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.
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.
