How many liters each of a 55% acid solution and a 80% acid solution must be used to produce 50 liters of a 60% acid solution? (Round to two decimal places if necessary.)
If there are x liters of the 55%, then the rest (50-x) is 80%.
SO, now just add up the acid in the parts -- it must equal the acis in the final mix:
.55x + .80(50-x) = .60*50
To solve this problem, let's set up a system of equations.
Let x represent the number of liters of the 55% acid solution.
Let y represent the number of liters of the 80% acid solution.
We know that the total volume of the acid solution is 50 liters, so our first equation is:
x + y = 50
We also know that the final solution should be 60% acid, so we can set up the following equation for the acid content:
0.55x + 0.80y = 0.60(50)
Let's solve this system of equations:
First, we can rewrite the first equation as x = 50 - y.
Substituting this expression for x in the second equation, we have:
0.55(50 - y) + 0.80y = 0.60(50)
Simplifying this equation:
27.5 - 0.55y + 0.80y = 30
Combining like terms:
0.25y = 2.5
Dividing both sides by 0.25:
y = 10
Now we can substitute this value of y back into the first equation to solve for x:
x + 10 = 50
Subtracting 10 from both sides:
x = 40
Therefore, we need 40 liters of the 55% acid solution and 10 liters of the 80% acid solution to produce 50 liters of a 60% acid solution.
To solve this problem, we can use the concept of a weighted average. The idea is that we can find the amount of acid in each solution and then find the average concentration.
Let's assume x liters of the 55% acid solution will be used. Since we need a total of 50 liters of the final solution, the amount of the 80% acid solution used will be (50 - x) liters.
Now, we can calculate the amount of acid in each solution:
Amount of acid in the 55% solution = 55% * x = 0.55x
Amount of acid in the 80% solution = 80% * (50 - x) = 0.8(50 - x) = 40 - 0.8x
To produce 50 liters of a 60% acid solution, the sum of the amounts of acid from each solution should be equal to 60% of the 50 liters:
0.55x + (40 - 0.8x) = 0.60 * 50
Now, we can solve the equation for x:
0.55x + 40 - 0.8x = 30
Combine like terms:
-0.25x + 40 = 30
Move 40 to the other side of the equation:
-0.25x = 30 - 40
-0.25x = -10
Divide both sides of the equation by -0.25:
x = -10 / -0.25
x = 40
Therefore, we need to use 40 liters of the 55% acid solution and (50 - 40) = 10 liters of the 80% acid solution to produce 50 liters of a 60% acid solution.