Fifteen percent of the pigment in paint color [tex]$ A $[/tex] is black. Sixty percent of the pigment in paint color [tex]$ B $[/tex] is black. An unknown amount of paint color [tex]$ B $[/tex] is mixed with [tex]$ 40 \, \text{ml} $[/tex] of paint color [tex]$ A $[/tex], resulting in a paint that contains [tex]$ 25\% $[/tex] black pigment.

Which equation can be used to solve for [tex]$ x $[/tex], the total amount of paint in the mixture of the two colors?

A. [tex]$ 0.15(40) + 0.6x = 0.25(40 + x) $[/tex]

B. [tex]$ 0.15(40) + 0.6(x - 40) = 0.25(x) $[/tex]

C. [tex]$ 0.15(40) + 0.6x = 0.25(40 - x) $[/tex]

D. [tex]$ 0.15(40) + 0.6(x + 40) = 0.25(x) $[/tex]

To determine the equation that can be used to solve for [tex]$ x $[/tex], the amount of paint color [tex]$ B $[/tex] in the mixture, we need to analyze the problem step by step.

1. Understand the given information:
- Amount of paint color [tex]$ A $[/tex]: [tex]$ 40 $[/tex] ml.
- Percentage of black pigment in paint color [tex]$ A $[/tex]: [tex]$ 15\% $[/tex] (or [tex]$ 0.15 $[/tex]).
- Percentage of black pigment in paint color [tex]$ B $[/tex]: [tex]$ 60\% $[/tex] (or [tex]$ 0.6 $[/tex]).
- Percentage of black pigment in the final mixture: [tex]$ 25\% $[/tex] (or [tex]$ 0.25 $[/tex]).
- Let [tex]$ x $[/tex] be the unknown amount of paint color [tex]$ B $[/tex].

2. Express the total amount of black pigment contributed by each paint:
- From paint color [tex]$ A $[/tex]: [tex]$ 0.15 \times 40 $[/tex].
- From paint color [tex]$ B $[/tex]: [tex]$ 0.6 \times x $[/tex].

3. Express the total amount of black pigment in the final mixture:
- Total volume of the mixture: [tex]$ 40 + x $[/tex] ml.
- The pigment percentage in the final mixture is [tex]$ 25\% $[/tex], so the total amount of black pigment in the mixture is [tex]$ 0.25 \times (40 + x) $[/tex].

4. Set up the equation based on the total black pigment:
- The amount of black pigment contributed by both paint colors should equal the black pigment in the final mixture.
- This gives us the equation:
[tex]\[ 0.15(40) + 0.6x = 0.25(40 + x) \][/tex]

Given the options:

1. [tex]$ 0.15(40) + 0.6x = 0.25(40 + x) $[/tex]
2. [tex]$ 0.15(40) + 0.6(x - 40) = 0.25(x) $[/tex]
3. [tex]$ 0.15(40) + 0.6x = 0.25(40 - x) $[/tex]
4. [tex]$ 0.15(40) + 0.6(x + 40) = 0.25(x) $[/tex]

The correct and logical equation from the analysis is:

[tex]\[ 0.15(40) + 0.6x = 0.25(40 + x) \][/tex]