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A store sells almonds for [tex]\$7[/tex] per pound, cashews for [tex]\$10[/tex] per pound, and walnuts for [tex]\$12[/tex] per pound. A customer buys 12 pounds of mixed nuts consisting of almonds, cashews, and walnuts for [tex]\$118[/tex]. The customer buys 2 more pounds of walnuts than cashews.

The matrix below represents this situation:

[tex]\[
\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}
\][/tex]

If the reduced row echelon form of this matrix represents the amount of each type of nut the customer buys, which statement is a possible interpretation of the results?

A. The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

B. The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.

C. The customer buys 0.5 more pound of walnuts than almonds and 2.5 more pounds of almonds than cashews.

D. The customer buys 6.5 more pounds of walnuts than almonds and 8.5 more pounds of almonds than cashews.

Answer :

To solve this problem, we're looking at how a customer's purchase of mixed nuts can be distributed across three types of nuts: almonds, cashews, and walnuts. We are given certain conditions about the prices and weights in the mix. Here's a breakdown of how we interpret the customer's purchase:

1. Define Variables:
- Let [tex]\( a \)[/tex] be the pounds of almonds.
- Let [tex]\( c \)[/tex] be the pounds of cashews.
- Let [tex]\( w \)[/tex] be the pounds of walnuts.

2. Translate the Problem into Equations:
- The total weight of nuts bought is 12 pounds:
[tex]\[
a + c + w = 12
\][/tex]

- The total cost is $118:
[tex]\[
7a + 10c + 12w = 118
\][/tex]

- The customer buys 2 more pounds of walnuts than cashews:
[tex]\[
w = c + 2
\][/tex]

3. Solve the System of Equations:
From the reduced row echelon form we have:

- [tex]\( a = 4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
- [tex]\( w = 5 \)[/tex]

This means:
- The customer buys 4 pounds of almonds.
- The customer buys 3 pounds of cashews.
- The customer buys 5 pounds of walnuts.

4. Interpret the Result:
- Walnuts (5 pounds) are 2 more pounds than cashews (3 pounds), which matches the condition given in the problem.
- This satisfies the requirement that the customer buys 2 more pounds of walnuts than cashews.

5. Conclusion:
Therefore, the results align with the statement regarding the nuts: The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.

The correct interpretation of the result is that the customer buys:
- 2 more pounds of walnuts than almonds
- 2 more pounds of almonds than cashews

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