College

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]
\left[
\begin{array}{ccc|c}
0 & -1 & 1 & 2 \\
7 & 10 & 12 & 118 \\
1 & 1 & 1 & 12
\end{array}
\right]
[/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, let's break it down step by step using the information and constraints provided.

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

2. Setting Up Equations Based on the Problem Statement:

- Equation for total weight:
The total weight of the almonds, cashews, and walnuts is 12 pounds:
[tex]\[
a + c + w = 12
\][/tex]

- Equation for total cost:
The total cost of the almonds, cashews, and walnuts is [tex]$118. The cost per pound for almonds is $[/tex]7, for cashews is [tex]$10, and for walnuts is $[/tex]12:
[tex]\[
7a + 10c + 12w = 118
\][/tex]

- Equation for the difference in pounds between walnuts and cashews:
The customer buys 2 more pounds of walnuts than cashews:
[tex]\[
w = c + 2
\][/tex]

3. Solving the System of Equations:
Substitute [tex]\( w = c + 2 \)[/tex] from the third equation into the other equations to solve for [tex]\( a \)[/tex], [tex]\( c \)[/tex], and [tex]\( w \)[/tex].

- Substituting into the weight equation:
[tex]\[
a + c + (c + 2) = 12 \implies a + 2c = 10
\][/tex]

- Substituting into the cost equation:
[tex]\[
7a + 10c + 12(c + 2) = 118 \implies 7a + 22c + 24 = 118 \implies 7a + 22c = 94
\][/tex]

Now, solve the two equations [tex]\( a + 2c = 10 \)[/tex] and [tex]\( 7a + 22c = 94 \)[/tex].

- From [tex]\( a + 2c = 10 \)[/tex], you can express [tex]\( a \)[/tex]:
[tex]\[
a = 10 - 2c
\][/tex]

- Substitute into the second equation:
[tex]\[
7(10 - 2c) + 22c = 94 \implies 70 - 14c + 22c = 94 \implies 8c = 24 \implies c = 3
\][/tex]

- Use [tex]\( c = 3 \)[/tex] in [tex]\( a = 10 - 2c \)[/tex]:
[tex]\[
a = 10 - 2(3) = 4
\][/tex]

- Use [tex]\( c = 3 \)[/tex] in [tex]\( w = c + 2 \)[/tex]:
[tex]\[
w = 3 + 2 = 5
\][/tex]

4. Interpret the Results:
- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- The difference in poundage between almonds and cashews is [tex]\( 4 - 3 = 1 \)[/tex].
- The difference in poundage between walnuts and almonds is [tex]\( 5 - 4 = 1 \)[/tex].

Therefore, the interpretation that matches these calculations is:
- The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.