Answer :
To solve this problem, we need to determine how many pounds of almonds, cashews, and walnuts the customer bought based on the conditions provided. Here’s how we can interpret the situation:
1. Initial Conditions and Given Data:
- The customer buys a total of 12 pounds of mixed nuts. This gives us the equation:
[tex]\[
A + C + W = 12
\][/tex]
where [tex]\( A \)[/tex] is the pounds of almonds, [tex]\( C \)[/tex] is the pounds of cashews, and [tex]\( W \)[/tex] is the pounds of walnuts.
2. Cost Equation:
- The total cost of the mixed nuts is [tex]$118, where almonds cost $[/tex]7 per pound, cashews [tex]$10 per pound, and walnuts $[/tex]12 per pound. This gives us the equation:
[tex]\[
7A + 10C + 12W = 118
\][/tex]
3. Relationship Between Walnuts and Cashews:
- The customer buys 2 more pounds of walnuts than cashews. This gives us:
[tex]\[
W = C + 2
\][/tex]
4. Setting Up the System of Equations:
Using the above three equations:
- [tex]\( A + C + W = 12 \)[/tex]
- [tex]\( 7A + 10C + 12W = 118 \)[/tex]
- [tex]\( W = C + 2 \)[/tex]
5. Solving the System:
- Let's substitute the third equation [tex]\( W = C + 2 \)[/tex] into the first two equations:
- [tex]\( A + C + (C + 2) = 12 \)[/tex] simplifies to [tex]\( A + 2C + 2 = 12 \)[/tex] or [tex]\( A + 2C = 10 \)[/tex]
- [tex]\( 7A + 10C + 12(C + 2) = 118 \)[/tex] simplifies to [tex]\( 7A + 10C + 12C + 24 = 118 \)[/tex] or [tex]\( 7A + 22C = 94 \)[/tex]
6. Solving for [tex]\( A \)[/tex], [tex]\( C \)[/tex], and [tex]\( W \)[/tex]:
- With the simplified system:
- Equation 1: [tex]\( A + 2C = 10 \)[/tex]
- Equation 2: [tex]\( 7A + 22C = 94 \)[/tex]
- Solve for [tex]\( A \)[/tex] in terms of [tex]\( C \)[/tex] from Equation 1:
[tex]\[
A = 10 - 2C
\][/tex]
- Substitute [tex]\( A = 10 - 2C \)[/tex] into Equation 2:
[tex]\[
7(10 - 2C) + 22C = 94
\][/tex]
[tex]\[
70 - 14C + 22C = 94
\][/tex]
[tex]\[
8C = 24
\][/tex]
[tex]\[
C = 3
\][/tex]
- Substitute [tex]\( C = 3 \)[/tex] back into [tex]\( A = 10 - 2C \)[/tex]:
[tex]\[
A = 10 - 2(3) = 4
\][/tex]
- Finally, find [tex]\( W \)[/tex]:
[tex]\[
W = C + 2 = 3 + 2 = 5
\][/tex]
7. Conclusion:
- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- This matches with the conditions provided in the problem.
Therefore, the solution shows:
- The customer buys 4 pounds of almonds (A), 3 pounds of cashews (C), and 5 pounds of walnuts (W). Since the statement description isn't verbatim from provided choices, ensure to align this answer with the closest correct interpretation.
1. Initial Conditions and Given Data:
- The customer buys a total of 12 pounds of mixed nuts. This gives us the equation:
[tex]\[
A + C + W = 12
\][/tex]
where [tex]\( A \)[/tex] is the pounds of almonds, [tex]\( C \)[/tex] is the pounds of cashews, and [tex]\( W \)[/tex] is the pounds of walnuts.
2. Cost Equation:
- The total cost of the mixed nuts is [tex]$118, where almonds cost $[/tex]7 per pound, cashews [tex]$10 per pound, and walnuts $[/tex]12 per pound. This gives us the equation:
[tex]\[
7A + 10C + 12W = 118
\][/tex]
3. Relationship Between Walnuts and Cashews:
- The customer buys 2 more pounds of walnuts than cashews. This gives us:
[tex]\[
W = C + 2
\][/tex]
4. Setting Up the System of Equations:
Using the above three equations:
- [tex]\( A + C + W = 12 \)[/tex]
- [tex]\( 7A + 10C + 12W = 118 \)[/tex]
- [tex]\( W = C + 2 \)[/tex]
5. Solving the System:
- Let's substitute the third equation [tex]\( W = C + 2 \)[/tex] into the first two equations:
- [tex]\( A + C + (C + 2) = 12 \)[/tex] simplifies to [tex]\( A + 2C + 2 = 12 \)[/tex] or [tex]\( A + 2C = 10 \)[/tex]
- [tex]\( 7A + 10C + 12(C + 2) = 118 \)[/tex] simplifies to [tex]\( 7A + 10C + 12C + 24 = 118 \)[/tex] or [tex]\( 7A + 22C = 94 \)[/tex]
6. Solving for [tex]\( A \)[/tex], [tex]\( C \)[/tex], and [tex]\( W \)[/tex]:
- With the simplified system:
- Equation 1: [tex]\( A + 2C = 10 \)[/tex]
- Equation 2: [tex]\( 7A + 22C = 94 \)[/tex]
- Solve for [tex]\( A \)[/tex] in terms of [tex]\( C \)[/tex] from Equation 1:
[tex]\[
A = 10 - 2C
\][/tex]
- Substitute [tex]\( A = 10 - 2C \)[/tex] into Equation 2:
[tex]\[
7(10 - 2C) + 22C = 94
\][/tex]
[tex]\[
70 - 14C + 22C = 94
\][/tex]
[tex]\[
8C = 24
\][/tex]
[tex]\[
C = 3
\][/tex]
- Substitute [tex]\( C = 3 \)[/tex] back into [tex]\( A = 10 - 2C \)[/tex]:
[tex]\[
A = 10 - 2(3) = 4
\][/tex]
- Finally, find [tex]\( W \)[/tex]:
[tex]\[
W = C + 2 = 3 + 2 = 5
\][/tex]
7. Conclusion:
- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- This matches with the conditions provided in the problem.
Therefore, the solution shows:
- The customer buys 4 pounds of almonds (A), 3 pounds of cashews (C), and 5 pounds of walnuts (W). Since the statement description isn't verbatim from provided choices, ensure to align this answer with the closest correct interpretation.
