Question

The owner of a shop wants to mix cashews and pecans to make 8 kg of mixed nuts worth $7.25 per kg. If the price of cashews is $8.00 per kg, and the price of pecans is $7.00 per kg, how many kilograms of each should he use?

Answer 1

To solve this problem, we can set up a system of equations based on the information provided. Let \( x \) be the amount of cashews in kilograms, and \( y \) be the amount of pecans in kilograms. We have the following equations:

1. The total weight of the mixed nuts:
\[
x + y = 8
\]

2. The total value of the mixed nuts:
Since the mixed nuts are worth $7.25 per kg, the total value of 8 kg is:
\[
7.25 \times 8 = 58.00 \text{ dollars}
\]
The value of the cashews and pecans can be expressed as:
\[
8x + 7y = 58
\]

Now we have the following system of equations:
\[
\begin{align*}
1) & \quad x + y = 8 \\
2) & \quad 8x + 7y = 58
\end{align*}
\]

We can solve this system of equations step by step.

**Step 1: Solve for \( y \) in terms of \( x \) from the first equation.**
\[
y = 8 - x
\]

**Step 2: Substitute \( y \) in the second equation.**
Replace \( y \) in the second equation:
\[
8x + 7(8 - x) = 58
\]
Expanding this gives:
\[
8x + 56 - 7x = 58
\]
Combining like terms, we get:
\[
x + 56 = 58
\]
Subtracting 56 from both sides, we find:
\[
x = 2
\]

**Step 3: Substitute \( x \) back to find \( y \).**
Now we can find \( y \):
\[
y = 8 - x = 8 - 2 = 6
\]

**Final Calculation:**
Thus, the owner should use 2 kg of cashews and 6 kg of pecans.

In conclusion, the amounts of each type of nut to be used are:
- **Cashews**: 2 kg
- **Pecans**: 6 kg