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Answer :
We first introduce variables for the amounts (in pounds) of each type of nut purchased:
- Let [tex]$a$[/tex] be the pounds of almonds.
- Let [tex]$c$[/tex] be the pounds of cashews.
- Let [tex]$w$[/tex] be the pounds of walnuts.
From the problem, we have the following three conditions:
1. The customer buys 2 more pounds of walnuts than cashews:
[tex]$$w - c = 2.$$[/tex]
2. The total cost is \[tex]$118. With almonds at \$[/tex]7 per pound, cashews at \[tex]$10 per pound, and walnuts at \$[/tex]12 per pound:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
3. The total weight of the nuts is 12 pounds:
[tex]$$a + c + w = 12.$$[/tex]
Step 1. Express one variable in terms of another.
From the first equation,
[tex]$$w = c + 2.$$[/tex]
Step 2. Substitute [tex]$w$[/tex] into the total weight equation.
Replace [tex]$w$[/tex] in the equation [tex]$a + c + w = 12$[/tex]:
[tex]$$a + c + (c + 2) = 12.$$[/tex]
Combine like terms:
[tex]$$a + 2c + 2 = 12.$$[/tex]
Subtract 2 from both sides:
[tex]$$a + 2c = 10.$$[/tex]
Solve for [tex]$a$[/tex]:
[tex]$$a = 10 - 2c.$$[/tex]
Step 3. Substitute [tex]$a$[/tex] and [tex]$w$[/tex] into the cost equation.
The cost equation is:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
Substitute [tex]$a = 10 - 2c$[/tex] and [tex]$w = c + 2$[/tex]:
[tex]$$7(10 - 2c) + 10c + 12(c + 2) = 118.$$[/tex]
Expand the terms:
[tex]$$70 - 14c + 10c + 12c + 24 = 118.$$[/tex]
Combine like terms:
[tex]$$70 + 24 + (-14c + 10c + 12c) = 118,$$[/tex]
[tex]$$94 + 8c = 118.$$[/tex]
Subtract 94 from both sides:
[tex]$$8c = 24.$$[/tex]
Divide both sides by 8:
[tex]$$c = 3.$$[/tex]
Step 4. Find [tex]$a$[/tex] and [tex]$w$[/tex].
Substitute [tex]$c = 3$[/tex] into [tex]$a = 10 - 2c$[/tex]:
[tex]$$a = 10 - 2(3) = 10 - 6 = 4.$$[/tex]
Substitute [tex]$c = 3$[/tex] into [tex]$w = c + 2$[/tex]:
[tex]$$w = 3 + 2 = 5.$$[/tex]
Step 5. Interpret the results.
The customer buys:
- 4 pounds of almonds,
- 3 pounds of cashews,
- 5 pounds of walnuts.
Now, compute the differences:
- Difference between walnuts and almonds:
[tex]$$w - a = 5 - 4 = 1.$$[/tex]
- Difference between almonds and cashews:
[tex]$$a - c = 4 - 3 = 1.$$[/tex]
This means the customer purchases 1 more pound of walnuts than almonds, and 1 more pound of almonds than cashews.
Conclusion:
The results correspond to the statement:
"The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
Thus, the correct interpretation is option 1.
- Let [tex]$a$[/tex] be the pounds of almonds.
- Let [tex]$c$[/tex] be the pounds of cashews.
- Let [tex]$w$[/tex] be the pounds of walnuts.
From the problem, we have the following three conditions:
1. The customer buys 2 more pounds of walnuts than cashews:
[tex]$$w - c = 2.$$[/tex]
2. The total cost is \[tex]$118. With almonds at \$[/tex]7 per pound, cashews at \[tex]$10 per pound, and walnuts at \$[/tex]12 per pound:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
3. The total weight of the nuts is 12 pounds:
[tex]$$a + c + w = 12.$$[/tex]
Step 1. Express one variable in terms of another.
From the first equation,
[tex]$$w = c + 2.$$[/tex]
Step 2. Substitute [tex]$w$[/tex] into the total weight equation.
Replace [tex]$w$[/tex] in the equation [tex]$a + c + w = 12$[/tex]:
[tex]$$a + c + (c + 2) = 12.$$[/tex]
Combine like terms:
[tex]$$a + 2c + 2 = 12.$$[/tex]
Subtract 2 from both sides:
[tex]$$a + 2c = 10.$$[/tex]
Solve for [tex]$a$[/tex]:
[tex]$$a = 10 - 2c.$$[/tex]
Step 3. Substitute [tex]$a$[/tex] and [tex]$w$[/tex] into the cost equation.
The cost equation is:
[tex]$$7a + 10c + 12w = 118.$$[/tex]
Substitute [tex]$a = 10 - 2c$[/tex] and [tex]$w = c + 2$[/tex]:
[tex]$$7(10 - 2c) + 10c + 12(c + 2) = 118.$$[/tex]
Expand the terms:
[tex]$$70 - 14c + 10c + 12c + 24 = 118.$$[/tex]
Combine like terms:
[tex]$$70 + 24 + (-14c + 10c + 12c) = 118,$$[/tex]
[tex]$$94 + 8c = 118.$$[/tex]
Subtract 94 from both sides:
[tex]$$8c = 24.$$[/tex]
Divide both sides by 8:
[tex]$$c = 3.$$[/tex]
Step 4. Find [tex]$a$[/tex] and [tex]$w$[/tex].
Substitute [tex]$c = 3$[/tex] into [tex]$a = 10 - 2c$[/tex]:
[tex]$$a = 10 - 2(3) = 10 - 6 = 4.$$[/tex]
Substitute [tex]$c = 3$[/tex] into [tex]$w = c + 2$[/tex]:
[tex]$$w = 3 + 2 = 5.$$[/tex]
Step 5. Interpret the results.
The customer buys:
- 4 pounds of almonds,
- 3 pounds of cashews,
- 5 pounds of walnuts.
Now, compute the differences:
- Difference between walnuts and almonds:
[tex]$$w - a = 5 - 4 = 1.$$[/tex]
- Difference between almonds and cashews:
[tex]$$a - c = 4 - 3 = 1.$$[/tex]
This means the customer purchases 1 more pound of walnuts than almonds, and 1 more pound of almonds than cashews.
Conclusion:
The results correspond to the statement:
"The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews."
Thus, the correct interpretation is option 1.
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