Answer :
Let's analyze the problem with the information provided. We need to determine how many pounds of almonds, cashews, and walnuts the customer buys, given the conditions.
### Given Conditions:
1. Price per pound:
- Almonds cost \[tex]$7 per pound.
- Cashews cost \$[/tex]10 per pound.
- Walnuts cost \[tex]$12 per pound.
2. Total weight and cost:
The customer buys a total of 12 pounds of mixed nuts for \$[/tex]118.
3. Relationships between types of nuts:
The customer buys 2 more pounds of walnuts than cashews.
### Setting up the Problem:
To set up the problem, let's define variables for the amounts in pounds:
- [tex]\( a \)[/tex] = pounds of almonds
- [tex]\( c \)[/tex] = pounds of cashews
- [tex]\( w \)[/tex] = pounds of walnuts
#### Formulating Equations:
1. From the relationship between walnuts and cashews:
[tex]\[
w = c + 2
\][/tex]
2. From the total weight:
[tex]\[
a + c + w = 12
\][/tex]
3. From the total cost:
[tex]\[
7a + 10c + 12w = 118
\][/tex]
### Solve the System of Equations:
Using the equations, we can substitute [tex]\( w \)[/tex] in the second and third equations:
1. Substitute [tex]\( w = c + 2 \)[/tex] into [tex]\( a + c + w = 12 \)[/tex]:
[tex]\[
a + c + (c + 2) = 12 \\
a + 2c + 2 = 12 \\
a + 2c = 10 \tag{Equation 1}
\][/tex]
2. Substitute [tex]\( w = c + 2 \)[/tex] into [tex]\( 7a + 10c + 12w = 118 \)[/tex]:
[tex]\[
7a + 10c + 12(c + 2) = 118 \\
7a + 10c + 12c + 24 = 118 \\
7a + 22c = 94 \tag{Equation 2}
\][/tex]
Now solve Equation 1 and Equation 2 simultaneously:
From Equation 1:
[tex]\[ a = 10 - 2c \][/tex]
Substitute [tex]\( a = 10 - 2c \)[/tex] into Equation 2:
[tex]\[
7(10 - 2c) + 22c = 94 \\
70 - 14c + 22c = 94 \\
8c = 24 \\
c = 3
\][/tex]
Find [tex]\( a \)[/tex] from [tex]\( a = 10 - 2c \)[/tex]:
[tex]\[
a = 10 - 2(3) = 4
\][/tex]
Find [tex]\( w \)[/tex] from [tex]\( w = c + 2 \)[/tex]:
[tex]\[
w = 3 + 2 = 5
\][/tex]
### Conclusion:
The customer buys:
- 4 pounds of almonds
- 3 pounds of cashews
- 5 pounds of walnuts
Therefore, none of the given options correctly describe the relation between the amounts of the types of nuts the customer buys based on the solution.
### Given Conditions:
1. Price per pound:
- Almonds cost \[tex]$7 per pound.
- Cashews cost \$[/tex]10 per pound.
- Walnuts cost \[tex]$12 per pound.
2. Total weight and cost:
The customer buys a total of 12 pounds of mixed nuts for \$[/tex]118.
3. Relationships between types of nuts:
The customer buys 2 more pounds of walnuts than cashews.
### Setting up the Problem:
To set up the problem, let's define variables for the amounts in pounds:
- [tex]\( a \)[/tex] = pounds of almonds
- [tex]\( c \)[/tex] = pounds of cashews
- [tex]\( w \)[/tex] = pounds of walnuts
#### Formulating Equations:
1. From the relationship between walnuts and cashews:
[tex]\[
w = c + 2
\][/tex]
2. From the total weight:
[tex]\[
a + c + w = 12
\][/tex]
3. From the total cost:
[tex]\[
7a + 10c + 12w = 118
\][/tex]
### Solve the System of Equations:
Using the equations, we can substitute [tex]\( w \)[/tex] in the second and third equations:
1. Substitute [tex]\( w = c + 2 \)[/tex] into [tex]\( a + c + w = 12 \)[/tex]:
[tex]\[
a + c + (c + 2) = 12 \\
a + 2c + 2 = 12 \\
a + 2c = 10 \tag{Equation 1}
\][/tex]
2. Substitute [tex]\( w = c + 2 \)[/tex] into [tex]\( 7a + 10c + 12w = 118 \)[/tex]:
[tex]\[
7a + 10c + 12(c + 2) = 118 \\
7a + 10c + 12c + 24 = 118 \\
7a + 22c = 94 \tag{Equation 2}
\][/tex]
Now solve Equation 1 and Equation 2 simultaneously:
From Equation 1:
[tex]\[ a = 10 - 2c \][/tex]
Substitute [tex]\( a = 10 - 2c \)[/tex] into Equation 2:
[tex]\[
7(10 - 2c) + 22c = 94 \\
70 - 14c + 22c = 94 \\
8c = 24 \\
c = 3
\][/tex]
Find [tex]\( a \)[/tex] from [tex]\( a = 10 - 2c \)[/tex]:
[tex]\[
a = 10 - 2(3) = 4
\][/tex]
Find [tex]\( w \)[/tex] from [tex]\( w = c + 2 \)[/tex]:
[tex]\[
w = 3 + 2 = 5
\][/tex]
### Conclusion:
The customer buys:
- 4 pounds of almonds
- 3 pounds of cashews
- 5 pounds of walnuts
Therefore, none of the given options correctly describe the relation between the amounts of the types of nuts the customer buys based on the solution.
