Purple Paint Is Made By Using Red Paint And Blue Paint In The Ratio 1:2. Write An Equation For { Y $}$ In Terms Of { X $}$ To Show The Relationship Between The Amount Of Red Paint ( { Y $}$) And The Amount Of Blue Paint
Introduction
In the world of art and color theory, understanding the relationship between different colors is crucial for creating harmonious and visually appealing compositions. In this article, we will explore the relationship between red and blue paint, specifically in the context of creating purple paint. We will derive an equation to show the relationship between the amount of red paint and the amount of blue paint required to produce purple paint.
The Ratio of Red to Blue Paint
According to the problem statement, purple paint is made by using red paint and blue paint in the ratio 1:2. This means that for every 1 part of red paint, 2 parts of blue paint are required to produce purple paint. Let's denote the amount of red paint as { y $}$ and the amount of blue paint as { x $}$.
Deriving the Equation
To derive the equation that shows the relationship between { y $}$ and { x $}$, we can use the ratio of red to blue paint. Since the ratio is 1:2, we can write the following equation:
{ \frac{y}{x} = \frac{1}{2} $}$
This equation states that the ratio of red paint to blue paint is equal to 1:2. However, we want to express { y $}$ in terms of { x $}$, so we can cross-multiply to get:
{ 2y = x $}$
Now, we can solve for { y $}$ by dividing both sides of the equation by 2:
{ y = \frac{x}{2} $}$
Interpretation of the Equation
The equation { y = \frac{x}{2} $}$ shows that the amount of red paint required to produce purple paint is half the amount of blue paint. This means that if we have 2 units of blue paint, we will need 1 unit of red paint to produce purple paint.
Example
Let's say we want to produce 12 units of purple paint. We can use the equation to determine the amount of red paint required. Since the ratio of red to blue paint is 1:2, we can set up the following equation:
{ \frac{y}{x} = \frac{1}{2} $}$
We know that the total amount of purple paint is 12 units, so we can set up the following equation:
{ x + y = 12 $}$
Substituting the equation { y = \frac{x}{2} $}$ into the previous equation, we get:
{ x + \frac{x}{2} = 12 $}$
Simplifying the equation, we get:
{ \frac{3x}{2} = 12 $}$
Multiplying both sides of the equation by 2/3, we get:
{ x = 8 $}$
Now that we know the amount of blue paint required, we can find the amount of red paint required by substituting the value of { x $}$ into the equation { y = \frac{x}{2} $}$:
{ y = \frac{8}{2} $}$
{ y = 4 $}$
Therefore, we will need 8 units of blue paint and 4 units of red paint to produce 12 units of purple paint.
Conclusion
Q: What is the ratio of red to blue paint required to produce purple paint?
A: The ratio of red to blue paint required to produce purple paint is 1:2. This means that for every 1 part of red paint, 2 parts of blue paint are required.
Q: How do I use the equation { y = \frac{x}{2} $}$ to determine the amount of red paint required?
A: To use the equation, simply substitute the value of { x $}$ (the amount of blue paint) into the equation. For example, if you have 8 units of blue paint, you can substitute { x = 8 $}$ into the equation to get:
{ y = \frac{8}{2} $}$
{ y = 4 $}$
Therefore, you will need 4 units of red paint to produce purple paint.
Q: What if I want to produce a different amount of purple paint? How do I adjust the equation?
A: If you want to produce a different amount of purple paint, you can adjust the equation by multiplying both sides of the equation by the desired amount of purple paint. For example, if you want to produce 12 units of purple paint, you can multiply both sides of the equation by 12:
{ 12y = 12 \times \frac{x}{2} $}$
{ 12y = 6x $}$
{ y = \frac{6x}{12} $}$
{ y = \frac{x}{2} $}$
This shows that the amount of red paint required is still half the amount of blue paint, but the total amount of purple paint is now 12 units.
Q: Can I use the equation to determine the amount of blue paint required?
A: Yes, you can use the equation to determine the amount of blue paint required. Simply substitute the value of { y $}$ (the amount of red paint) into the equation and solve for { x $}$. For example, if you have 4 units of red paint, you can substitute { y = 4 $}$ into the equation to get:
{ 4 = \frac{x}{2} $}$
{ 8 = x $}$
Therefore, you will need 8 units of blue paint to produce purple paint.
Q: What if I want to produce a different ratio of red to blue paint? How do I adjust the equation?
A: If you want to produce a different ratio of red to blue paint, you can adjust the equation by changing the ratio. For example, if you want to produce a ratio of 2:3, you can change the equation to:
{ \frac{y}{x} = \frac{2}{3} $}$
{ 3y = 2x $}$
{ y = \frac{2x}{3} $}$
This shows that the amount of red paint required is now 2/3 the amount of blue paint.
Q: Can I use the equation to determine the amount of purple paint produced?
A: Yes, you can use the equation to determine the amount of purple paint produced. Simply add the amount of red paint and the amount of blue paint to get the total amount of purple paint. For example, if you have 4 units of red paint and 8 units of blue paint, you can add them together to get:
{ 4 + 8 = 12 $}$
Therefore, you will produce 12 units of purple paint.
Conclusion
In this article, we answered some frequently asked questions about the relationship between red and blue paint. We provided examples and explanations to help you understand how to use the equation to determine the amount of red paint required, the amount of blue paint required, and the amount of purple paint produced.