College

We appreciate your visit to 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. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

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 pounds 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 need to find out how many pounds of almonds, cashews, and walnuts the customer buys. Here's the problem breakdown and solution step-by-step:

The store sells:
- Almonds for [tex]$7 per pound,
- Cashews for $[/tex]10 per pound,
- Walnuts for [tex]$12 per pound.

The customer buys 12 pounds of mixed nuts for $[/tex]118, with 2 more pounds of walnuts than cashews.

Firstly, let's define:
- [tex]\( a \)[/tex] as the pounds of almonds,
- [tex]\( c \)[/tex] as the pounds of cashews,
- [tex]\( w \)[/tex] as the pounds of walnuts.

We will create equations based on the given information:

1. Total Weight Equation:
Since the total weight of the nuts is 12 pounds, we have:
[tex]\[
a + c + w = 12
\][/tex]

2. Total Cost Equation:
The total cost is $118:
[tex]\[
7a + 10c + 12w = 118
\][/tex]

3. Walnuts and Cashews Relationship:
There are 2 more pounds of walnuts than cashews:
[tex]\[
w = c + 2
\][/tex]

Now, let's use these equations to find the amounts.

Step 1: Substitute [tex]\( w = c + 2 \)[/tex] in the Total Weight Equation
[tex]\[
a + c + (c + 2) = 12
\][/tex]
Simplify:
[tex]\[
a + 2c + 2 = 12
\][/tex]
[tex]\[
a + 2c = 10 \quad \Rightarrow \quad a = 10 - 2c
\][/tex]

Step 2: Substitute [tex]\( a = 10 - 2c \)[/tex] and [tex]\( w = c + 2 \)[/tex] in the Total Cost Equation
[tex]\[
7(10 - 2c) + 10c + 12(c + 2) = 118
\][/tex]
Simplify the expression:
[tex]\[
70 - 14c + 10c + 12c + 24 = 118
\][/tex]
Combine like terms:
[tex]\[
70 + 24 + 8c = 118
\][/tex]
[tex]\[
8c = 118 - 94
\][/tex]
[tex]\[
8c = 24
\][/tex]
[tex]\[
c = 3
\][/tex]

Step 3: Find [tex]\( a \)[/tex] and [tex]\( w \)[/tex] using [tex]\( c = 3 \)[/tex]
Substitute [tex]\( c = 3 \)[/tex] into [tex]\( a = 10 - 2c \)[/tex]:
[tex]\[
a = 10 - 2 \times 3 = 4
\][/tex]

Substitute [tex]\( c = 3 \)[/tex] into [tex]\( w = c + 2 \)[/tex]:
[tex]\[
w = 3 + 2 = 5
\][/tex]

Thus, the customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.

Interpretation of Results:
From our calculations, we see that:
- The customer indeed buys 2 more pounds of walnuts than cashews.

Therefore, the correct statement that fits this situation is:
"The customer buys 2 more pounds of walnuts than cashews."

Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada