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]
