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

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

What answer should he come up with?

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 $95 (expensive) and makes up 84% of the cheap mixture. This makes sense because you need <em>more of the expensive ingredient</em> to reach the target price of$ 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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