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Answer :
Let's break down the situation described in the problem:
1. Understand What We Have:
- We know that almonds cost \[tex]$7 per pound, cashews cost \$[/tex]10 per pound, and walnuts cost \[tex]$12 per pound.
- A customer bought a total of 12 pounds of nuts for \$[/tex]118.
- The amount of walnuts bought is 2 more pounds than the amount of cashews.
2. Set Up the Equations:
- Let [tex]\( a \)[/tex] be the number of pounds of almonds, [tex]\( c \)[/tex] be the number of pounds of cashews, and [tex]\( w \)[/tex] be the number of pounds of walnuts.
- From the problem, we can form these equations:
- Total weight equation: [tex]\( a + c + w = 12 \)[/tex]
- Total cost equation: [tex]\( 7a + 10c + 12w = 118 \)[/tex]
- Relationship between walnuts and cashews: [tex]\( w = c + 2 \)[/tex]
3. Solve the System of Equations:
- Substituting [tex]\( w = c + 2 \)[/tex] into the total weight equation:
[tex]\( a + c + (c + 2) = 12 \)[/tex]
Simplifies to [tex]\( a + 2c + 2 = 12 \)[/tex]
Further simplifies to [tex]\( a + 2c = 10 \)[/tex] (Equation 1)
- Substitute [tex]\( w = c + 2 \)[/tex] into the total cost equation:
[tex]\( 7a + 10c + 12(c + 2) = 118 \)[/tex]
Simplifies to [tex]\( 7a + 10c + 12c + 24 = 118 \)[/tex]
Which simplifies to [tex]\( 7a + 22c = 94 \)[/tex] (Equation 2)
4. Using the Equations:
- Now, solve Equation 1 and Equation 2 together:
- Equation 1: [tex]\( a + 2c = 10 \)[/tex]
- Equation 2: [tex]\( 7a + 22c = 94 \)[/tex]
- Solve Equation 1 for [tex]\( a \)[/tex]:
[tex]\( a = 10 - 2c \)[/tex]
- Substitute [tex]\( a = 10 - 2c \)[/tex] into Equation 2:
[tex]\( 7(10 - 2c) + 22c = 94 \)[/tex]
[tex]\( 70 - 14c + 22c = 94 \)[/tex]
[tex]\( 8c = 24 \)[/tex]
[tex]\( c = 3 \)[/tex]
- Now, find [tex]\( a \)[/tex] using [tex]\( c = 3 \)[/tex]:
[tex]\( a = 10 - 2(3) = 4 \)[/tex]
- And find [tex]\( w \)[/tex] using [tex]\( c = 3 \)[/tex]:
[tex]\( w = c + 2 = 3 + 2 = 5 \)[/tex]
5. Verify the Interpretation:
- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- The statement "The customer buys 2 more pounds of walnuts than cashews and 1 more pound of almonds than cashews" matches the quantities calculated:
- Walnuts (5) are 2 more than cashews (3).
- Almonds (4) are 1 more than cashews (3).
Hence, the statement "The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews" is incorrect based on our calculations.
However, the correct interpretation based on our calculations is: "The customer buys 2 more pounds of walnuts than cashews and 1 more pound of almonds than cashews."
1. Understand What We Have:
- We know that almonds cost \[tex]$7 per pound, cashews cost \$[/tex]10 per pound, and walnuts cost \[tex]$12 per pound.
- A customer bought a total of 12 pounds of nuts for \$[/tex]118.
- The amount of walnuts bought is 2 more pounds than the amount of cashews.
2. Set Up the Equations:
- Let [tex]\( a \)[/tex] be the number of pounds of almonds, [tex]\( c \)[/tex] be the number of pounds of cashews, and [tex]\( w \)[/tex] be the number of pounds of walnuts.
- From the problem, we can form these equations:
- Total weight equation: [tex]\( a + c + w = 12 \)[/tex]
- Total cost equation: [tex]\( 7a + 10c + 12w = 118 \)[/tex]
- Relationship between walnuts and cashews: [tex]\( w = c + 2 \)[/tex]
3. Solve the System of Equations:
- Substituting [tex]\( w = c + 2 \)[/tex] into the total weight equation:
[tex]\( a + c + (c + 2) = 12 \)[/tex]
Simplifies to [tex]\( a + 2c + 2 = 12 \)[/tex]
Further simplifies to [tex]\( a + 2c = 10 \)[/tex] (Equation 1)
- Substitute [tex]\( w = c + 2 \)[/tex] into the total cost equation:
[tex]\( 7a + 10c + 12(c + 2) = 118 \)[/tex]
Simplifies to [tex]\( 7a + 10c + 12c + 24 = 118 \)[/tex]
Which simplifies to [tex]\( 7a + 22c = 94 \)[/tex] (Equation 2)
4. Using the Equations:
- Now, solve Equation 1 and Equation 2 together:
- Equation 1: [tex]\( a + 2c = 10 \)[/tex]
- Equation 2: [tex]\( 7a + 22c = 94 \)[/tex]
- Solve Equation 1 for [tex]\( a \)[/tex]:
[tex]\( a = 10 - 2c \)[/tex]
- Substitute [tex]\( a = 10 - 2c \)[/tex] into Equation 2:
[tex]\( 7(10 - 2c) + 22c = 94 \)[/tex]
[tex]\( 70 - 14c + 22c = 94 \)[/tex]
[tex]\( 8c = 24 \)[/tex]
[tex]\( c = 3 \)[/tex]
- Now, find [tex]\( a \)[/tex] using [tex]\( c = 3 \)[/tex]:
[tex]\( a = 10 - 2(3) = 4 \)[/tex]
- And find [tex]\( w \)[/tex] using [tex]\( c = 3 \)[/tex]:
[tex]\( w = c + 2 = 3 + 2 = 5 \)[/tex]
5. Verify the Interpretation:
- The customer buys 4 pounds of almonds, 3 pounds of cashews, and 5 pounds of walnuts.
- The statement "The customer buys 2 more pounds of walnuts than cashews and 1 more pound of almonds than cashews" matches the quantities calculated:
- Walnuts (5) are 2 more than cashews (3).
- Almonds (4) are 1 more than cashews (3).
Hence, the statement "The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews" is incorrect based on our calculations.
However, the correct interpretation based on our calculations is: "The customer buys 2 more pounds of walnuts than cashews and 1 more pound of almonds than cashews."
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