First, let's define the variables and set up our equation based on the given problem:
- Let [tex]\( x \)[/tex] represent the volume of Solution B (in milliliters) that we need to find out.
We know the following information:
- Solution A is 12% alcohol and Manuel uses 2000 milliliters of it.
- Solution B is 40% alcohol.
- The resulting mixture should be 20% alcohol.
To solve this, we need to set up an equation based on the total amount of alcohol in the mixture. Here’s the step-by-step approach:
1. Calculate the amount of alcohol in Solution A:
- Solution A has 12% alcohol, so the amount of alcohol in 2000 milliliters of Solution A is:
[tex]\[
0.12 \times 2000 = 240 \text{ milliliters}
\][/tex]
2. Set up the equation for the total alcohol content in the mixture:
- Let [tex]\( x \)[/tex] be the amount of Solution B in milliliters. Solution B is 40% alcohol, so the amount of alcohol in [tex]\( x \)[/tex] milliliters of Solution B is:
[tex]\[
0.40 \times x = 0.40x \text{ milliliters}
\][/tex]
3. Formulate the equation for the mixture:
- The total volume of the mixture is [tex]\( (2000 + x) \)[/tex] milliliters.
- The mixture should contain 20% alcohol. Therefore, the amount of alcohol in the mixture is:
[tex]\[
0.20 \times (2000 + x) \text{ milliliters}
\][/tex]
4. Combine these expressions into an equation to represent the total alcohol content:
[tex]\[
240 + 0.40x = 0.20 \times (2000 + x)
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
- Expand the right-hand side of the equation:
[tex]\[
0.20 \times 2000 + 0.20 \times x = 400 + 0.20x
\][/tex]
- The equation now looks like this:
[tex]\[
240 + 0.40x = 400 + 0.20x
\][/tex]
- Simplify by isolating [tex]\( x \)[/tex]:
[tex]\[
0.40x - 0.20x = 400 - 240
\][/tex]
[tex]\[
0.20x = 160
\][/tex]
- Divide both sides by 0.20 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{160}{0.20}
\][/tex]
[tex]\[
x = 800
\][/tex]
Therefore, Manuel needs to use 800 milliliters of Solution B to achieve the desired 20% alcohol concentration in the resulting mixture.
