One hundred liters of a 57.84% acid solution is obtained by mixing a 75% solution with a 42% solution. How many liters of each must be used to obtain the desired mixture?
A) 40 liters of 75% acid and 60 liters of 42% acid
B) 53 liters of 75% acid and 47 liters of 42% acid
C) 58 liters of 75% acid and 42 liters of 42% acid
D) 43 liters of 75% acid and 57 liters of 42% acid
E) 48 liters of 75% acid and 52 liters of 42% acid
To solve this problem, we can set up a system of equations using the equation for the amount of acid in each solution.
Let x represent the number of liters of the 75% acid solution and y represent the number of liters of the 42% acid solution.
The equation for the amount of acid in the 75% solution would be: 0.75x (since the solution is 75% acid).
The equation for the amount of acid in the 42% solution would be: 0.42y (since the solution is 42% acid).
Since we want to obtain 100 liters of a 57.84% acid solution, the equation for the total amount of acid in the desired mixture would be: 0.5784(100) = 57.84 liters.
The equation for the total amount of acid in the mixture can be expressed as: 0.75x + 0.42y = 57.84.
Next, we can set up an equation for the total amount of solution (since we need to obtain 100 liters): x + y = 100.
We now have a system of equations:
0.75x + 0.42y = 57.84
x + y = 100
We can solve this system of equations using substitution or elimination.
Using substitution, we solve the second equation for x:
x = 100 - y.
Substituting this value of x into the first equation:
0.75(100 - y) + 0.42y = 57.84.
Simplifying:
75 - 0.75y + 0.42y = 57.84,
75 + 0.42y - 0.75y = 57.84,
-0.33y = -17.16,
y = -17.16 / -0.33,
y ≈ 52.
Substituting the value of y back into the equation for x:
x = 100 - 52,
x = 48.
The solution is therefore 48 liters of the 75% acid solution and 52 liters of the 42% acid solution.
Therefore, the answer is E) 48 liters of 75% acid and 52 liters of 42% acid.
To solve this problem, let's assume that x liters of the 75% acid solution and y liters of the 42% acid solution are mixed to obtain the desired mixture.
According to the problem, the total volume of the mixture is 100 liters, so we have the equation:
x + y = 100
Now let's consider the acid content in the mixture. The acid content in the 75% acid solution is 75% of x liters, and the acid content in the 42% acid solution is 42% of y liters. The total acid content in the mixture is 57.84%, so we have the equation:
0.75x + 0.42y = 0.5784 * 100
Simplifying the equation:
0.75x + 0.42y = 57.84
To solve this system of equations, we can use the method of substitution:
From the first equation, we have x = 100 - y.
Substituting x in the second equation, we get:
0.75(100 - y) + 0.42y = 57.84
Simplifying the equation:
75 - 0.75y + 0.42y = 57.84
Combining like terms:
-0.33y = 57.84 - 75
-0.33y = -17.16
Dividing by -0.33:
y = 52
Substituting y into the first equation, we can find x:
x + 52 = 100
x = 100 - 52
x = 48
So, the solution is 48 liters of 75% acid and 52 liters of 42% acid, which corresponds to option E. Therefore, the answer is option E) 48 liters of 75% acid and 52 liters of 42% acid.