Let's analyze the situation step by step to determine which statements are correct.
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Let a be the cost per pound of almonds, and since the pistachios cost $4 more per pound, the cost per pound of pistachios will be a + 4.
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Venetta buys:
- 2 pounds of pistachios: \( 2(a + 4) \)
- 3 pounds of almonds: \( 3a \)
She pays a total of $48: \[ 2(a + 4) + 3a = 48 \] Expanding this equation: \[ 2a + 8 + 3a = 48 \ 5a + 8 = 48 \ 5a = 40 \ a = 8 \] So the cost per pound of almonds is \( a = 8 \), and the cost per pound of pistachios is \( a + 4 = 12 \).
Now we can evaluate the statements provided:
A. One pound of pistachios plus 1 pound of almonds cost $20.
- Pistachios = $12 and Almonds = $8, so \( 12 + 8 = 20 \).
- This statement is correct.
B. The pistachios cost twice as much per pound as the almonds.
- Pistachios = $12 and Almonds = $8. Thus, \( 12 \neq 2 \times 8 \).
- This statement is incorrect.
C. Reducing the number of pounds of almonds by one results in a total cost of $40.
- If Venetta buys 2 pounds of pistachios and 2 pounds of almonds (1 pound less than before): \[ 2(12) + 2(8) = 24 + 16 = 40 \]
- This statement is correct.
D. The cost a, in dollars, of 1 pound of almonds is modeled by the equation \(2(a - 4) + 3a = 48\).
- This equation would imply \(2a - 8 + 3a = 48\) which simplifies to \(5a - 8 = 48\). Thus, \(5a = 56\) and \(a = 11.2\) which is incorrect since we know \(a = 8\).
- This statement is incorrect.
E. The cost p, in dollars, of 1 pound of pistachios is modeled by the equation \(2p + 3(p - 4) = 48\).
- This expands to \(2p + 3p - 12 = 48\) which simplifies to \(5p - 12 = 48\). Thus, \(5p = 60\) and \(p = 12\), which is correct since we found \(p = 12\) earlier.
- This statement is correct.
Summary of Correct Statements:
- A. Correct
- C. Correct
- E. Correct
So the correct answers are:
- A, C, and E.
3 answers