

A store creates a mixture using only peanuts and almonds.

1. There are 20 pounds of the mixture.
2. Peanuts cost $2.95 per pound.
3 almonds cost $5.95 per pound.
4. The mixture costs $4.00 per pound.
How many pounds of peanuts are in the mixture?

Answer 1

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:

1. The total weight of the mixture is 20 pounds:
\[
p + a = 20
\]

2. 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:
\[
20 \times 4.00 = 80.00
\]

3. 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:

1. \( p + a = 20 \)
2. \( 2.95p + 5.95a = 80.00 \)

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.