Terrence, a chemist at a chemical supply company, has carefully measured out 400 litres of a solution containing 5% acetone. He also has access to unlimited quantities of a 20% acetone solution. How many litres of the 20% acetone solution must Terrence add to the 5% acetone solution to create a solution that contains 14% acetone?
.20x + .05(400) = .14(x+400)
To find out how many liters of the 20% acetone solution Terrence needs to add, we can set up an equation based on the amount of acetone in each solution.
Let x represent the amount of 20% acetone solution in liters that Terrence needs to add.
The amount of acetone in the 5% solution is 5% of 400 liters, which can be calculated as:
Amount of acetone in 5% solution = 5/100 * 400 = 20 liters.
The amount of acetone in the 20% solution that Terrence adds is 20% of x liters, which can be calculated as:
Amount of acetone in 20% solution = 20/100 * x = 0.2x.
When Terrence combines the two solutions, the total amount of acetone is 20 + 0.2x.
Terrence wants the final solution to contain 14% acetone. So, setting up the equation:
0.14(400 + x) = 20 + 0.2x.
Now, we can solve this equation step by step.
Step 1: Distribute 0.14 to both terms inside the parentheses.
56 + 0.14x = 20 + 0.2x.
Step 2: Move all terms involving x to one side of the equation by subtracting 0.14x and adding 0.2x to both sides.
56 - 20 = 0.2x - 0.14x.
36 = 0.06x.
Step 3: Divide both sides of the equation by 0.06 to isolate x.
x = 36 / 0.06.
x = 600.
Therefore, Terrence needs to add 600 liters of the 20% acetone solution to the 5% acetone solution to create a solution containing 14% acetone.
To solve this problem, we can use the concept of the concentration of acetone in the solution.
Let's break down the problem:
1. Terrence already has 400 liters of a 5% acetone solution.
2. He needs to add some amount of a 20% acetone solution to this 5% acetone solution.
3. The resulting solution should contain 14% acetone.
To find the amount of 20% acetone solution required, we need to consider the amount of acetone in both the 5% and 20% solutions.
Let's assume Terrence needs to add x liters of the 20% acetone solution.
The amount of acetone in the 5% solution is equal to (400 liters) x (5/100) = 20 liters.
The amount of acetone in the x liters of 20% acetone solution is equal to (x liters) x (20/100) = (20x/100) = (1/5)x liters.
In the resulting solution, the total amount of acetone is the sum of acetone from both the 5% solution and the x liters of 20% solution.
So, the total amount of acetone in the resulting solution is 20 liters + (1/5)x liters.
According to the given condition, this resulting solution should contain 14% acetone. We can express this as (14/100) times the total volume of the resulting solution.
Therefore, we can set up an equation using the amount of acetone in the resulting solution:
(20 liters + (1/5)x liters) = (14/100) times the total volume of the resulting solution.
Simplifying the equation:
20 + (1/5)x = (14/100) times the total volume of the resulting solution.
We know that the total volume of the resulting solution is 400 liters + x liters.
Substituting this in the equation:
20 + (1/5)x = (14/100) times (400 + x).
Now, we can solve this equation to find the value of x, which represents the number of liters of the 20% acetone solution that Terrence needs to add to the 5% acetone solution to create a solution with 14% acetone.
I hope this explanation helps you understand the problem and how to approach solving it!