Answer :
Let the number of pounds of almonds, cashews, and walnuts be [tex]$a$[/tex], [tex]$c$[/tex], and [tex]$w$[/tex], respectively. We are given the following information:
1. The total weight is 12 pounds:
[tex]$$a + c + w = 12.$$[/tex]
2. The total cost is \[tex]$118, with prices \$[/tex]7, \[tex]$10, and \$[/tex]12 per pound for almonds, cashews, and walnuts, respectively:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
3. There are 2 more pounds of walnuts than cashews:
[tex]$$w - c = 2.$$[/tex]
Step 1. Express [tex]$w$[/tex] in terms of [tex]$c$[/tex].
From the third equation,
[tex]$$w = c + 2.$$[/tex]
Step 2. Substitute [tex]$w = c + 2$[/tex] into the total weight equation.
The first equation becomes:
[tex]\[
a + c + (c + 2) = 12.
\][/tex]
Simplify the equation:
[tex]\[
a + 2c + 2 = 12 \quad \Longrightarrow \quad a + 2c = 10.
\][/tex]
Solve for [tex]$a$[/tex]:
[tex]\[
a = 10 - 2c.
\][/tex]
Step 3. Substitute [tex]$a = 10 - 2c$[/tex] and [tex]$w = c + 2$[/tex] into the cost equation.
The cost equation becomes:
[tex]\[
7(10 - 2c) + 10c + 12(c + 2) = 118.
\][/tex]
Expand and simplify:
[tex]\[
70 - 14c + 10c + 12c + 24 = 118.
\][/tex]
Combine like terms:
[tex]\[
70 + 24 + (-14c + 10c + 12c) = 118 \quad \Longrightarrow \quad 94 + 8c = 118.
\][/tex]
Solve for [tex]$c$[/tex]:
[tex]\[
8c = 118 - 94 \quad \Longrightarrow \quad 8c = 24 \quad \Longrightarrow \quad c = 3.
\][/tex]
Step 4. Determine [tex]$a$[/tex] and [tex]$w$[/tex].
Using [tex]$a = 10 - 2c$[/tex]:
[tex]\[
a = 10 - 2(3) = 10 - 6 = 4.
\][/tex]
Using [tex]$w = c + 2$[/tex]:
[tex]\[
w = 3 + 2 = 5.
\][/tex]
Step 5. Calculate the differences between the quantities.
Difference between walnuts and almonds:
[tex]\[
w - a = 5 - 4 = 1.
\][/tex]
Difference between almonds and cashews:
[tex]\[
a - c = 4 - 3 = 1.
\][/tex]
Conclusion:
The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews. This interpretation corresponds to the statement:
"The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
1. The total weight is 12 pounds:
[tex]$$a + c + w = 12.$$[/tex]
2. The total cost is \[tex]$118, with prices \$[/tex]7, \[tex]$10, and \$[/tex]12 per pound for almonds, cashews, and walnuts, respectively:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
3. There are 2 more pounds of walnuts than cashews:
[tex]$$w - c = 2.$$[/tex]
Step 1. Express [tex]$w$[/tex] in terms of [tex]$c$[/tex].
From the third equation,
[tex]$$w = c + 2.$$[/tex]
Step 2. Substitute [tex]$w = c + 2$[/tex] into the total weight equation.
The first equation becomes:
[tex]\[
a + c + (c + 2) = 12.
\][/tex]
Simplify the equation:
[tex]\[
a + 2c + 2 = 12 \quad \Longrightarrow \quad a + 2c = 10.
\][/tex]
Solve for [tex]$a$[/tex]:
[tex]\[
a = 10 - 2c.
\][/tex]
Step 3. Substitute [tex]$a = 10 - 2c$[/tex] and [tex]$w = c + 2$[/tex] into the cost equation.
The cost equation becomes:
[tex]\[
7(10 - 2c) + 10c + 12(c + 2) = 118.
\][/tex]
Expand and simplify:
[tex]\[
70 - 14c + 10c + 12c + 24 = 118.
\][/tex]
Combine like terms:
[tex]\[
70 + 24 + (-14c + 10c + 12c) = 118 \quad \Longrightarrow \quad 94 + 8c = 118.
\][/tex]
Solve for [tex]$c$[/tex]:
[tex]\[
8c = 118 - 94 \quad \Longrightarrow \quad 8c = 24 \quad \Longrightarrow \quad c = 3.
\][/tex]
Step 4. Determine [tex]$a$[/tex] and [tex]$w$[/tex].
Using [tex]$a = 10 - 2c$[/tex]:
[tex]\[
a = 10 - 2(3) = 10 - 6 = 4.
\][/tex]
Using [tex]$w = c + 2$[/tex]:
[tex]\[
w = 3 + 2 = 5.
\][/tex]
Step 5. Calculate the differences between the quantities.
Difference between walnuts and almonds:
[tex]\[
w - a = 5 - 4 = 1.
\][/tex]
Difference between almonds and cashews:
[tex]\[
a - c = 4 - 3 = 1.
\][/tex]
Conclusion:
The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews. This interpretation corresponds to the statement:
"The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
