A vat contains 8080 gallons of a fruit juice mixture that is 30%30% orange juice. How many gallons of the mixture should be drained and replaced with pure orange juice to make the mixture 80%80% orange juice? Round your answer to two decimal places, if necessary.
[.30(8080 - x)] + [1.0(x)] = .80(8080)
To solve this problem, follow these steps:
Step 1: Determine the amount of orange juice in the initial mixture
Given that the initial mixture is 30% orange juice and the total volume is 8080 gallons, we can find the amount of orange juice in the mixture.
Orange juice in the initial mixture = 30% of 8080 gallons
= (30/100) * 8080 gallons
= 2424 gallons
Step 2: Set up the equation
Let's assume x gallons need to be drained and replaced with pure orange juice. The amount of orange juice in the vat will increase by x gallons.
Therefore, the amount of orange juice in the final mixture will be:
2424 gallons + x gallons
Step 3: Calculate the total volume in the vat after replacing x gallons
Since x gallons were drained and replaced with pure orange juice, the total volume in the vat will remain constant at 8080 gallons.
Therefore, the equation for the total volume in the vat after replacing x gallons is:
8080 gallons = amount of orange juice (2424 gallons) + x gallons
Step 4: Solve the equation
Let's solve the equation for x to find how many gallons need to be drained and replaced.
8080 = 2424 + x
x = 8080 - 2424
x = 5656 gallons
Therefore, 5656 gallons of the mixture should be drained and replaced with pure orange juice to make the mixture 80% orange juice.