Fifteen percent of the pigment in paint color A is black. Sixty percent of the pigment in paint color B is black. An unknown amount of paint color B is mixed with 40 ml of paint color A, resulting in a paint that contains $25 \%$ black pigment.
Which equation can be used to solve for $x$, the total amount of paint in the mixture of the two colors?
$0.15(40)+0.6 x=0.25(40+x)$
$0.15(40)+0.6(x-40)=0.25(x)$
$0.15(40)+0.6 x=0.25(40-x)$
$0.15(40)+0.6(x+40)=0.25(x)$
Answer (1)
Define x as the total amount of paint in the mixture.
Express the amount of paint B as x − 40 .
Calculate the amount of black pigment in paint A: 0.15 × 40 = 6 .
Calculate the amount of black pigment in paint B: 0.60 ( x − 40 ) .
Set up the equation: 0.15 ( 40 ) + 0.60 ( x − 40 ) = 0.25 x .
The equation to solve for x is: 0.15 ( 40 ) + 0.6 ( x − 40 ) = 0.25 ( x ) .
Explanation
Problem Analysis Let's analyze the problem. We have two paint colors, A and B, with different percentages of black pigment. We mix them to get a new paint with a specific percentage of black pigment. We need to find the equation that relates the total amount of the mixture to the given percentages and volumes.
Defining Variables Let x be the total amount of paint in the mixture (in ml). We know that 40 ml of paint A is used. Therefore, the amount of paint B used is x − 40 ml.
Pigment in Paint A The amount of black pigment in paint A is 15% of 40 ml, which is 0.15 × 40 = 6 ml.
Pigment in Paint B The amount of black pigment in paint B is 60% of ( x − 40 ) ml, which is 0.60 ( x − 40 ) ml.
Pigment in Mixture The total amount of black pigment in the mixture is 25% of the total volume x , which is 0.25 x ml.
Setting up the Equation Now, we can set up the equation. The total amount of black pigment in the mixture is the sum of the black pigment in paint A and paint B. So, we have: 0.15 ( 40 ) + 0.60 ( x − 40 ) = 0.25 x
Final Equation Comparing this equation with the given options, we see that it matches the second option: 0.15 ( 40 ) + 0.6 ( x − 40 ) = 0.25 ( x )
Examples
Imagine you're mixing different fruit juices to create a new blend. Each juice has a different concentration of sugar. This problem is similar to figuring out how much of each juice to mix so that the final blend has a specific sugar concentration. Understanding how to set up and solve these equations helps in various real-life scenarios, such as mixing chemicals, blending ingredients in cooking, or even managing investments with different interest rates. By mastering these concepts, you can make informed decisions to achieve the desired outcome in your mixtures.
