Calculate the Paint Ratio: Solving for Percentages in 0.7x + 0.95(1-x) = 0.91

Q: How do I know which paint to call x?

It doesn't matter! You can let x = fraction of blue or x = fraction of red. Just be consistent throughout the problem and clearly state your choice.

Q: Why do we use (1-x) for the other paint?

Since the two paints must add up to 100% of the mixture, if one paint is x, then the other must be (1-x). This ensures the total is always 1 or 100%.

Q: How do I convert 0.84 to a percentage?

Multiply by 100: 0.84 × 100 = 84%. Remember that decimals represent parts of 1, while percentages represent parts of 100.

Weighted Averages with Percentage Calculations

An employee at a paint shop creates the color purple using the colors red and blue.

The red paint costs $70 per liter, while the blue paint costs $95 per liter.

What percentage of blue and red paint are used if the price of the purple paint is $91 per liter?

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Understand the problem

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Set up the equation using the weighted average formula
Step 2: Solve the equation for $x$
Step 3: Calculate the percentage of blue and red paint

Now, let's work through each step:

Step 1: The equation for the weighted average is:
$C_p = x \cdot C_b + (1-x) \cdot C_r$

Substituting the given values:
$91 = x \cdot 95 + (1-x) \cdot 70$

Step 2: Simplify and solve for $x$ :
$91 = 95x + 70 - 70x$
Combine terms:
$91 = 25x + 70$
Subtract 70 from both sides:
$21 = 25x$
Divide by 25:
$x = \frac{21}{25} = 0.84$

Step 3: $x = 0.84$ means 84% of the paint is blue:
The percentage of red paint is $1 - x = 0.16$ or 16%.

Therefore, the solution to the problem is blue: 84% red: 16%.

Final Answer

blue: 84% red: 16%

Key Points to Remember

Essential concepts to master this topic

Formula: Price equals weighted sum of component costs by proportions
Setup: Let x = blue fraction, then $91 = 95x + 70(1-x)$
Check: Substitute back: $95(0.84) + 70(0.16) = 79.8 + 11.2 = 91$ ✓

Common Mistakes

Avoid these frequent errors

Setting up the equation with wrong variable definitions
Don't randomly assign x to red or blue without being consistent = mixed up percentages! Students often switch which paint x represents mid-problem, leading to backwards answers. Always clearly define what x represents at the start and stick with it throughout the entire solution.

Practice Quiz

Test your knowledge with interactive questions

A hotel's overall rating is determined according to a weighted average of several categories. Each category is given a rating and a weighted factor. Below are the ratings for the "Happy Tourist" hotel:

Determine the hotel's overall rating?

FAQ

Everything you need to know about this question

How do I know which paint to call x?

It doesn't matter! You can let x = fraction of blue or x = fraction of red. Just be consistent throughout the problem and clearly state your choice.

Why do we use (1-x) for the other paint?

Since the two paints must add up to 100% of the mixture, if one paint is x, then the other must be (1-x). This ensures the total is always 1 or 100%.

What if I get the percentages backwards?

Check which paint costs more! Blue costs $95 (expensive) and makes up 84% of the cheap mixture. This makes sense because you need more of the expensive ingredient to reach the target price of $91.

Can I solve this without algebra?

You could guess and check, but algebra is much faster! The weighted average formula $C_p = x \cdot C_1 + (1-x) \cdot C_2$ works for any mixture problem.

How do I convert 0.84 to a percentage?

Multiply by 100: 0.84 × 100 = 84%. Remember that decimals represent parts of 1, while percentages represent parts of 100.

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