How many gallons of 30% alcohol solution and 45% alcohol solution must be mixed to get 12 gallons of 35% alcohol solution?
see related questions below, or the mixture problems posted after this one.
To solve this problem, we can use a mixture equation.
Let's assume that "x" gallons of 30% alcohol solution and "y" gallons of 45% alcohol solution are mixed to obtain 12 gallons of a 35% alcohol solution.
The total gallons of the mixture is the sum of the individual gallons of the two solutions:
x + y = 12 (Equation 1)
The total amount of alcohol in the mixture is the sum of the alcohol in the two solutions:
0.30x + 0.45y = 0.35 * 12 (Equation 2)
Now we have a system of equations that we can solve simultaneously.
To solve for x and y, we can use substitution or elimination.
Let's solve using elimination:
Multiply Equation 1 by 0.30 and Equation 2 by 100 to eliminate the decimals:
0.30x + 0.30y = 0.30 * 12
30x + 45y = 35 * 12
Now, multiply Equation 1 by -30 to make the x's cancel:
-30x - 30y = -30 * 12
Now we have:
-30x - 30y = -360
30x + 45y = 420
Add the equations together:
(-30x + 30x) + (-30y + 45y) = (-360 + 420)
15y = 60
Divide both sides by 15:
y = 4
Now, substitute the value of y back into Equation 1 to solve for x:
x + 4 = 12
x = 8
Therefore, you will need 8 gallons of 30% alcohol solution and 4 gallons of 45% alcohol solution to obtain 12 gallons of 35% alcohol solution.