How many liters each of 25% and 10% acid should be mixed to obtain 15 liters of 12% acid?
Kind of Solution
Liters of Solution
Amount of Pure Acid
0.25
x
0.1
y
0.12
15
Let's label the amount of 25% acid as x and the amount of 10% acid as y.
We want to find the values of x and y that satisfy the equation:
0.25x + 0.1y = 0.12(15)
This equation represents the total amount of pure acid in the mixture.
Let's solve for x:
0.25x + 0.1y = 1.8
Multiply both sides of the equation by 100 to get rid of the decimals:
25x + 10y = 180
Now let's solve for y in terms of x:
y = (180 - 25x)/10
We also know that the total amount of the mixture is 15 liters, so:
x + y = 15
Substituting the expression for y:
x + (180 - 25x)/10 = 15
Multiplying both sides of the equation by 10 to get rid of the fraction:
10x + 180 - 25x = 150
Combine like terms:
-15x = -30
Divide both sides of the equation by -15:
x = 2
Now let's substitute this value of x back into the equation for y:
y = (180 - 25(2))/10 = (180 - 50)/10 = 13
Therefore, we need 2 liters of 25% acid and 13 liters of 10% acid to obtain 15 liters of 12% acid.
To solve this problem, we can set up a system of equations based on the given information.
Let x represent the number of liters of 25% acid, and y represent the number of liters of 10% acid.
The amount of pure acid in the 25% solution is 0.25x liters, and the amount of pure acid in the 10% solution is 0.1y liters.
Since we want to obtain 15 liters of a 12% acid solution, the amount of pure acid in this solution is 0.12 * 15 = 1.8 liters.
So, we can now set up the following system of equations:
x + y = 15 (equation representing the total number of liters)
0.25x + 0.1y = 1.8 (equation representing the amount of pure acid)
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method.
From the first equation, we can rewrite it as x = 15 - y.
Substituting this into the second equation, we get:
0.25(15 - y) + 0.1y = 1.8
3.75 - 0.25y + 0.1y = 1.8
Combine like terms:
-0.15y + 3.75 = 1.8
Now, isolate y by subtracting 3.75 from both sides:
-0.15y = 1.8 - 3.75
-0.15y = -1.95
Divide both sides by -0.15:
y = -1.95 / -0.15
y = 13
Now substitute the value of y back into the first equation:
x + 13 = 15
x = 15 - 13
x = 2
Therefore, you need 2 liters of 25% acid and 13 liters of 10% acid to obtain 15 liters of 12% acid.
To solve this problem, we can set up a system of equations based on the amounts of pure acid in the two solutions.
Let's assume that we need x liters of the 25% acid solution and y liters of the 10% acid solution to obtain 15 liters of 12% acid solution.
Since the concentration of acid is defined as the ratio of the amount of pure acid to the total volume, we can write the first equation as:
0.25x + 0.10y = 0.12(15)
This equation represents the amount of pure acid in the 25% acid solution (0.25x) plus the amount of pure acid in the 10% acid solution (0.10y) equals the amount of pure acid in the 12% acid solution (0.12 multiplied by the total volume, which is 15 liters).
The second equation is based on the total volume of the mixture:
x + y = 15
This equation represents the fact that the sum of the volumes of the two solutions should equal the total volume of the mixture, which is 15 liters.
We now have a system of two equations:
0.25x + 0.10y = 0.12(15)
x + y = 15
To solve this system, we can use one of the methods such as substitution or elimination.
Let's use the elimination method to solve these equations:
Multiply the second equation by 0.10 to make the coefficients of y in both equations equal:
0.25x + 0.10y = 0.12(15)
0.10x + 0.10y = 0.10(15)
Now, subtract the second equation from the first equation to eliminate y:
0.25x + 0.10y - (0.10x + 0.10y) = 0.12(15) - 0.10(15)
0.25x - 0.10x = 1.8 - 1.5
0.15x = 0.3
Divide both sides of the equation by 0.15:
x = 0.3 / 0.15
x = 2
Now, substitute the value of x into the second equation to find y:
2 + y = 15
y = 15 - 2
y = 13
So, to obtain 15 liters of 12% acid solution, you would need to mix 2 liters of the 25% acid solution and 13 liters of the 10% acid solution.
I hope this explanation helps you understand how to solve this type of problem!