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Answer :
Let $a$, $c$, and $w$ represent the pounds of almonds, cashews, and walnuts respectively. We are given the following information:
1. Walnuts: There are 2 more pounds of walnuts than cashews, which gives
$$w = c + 2.$$
2. Total weight: The customer buys 12 pounds of nuts, so
$$a + c + w = 12.$$
3. Total cost: Almonds cost \$7 per pound, cashews \$10 per pound, and walnuts \$12 per pound, with a total cost of \$118. Thus,
$$7a + 10c + 12w = 118.$$
**Step 1. Substitute the Relation for Walnuts**
Since $w = c + 2$, substitute this into the total weight equation:
\[
a + c + (c + 2) = 12.
\]
Combine like terms:
\[
a + 2c + 2 = 12 \quad \Longrightarrow \quad a + 2c = 10.
\]
Solve for $a$:
\[
a = 10 - 2c.
\]
**Step 2. Substitute into the Total Cost Equation**
Replace $a$ and $w$ in the cost equation with their expressions in terms of $c$:
\[
7(10 - 2c) + 10c + 12(c + 2) = 118.
\]
Expand each term:
\[
70 - 14c + 10c + 12c + 24 = 118.
\]
Combine like terms:
\[
(70 + 24) + (-14c + 10c + 12c) = 118 \quad \Longrightarrow \quad 94 + 8c = 118.
\]
Solve for $c$:
\[
8c = 118 - 94 = 24 \quad \Longrightarrow \quad c = 3.
\]
**Step 3. Find $a$ and $w$**
Substitute $c = 3$ back into the expressions for $a$ and $w$:
\[
a = 10 - 2(3) = 10 - 6 = 4,
\]
\[
w = c + 2 = 3 + 2 = 5.
\]
**Step 4. Interpret the Answer**
Now we have:
\[
a = 4, \quad c = 3, \quad w = 5.
\]
Next, we compute the differences:
\[
w - a = 5 - 4 = 1,\quad \text{and}\quad a - c = 4 - 3 = 1.
\]
This means the customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
Thus, the correct possible interpretation is:
$$\textbf{The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.}$$
1. Walnuts: There are 2 more pounds of walnuts than cashews, which gives
$$w = c + 2.$$
2. Total weight: The customer buys 12 pounds of nuts, so
$$a + c + w = 12.$$
3. Total cost: Almonds cost \$7 per pound, cashews \$10 per pound, and walnuts \$12 per pound, with a total cost of \$118. Thus,
$$7a + 10c + 12w = 118.$$
**Step 1. Substitute the Relation for Walnuts**
Since $w = c + 2$, substitute this into the total weight equation:
\[
a + c + (c + 2) = 12.
\]
Combine like terms:
\[
a + 2c + 2 = 12 \quad \Longrightarrow \quad a + 2c = 10.
\]
Solve for $a$:
\[
a = 10 - 2c.
\]
**Step 2. Substitute into the Total Cost Equation**
Replace $a$ and $w$ in the cost equation with their expressions in terms of $c$:
\[
7(10 - 2c) + 10c + 12(c + 2) = 118.
\]
Expand each term:
\[
70 - 14c + 10c + 12c + 24 = 118.
\]
Combine like terms:
\[
(70 + 24) + (-14c + 10c + 12c) = 118 \quad \Longrightarrow \quad 94 + 8c = 118.
\]
Solve for $c$:
\[
8c = 118 - 94 = 24 \quad \Longrightarrow \quad c = 3.
\]
**Step 3. Find $a$ and $w$**
Substitute $c = 3$ back into the expressions for $a$ and $w$:
\[
a = 10 - 2(3) = 10 - 6 = 4,
\]
\[
w = c + 2 = 3 + 2 = 5.
\]
**Step 4. Interpret the Answer**
Now we have:
\[
a = 4, \quad c = 3, \quad w = 5.
\]
Next, we compute the differences:
\[
w - a = 5 - 4 = 1,\quad \text{and}\quad a - c = 4 - 3 = 1.
\]
This means the customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.
Thus, the correct possible interpretation is:
$$\textbf{The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.}$$
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