Question
In a seafood stew, Erin wants to use lobster, crab, and shrimp. She wants one part lobster, two parts crab, and three parts shrimp. lobster cost $20 per pound, crab cost $15 per pound, and shrimp cost $10 per pound. Erin has $40 to spend on the seafood mix. The equation 20x + 15(2x) + 10(3x) = 40 models the pounds, x. How many pounds of the seafood mix can Erin buy?
Responses
A 22
B 33
C 44
D 6
Responses
A 22
B 33
C 44
D 6
Answers
GPT-4o mini
To determine how many pounds of the seafood mix Erin can buy, we start with the equation she provided:
\[ 20x + 15(2x) + 10(3x) = 40. \]
Let's simplify this step by step.
1. Distribute the quantities in the equation:
\[ 20x + 30x + 30x = 40. \]
2. Combine like terms:
\[ (20x + 30x + 30x) = 80x. \]
So, the equation simplifies to:
\[ 80x = 40. \]
3. Now, solve for \( x \):
\[ x = \frac{40}{80} = \frac{1}{2}. \]
Now, recall that \( x \) represents the part of lobster, meaning for every 1 part of lobster, there are 2 parts of crab and 3 parts of shrimp. Therefore, the total amount in terms of parts can be calculated:
- Total parts = 1 part lobster + 2 parts crab + 3 parts shrimp = 6 parts.
To find the total weight of the seafood mix, we multiply \( x \) (which is 0.5 pounds) by the total number of parts:
- Total pounds of seafood mix = \( 6 \times x = 6 \times \frac{1}{2} = 3 \) pounds.
Now, let's check how much Erin spends for this mix:
- Lobster: \( 20 \times \frac{1}{2} = 10 \)
- Crab: \( 15 \times 1 = 15 \)
- Shrimp: \( 10 \times \frac{3}{2} = 15 \)
Total spent:
\[ 10 + 15 + 15 = 40 \].
Since this aligns with her budget, we conclude the correct response:
Erin can buy a total of **3 pounds** of the seafood mix. However, this option was not provided. Assuming the list options corresponded wrongly or there's a misunderstanding, it's simply not listed. But according to the calculations, it is 3 pounds that she could buy, which means
**Answer: 3 pounds (not provided in the answer options).**
\[ 20x + 15(2x) + 10(3x) = 40. \]
Let's simplify this step by step.
1. Distribute the quantities in the equation:
\[ 20x + 30x + 30x = 40. \]
2. Combine like terms:
\[ (20x + 30x + 30x) = 80x. \]
So, the equation simplifies to:
\[ 80x = 40. \]
3. Now, solve for \( x \):
\[ x = \frac{40}{80} = \frac{1}{2}. \]
Now, recall that \( x \) represents the part of lobster, meaning for every 1 part of lobster, there are 2 parts of crab and 3 parts of shrimp. Therefore, the total amount in terms of parts can be calculated:
- Total parts = 1 part lobster + 2 parts crab + 3 parts shrimp = 6 parts.
To find the total weight of the seafood mix, we multiply \( x \) (which is 0.5 pounds) by the total number of parts:
- Total pounds of seafood mix = \( 6 \times x = 6 \times \frac{1}{2} = 3 \) pounds.
Now, let's check how much Erin spends for this mix:
- Lobster: \( 20 \times \frac{1}{2} = 10 \)
- Crab: \( 15 \times 1 = 15 \)
- Shrimp: \( 10 \times \frac{3}{2} = 15 \)
Total spent:
\[ 10 + 15 + 15 = 40 \].
Since this aligns with her budget, we conclude the correct response:
Erin can buy a total of **3 pounds** of the seafood mix. However, this option was not provided. Assuming the list options corresponded wrongly or there's a misunderstanding, it's simply not listed. But according to the calculations, it is 3 pounds that she could buy, which means
**Answer: 3 pounds (not provided in the answer options).**