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Difficulty: Very hardRatios, Rates, and Proportions

An artist mixes blue, yellow, and white paint in a ratio of 5:6:35:6:3, respectively. To create a new shade, she adds 8 ounces8\text{ ounces} of blue paint and 8 ounces8\text{ ounces} of white paint to the mixture, but adds no yellow paint. If the ratio of yellow paint to white paint in the new mixture is 6:56:5, how many ounces of blue paint are in the new mixture?

  1. A
    20
  2. B
    24
  3. 28Answer
  4. D
    52
  5. E
    72

Answer

28
The correct answer is 2828 ounces. Let the initial quantities of blue, yellow, and white paint be 5x5x, 6x6x, and 3x3x, respectively. Adding 88 ounces of blue and 88 ounces of white paint gives new amounts of 5x+85x + 8 for blue and 3x+83x + 8 for white. The amount of yellow paint remains 6x6x. The new ratio of yellow to white paint is 6x3x+8=65\frac{6x}{3x + 8} = \frac{6}{5}. Cross-multiplying yields 30x=18x+4830x = 18x + 48, which simplifies to 12x=4812x = 48, giving x=4x = 4. Substituting x=4x = 4 into the expression for the new amount of blue paint gives 5(4)+8=285(4) + 8 = 28 ounces.

Step-by-Step Solution

1
Define variables for the initial quantities of paint using a common multiplier xx.
Blue paint = 5x5x, yellow paint = 6x6x, and white paint = 3x3x.
Since the ratio of blue to yellow to white paint is 5:6:35:6:3, the actual amounts are multiples of these ratio parts.
2
Express the new amounts of paint after adding the additional ounces.
New blue paint = 5x+85x + 8, new yellow paint = 6x6x, and new white paint = 3x+83x + 8.
The artist adds 8 ounces8\text{ ounces} of blue and 8 ounces8\text{ ounces} of white paint, with no change to the yellow paint.
3
Set up a proportion using the ratio of yellow paint to white paint in the new mixture.
6x3x+8=65\frac{6x}{3x + 8} = \frac{6}{5}
The problem states that the ratio of yellow paint to white paint in the new mixture is 6:56:5.
4
Solve the equation for the multiplier xx.
30x=6(3x+8)  ⟹  30x=18x+48  ⟹  12x=48  ⟹  x=430x = 6(3x + 8) \implies 30x = 18x + 48 \implies 12x = 48 \implies x = 4.
Cross-multiplying allows us to solve the linear equation for the scale factor.
5
Find the final amount of blue paint in the new mixture by substituting x=4x = 4.
5(4)+8=28 ounces5(4) + 8 = 28\text{ ounces}.
The question asks for the ounces of blue paint in the new mixture, which is represented by 5x+85x + 8.

Key Concept

Setting up and solving algebraic proportions from multi-part ratios when quantities are modified.
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Topics

Ratios, Rates, and Proportions

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