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
.75 x + .42 y = .5784 (100)
x+y = 100
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.75 x + .42 y = 57.84
.75 x + .75 y = 75
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-.33 y = - 17.16
y = 52
x = 48
E
To solve this problem, we need to determine the volumes of the 75% and 42% acid solutions required to obtain 100 liters of a 57.84% acid solution. Let's denote the volume of the 75% acid solution as x liters and the volume of the 42% acid solution as y liters.
We know that the total volume of the mixture is 100 liters, so we can write the equation:
x + y = 100 (Equation 1)
We also know that the total amount of acid in the mixture can be calculated by multiplying the volume of the solution by the concentration of acid. For example, for the 75% acid solution, the amount of acid can be calculated as 0.75x (since 75% is equivalent to 0.75 when expressed as a decimal). Similarly, for the 42% acid solution, the amount of acid is 0.42y.
We want the resulting mixture to have a concentration of 57.84% acid. This means that the total amount of acid in the mixture should be 57.84% of the total volume. The total amount of acid in the mixture is given by the sum of the acid from the two solutions:
0.75x + 0.42y = 57.84 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find x and y.
To solve the system of equations, we can use the substitution or elimination method. However, in this case, let's use the substitution method.
From Equation 1, we can isolate x:
x = 100 - y
Now, substitute this expression for x into Equation 2:
0.75(100 - y) + 0.42y = 57.84
75 - 0.75y + 0.42y = 57.84
-0.33y = 57.84 - 75
-0.33y = -17.16
y = (-17.16) / (-0.33)
y ≈ 52
Now, substitute the value of y back into Equation 1 to solve for x:
x + 52 = 100
x = 100 - 52
x ≈ 48
Therefore, the solution is approximately 48 liters of 75% acid and 52 liters of 42% acid, which matches option E).