A lab technician has a 15 percent alcohol solution and a 35 percent alcohol solution. She wants to make 100 gallons of a 29 percent alcohol solution. How much of the 15 percent solution should she use?
Start by defining variables.
Let x = the amount of the 15% alcohol solution.
Let y = the amount of the 35% alcohol solution.
We want to find x.
Now define equations from the word problems.
x+y = 100 to have 100 gallons of solution.
0.15x + 0.35y = 29 to have a 29% solution.
Solve the system of equations. x=?
A lab technician has a 15% alcohol solution and a 35% alcohol solution.
She wants to make 100 gallons of a 29% alcohol solution. How many
gallons of each solution should she use?
To find out how much of the 15 percent solution the lab technician should use, we can set up an equation based on the following information:
Let x be the amount of the 15 percent alcohol solution in gallons.
The lab technician wants to make 100 gallons of a 29 percent alcohol solution, so the amount of the 35 percent solution would be (100 - x) gallons.
The equation can be set up using the alcohol content percentages:
0.15x + 0.35(100 - x) = 0.29(100)
Now, let's solve the equation step by step:
Step 1: Distribute the 0.35 to simplify the equation:
0.15x + 35 - 0.35x = 29
Step 2: Combine like terms:
(0.15 - 0.35)x + 35 = 29
Step 3: Simplify further:
-0.2x + 35 = 29
Step 4: Move the constant term to the other side of the equation:
-0.2x = 29 - 35
Step 5: Evaluate:
-0.2x = -6
Step 6: Divide both sides of the equation by -0.2:
x = -6 / -0.2
Step 7: Simplify:
x = 30
Therefore, the lab technician should use 30 gallons of the 15 percent alcohol solution to make 100 gallons of a 29 percent alcohol solution.
To determine the amount of the 15 percent alcohol solution the lab technician should use, we can set up a mathematical equation based on the alcohol content in the solutions and the desired alcohol percentage in the final solution.
Let's assume that x represents the amount of the 15 percent alcohol solution the lab technician should use (in gallons).
To find the amount of alcohol in the 15 percent solution, we multiply the alcohol percentage (15%) by the amount used (x) to get 0.15x gallons of alcohol.
Similarly, for the 35 percent alcohol solution, the amount of alcohol can be calculated as 0.35(100 - x) gallons. Here, (100 - x) represents the amount of the 35 percent alcohol solution being used (in gallons) since the total volume of the final solution is 100 gallons.
To find the amount of alcohol in the final solution, we multiply the desired alcohol percentage (29%) by the total volume of the final solution (100 gallons), which gives us 0.29(100) = 29 gallons of alcohol.
Since the amount of alcohol in the final solution should be equal to the sum of the alcohol in the two solutions used, we can write the equation:
0.15x + 0.35(100 - x) = 29
Now, we can solve the equation to find the value of x, which represents the amount of the 15 percent alcohol solution the lab technician should use.