a chemist wants to mix a 22 acid solution with a 46% acid solution to get 32 gallons of 37% acid solution. How many gallons of the 22% acid solution and how many gallons of the 46% acid solution should be mixed?
just add up all the acid:
.22x + .46(32-x) = .37*32
A chemist needs to mix up 80 gallons of a 60% acid solution. She only has 45% and 70% solutions on hand.
How many gallons of each should she mix together?
To solve this problem, we can use a mixture equation.
Let's assume x represents the number of gallons of the 22% acid solution that needs to be mixed, and y represents the number of gallons of the 46% acid solution.
Step 1: Set up the equation for the total amount of solution:
x + y = 32 (equation 1)
Step 2: Set up the equation for the total amount of acid in the mixture:
0.22x + 0.46y = 0.37(32) (equation 2)
Step 3: Simplify equation 2:
0.22x + 0.46y = 11.84
Now we have a system of two equations with two variables.
Step 4: Solve the system of equations using substitution or elimination method.
Using substitution method:
From equation 1, we have x = 32 - y
Substitute x in equation 2:
0.22(32 - y) + 0.46y = 11.84
Simplify and solve for y:
7.04 - 0.22y + 0.46y = 11.84
0.24y = 4.8
y = 20
Step 5: Calculate x using equation 1:
x = 32 - y
x = 32 - 20
x = 12
Therefore, the chemist should mix 12 gallons of the 22% acid solution and 20 gallons of the 46% acid solution to get 32 gallons of a 37% acid solution.
To determine the amount of each solution needed, we can use a basic equation to calculate the concentration of the resulting solution:
(amount of 22% acid solution * concentration of 22% acid solution) + (amount of 46% acid solution * concentration of 46% acid solution) = (total amount of resulting solution * desired concentration)
In this case, we have the following information:
Concentration of the 22% acid solution: 22%
Concentration of the 46% acid solution: 46%
Desired concentration of the resulting solution: 37%
Total amount of resulting solution: 32 gallons
Let's assume x represents the number of gallons of the 22% acid solution and y represents the number of gallons of the 46% acid solution.
We can now set up the equation:
(x * 22) + (y * 46) = 32 * 37
Simplifying it further:
22x + 46y = 1184
Now we need another equation to solve for x and y. The second equation is the total volume equation:
x + y = 32
We have two equations with two unknowns, so we can solve this system of equations using various methods, such as substitution or elimination.
Let's use substitution. Rearrange the second equation for x:
x = 32 - y
Substitute this value into the first equation:
22(32 - y) + 46y = 1184
Now, solve for y:
704 - 22y + 46y = 1184
24y = 480
y = 20
Now substitute this value back into the second equation to solve for x:
x + 20 = 32
x = 12
Therefore, to obtain 32 gallons of 37% acid solution, the chemist should mix 12 gallons of the 22% acid solution with 20 gallons of the 46% acid solution.