The boys mixed 10 gallons of the 20% pure lemon juice mix and 10 gallons of the 70% pure lemon juice mix, because they wanted 20 gallons of lemonade at a 40% lemon juice mixture. They thought that because 40% was almost halfway between 20% and 70%, they should just mix equal parts of both, but the lemonade turned out too tart. How much of each should they have used to get a final mixture of 20 gallons at 40% lemon juice? Write your answer in complete sentences and show all work.

Where is your attempt to "Write your answer in complete sentences and show all work." ?

To solve this problem, we'll need to use the concept of weighted averages. Let's break down the problem step by step.

Step 1: Determine the amount of lemon juice in the desired 40% mixture.
Since we want a final mixture of 20 gallons at 40% lemon juice, we can calculate the total amount of lemon juice needed using the formula:
Amount of lemon juice = Total mixture volume * Lemon juice percentage.
So, the amount of lemon juice in 20 gallons at 40% is:
Lemon juice = 20 gallons * 0.4 = 8 gallons

Step 2: Set up a system of equations to represent the problem.
Let's assume x represents the amount of 20% lemon juice mix used and y represents the amount of 70% lemon juice mix used. We can set up two equations based on the given information:
Equation 1: x + y = 20 (total mixture volume)
Equation 2: 0.2x + 0.7y = 8 (total amount of lemon juice)

Step 3: Solve the system of equations.
We have the following system of equations:
x + y = 20
0.2x + 0.7y = 8

We can solve the system using substitution or elimination method. Let's solve it using substitution.

From Equation 1, we can express x as a function of y:
x = 20 - y

Substituting this into Equation 2, we get:
0.2(20 - y) + 0.7y = 8
4 - 0.2y + 0.7y = 8
0.5y = 4
y = 4 / 0.5
y = 8

Now, substitute the value of y back into Equation 1:
x + 8 = 20
x = 20 - 8
x = 12

So, they should have used 12 gallons of the 20% lemon juice mix and 8 gallons of the 70% lemon juice mix to get a final mixture of 20 gallons at 40% lemon juice.