To obtain a 45% alcohol solution, you need to mix the 20% and 55% alcohol solutions. Let's solve this step-by-step.
Let's say you need x liters of the 20% alcohol solution and y liters of the 55% alcohol solution to obtain the desired 7 liters of the 45% alcohol solution.
To calculate the amount of alcohol in the final solution, we multiply the volume of each solution by its respective alcohol percentage.
For the 20% alcohol solution, the amount of alcohol is 0.2x liters (20% of x liters).
For the 55% alcohol solution, the amount of alcohol is 0.55y liters (55% of y liters).
To find the total amount of alcohol in the final solution, we add these two quantities:
0.2x + 0.55y = total amount of alcohol in the final solution
Since we want a 45% alcohol solution, the total amount of alcohol should be 45% of the total volume of the solution. So, the amount of alcohol is 0.45 * 7 = 3.15 liters.
Now we can set up an equation using the previous information:
0.2x + 0.55y = 3.15
We also have the constraint that the total volume of the solution is 7 liters:
x + y = 7
Now, we can solve this system of equations using substitution or elimination to find the values of x and y.
Let's use the substitution method:
From the constraint equation, we can express x in terms of y as x = 7 - y.
Substituting this value of x into the first equation:
0.2(7 - y) + 0.55y = 3.15
Simplifying the equation:
1.4 - 0.2y + 0.55y = 3.15
Combine like terms:
0.35y = 1.75
Divide both sides by 0.35:
y = 5
Now substitute the value of y back into the equation x = 7 - y:
x = 7 - 5
x = 2
Therefore, you need 2 liters of the 20% alcohol solution and 5 liters of the 55% alcohol solution to obtain 7 liters of a 45% alcohol solution.