a chemist needs to mix a 70 alcohol solution with a 40 alcohol solution to obtain 75 liters of a 50% alcohol solution. how many liters of each alcohol solution must be used
To determine the number of liters of each alcohol solution needed, we can use a simple algebraic approach.
Let's denote the volume of the 70% alcohol solution as 'x' liters and the volume of the 40% alcohol solution as 'y' liters.
1. Create an equation for the total volume of the solution:
x + y = 75 (equation 1)
2. Create an equation for the total amount of alcohol in the solution:
0.70x + 0.40y = 0.50 * 75 (equation 2)
3. Simplify equation 2:
0.70x + 0.40y = 37.5
Now, we have a system of two equations:
x + y = 75 (equation 1)
0.70x + 0.40y = 37.5 (equation 2)
We can solve this system using a substitution method or elimination method.
Let's use the substitution method here.
From equation 1, solve for x in terms of y:
x = 75 - y
Substitute this value of x into equation 2:
0.70(75 - y) + 0.40y = 37.5
Simplify the equation:
52.5 - 0.70y + 0.40y = 37.5
Combine like terms:
-0.30y = 37.5 - 52.5
-0.30y = -15
Divide both sides by -0.30 to solve for y:
y = (-15) / (-0.30)
y = 50 liters
Substitute this value of y back into equation 1 to solve for x:
x + 50 = 75
x = 75 - 50
x = 25 liters
So, the chemist should mix 25 liters of the 70% alcohol solution with 50 liters of the 40% alcohol solution to obtain 75 liters of a 50% alcohol solution.
To find out how many liters of each alcohol solution should be used, we can set up an equation based on the information given.
Let's assume that x liters of the 70% alcohol solution is used.
Since the total solution volume required is 75 liters, then (75 - x) liters of the 40% alcohol solution must be used.
Now, let's calculate the amount of alcohol in each solution.
In the 70% alcohol solution, there will be 0.70x liters of alcohol.
In the 40% alcohol solution, there will be 0.40(75 - x) = 30 - 0.40x liters of alcohol.
According to the problem, the final solution should have a total volume of 75 liters with 50% alcohol content.
So, the amount of alcohol in the final solution would be 0.50(75) = 37.5 liters.
To solve the equation, we can equate the total alcohol content in each solution with the final solution:
0.70x + 30 - 0.40x = 37.5
Combine like terms:
0.30x + 30 = 37.5
Subtract 30 from both sides:
0.30x = 7.5
Divide by 0.30:
x = 7.5 / 0.30 = 25
So, 25 liters of the 70% alcohol solution should be used, and (75 - 25) = 50 liters of the 40% alcohol solution should be used to obtain a 75 liter solution with 50% alcohol content.