Solution A is 60% alcohol and Solution B is 30% alcohol. How much of each is needed to make 30 gallons of a solution that is 40% alcohol

A + B = 30

0.6A + 0.3B = 0.4(30)

A + B = 30
6A + 3B = 120

-3A -3B = -90
6A + 3B = 120

3A = 30
3A/3 = 30/3
A = 10
10 + B = 30
10-10 + B = 30-10
B = 20

Or
A = x B = 3-x , 0.4(30)
0.6x + 0.3(30-x) = 0.4(30)
0.6x + 9 -0.3x = 12
0.3x + 9 = 12
0.3x + 9-9 = 12-9
0.3x = 3
0.3x/0.3 = 3/0.3
x = 10
A = 10
B =30-10 =20
10 at 60% and 20 at 30%.

To find out the amount of each solution needed to make the desired solution, we can set up a system of two equations.

Let's assume x represents the amount of Solution A (60% alcohol) needed and y represents the amount of Solution B (30% alcohol) needed.

Equation 1: x + y = 30
This equation represents the total amount of solution needed, which is 30 gallons.

Equation 2: 0.6x + 0.3y = 0.4 * 30
Since we want the resulting solution to be 40% alcohol, we multiply the alcohol content by the total volume.

Now, let's solve the system of equations:

From Equation 1, we have:
x = 30 - y

Substituting x in Equation 2, we get:
0.6(30 - y) + 0.3y = 12

18 - 0.6y + 0.3y = 12
-0.3y = 12 - 18
-0.3y = -6
y = -6 / -0.3
y = 20

Substituting this value of y back into Equation 1, we have:
x + 20 = 30
x = 30 - 20
x = 10

So, to make a 40% alcohol solution, you would need to mix 10 gallons of Solution A (60% alcohol) with 20 gallons of Solution B (30% alcohol).

To determine how much of each solution is needed to make a 40% alcohol solution, we can solve this problem using the method of mixtures.

Let's assume that we will need "x" gallons of Solution A and "y" gallons of Solution B.

Given:
Solution A is 60% alcohol, which means it contains 60/100 = 0.6 alcohol content.
Solution B is 30% alcohol, which means it contains 30/100 = 0.3 alcohol content.
We need to make 30 gallons of a solution that is 40% alcohol.

To solve this problem, we can create an equation based on the alcohol content as follows:

0.6x + 0.3y = 0.4 * 30

Now, let's solve this equation to find the values of x and y.

0.6x + 0.3y = 12

Multiply the equation by 10 to get rid of the decimal point:

6x + 3y = 120

To further simplify the equation, we can divide it by 3:

2x + y = 40

Now, we have two equations:
(1) 2x + y = 40
(2) x + y = 30 (since we need a total of 30 gallons)

To solve these equations simultaneously, we can either use substitution or elimination method. Let's use the elimination method.

Multiply equation (2) by -1, so we can eliminate the "y" term:

-1(x + y) = -1(30)
-x - y = -30

Now, add the two equations together:

(2x + y) + (-x - y) = 40 + (-30)
x = 10

Substitute the value of x into equation (2):

10 + y = 30
y = 30 - 10
y = 20

Therefore, to make a 30-gallon solution that is 40% alcohol, you would need 10 gallons of Solution A (60% alcohol) and 20 gallons of Solution B (30% alcohol).