Maggie makes some fruit punch. She mixes 2 1/2 quarts of grape juice with 1 1/2 quarts of orange juice. One quart of grape juice costs $1 less than one quart of orange juice. She finds that the total cost of making the fruit punch is $12.50. Calculate the cost of each quart of grape juice and each quart of orange juice.
cost per quart for orange --- x
cost per quart for grapejuice -- x-1
(5/2)(x-1) + (3/2)x = 12.5
times 2
5(x-1) + 3x = 25
8x = 30
x = 3.75
x-1 = 2.75
orange juice costs $3.75 per quart and grape juice costs $2.75 per quart
I think it's 3 cups idrk
Step 1: We know that Maggie mixes 2 1/2 quarts of grape juice with 1 1/2 quarts of orange juice.
Step 2: Let's calculate the total amount of juice Maggie mixed:
- 2 1/2 quarts of grape juice = 2.5 quarts
- 1 1/2 quarts of orange juice = 1.5 quarts
So, the total amount of juice mixed is 2.5 + 1.5 = 4 quarts.
Step 3: Now, let's assign variables to the cost of each quart of grape juice and orange juice.
- Let's assume the cost of each quart of orange juice is x dollars.
- According to the problem, the cost of each quart of grape juice is $1 less than the cost of each quart of orange juice. So, the cost of each quart of grape juice = x - $1.
Step 4: We can write the equation for the total cost of making the fruit punch:
- Total cost = (cost per quart of grape juice) × (quarts of grape juice) + (cost per quart of orange juice) × (quarts of orange juice).
Given that the total cost is $12.50 and the total amount of juice mixed is 4 quarts, we get the equation:
- $12.50 = (x - $1) × 2.5 + x × 1.5.
Step 5: Simplify the equation:
- $12.50 = 2.5x - $2.5 + 1.5x.
Step 6: Combine like terms:
- $12.50 = 4x - $2.5.
Step 7: Move the constant term to the other side:
- $12.50 + $2.50 = 4x.
- $15 = 4x.
Step 8: Solve for x:
- Divide both sides of the equation by 4:
- $15 / 4 = x.
- $3.75 = x.
Step 9: Therefore, the cost of each quart of orange juice is $3.75.
Step 10: To find the cost of each quart of grape juice, we subtract $1 from the cost of each quart of orange juice:
- Cost of each quart of grape juice = $3.75 - $1 = $2.75.
Therefore, the cost of each quart of grape juice is $2.75 and the cost of each quart of orange juice is $3.75.
To solve this problem, we need to set up equations based on the given information. Let's start by assigning variables to the unknowns.
Let's say the cost of one quart of orange juice is x dollars. Since one quart of grape juice costs $1 less, we can say that the cost of one quart of grape juice is x - $1.
Next, let's calculate the total quantity of grape juice and orange juice used. According to the problem, Maggie mixes 2 1/2 quarts of grape juice and 1 1/2 quarts of orange juice. We can convert these mixed numbers into improper fractions for easier calculations:
2 1/2 quarts of grape juice is equal to (2 * 2) + 1/2 = 5/2 quarts of grape juice.
1 1/2 quarts of orange juice is equal to (1 * 2) + 1/2 = 3/2 quarts of orange juice.
Now we can write an equation to represent the total cost of making the fruit punch. The cost of each quart of grape juice multiplied by the total quantity of grape juice, plus the cost of each quart of orange juice multiplied by the total quantity of orange juice, equals the total cost of $12.50:
(x - $1) * (5/2) + x * (3/2) = $12.50
Now, simplify the equation:
(5/2)x - (5/2) + (3/2)x = $12.50
(8/2)x - (5/2) = $12.50
4x - 5 = $12.50
To solve for x, we can isolate x on one side of the equation by adding 5 to both sides:
4x - 5 + 5 = $12.50 + $5
4x = $17.50
Finally, we divide both sides by 4 to solve for x:
4x/4 = $17.50/4
x = $4.375
Therefore, the cost of one quart of orange juice (x) is $4.375, and the cost of one quart of grape juice (x - $1) is $3.375.