Question

A painter mixed red, blue and white paint to form a mixture of purple paint. 1/4 of the purple paint was made up of red paint. The ratio of the volume of blue paint to the volume of red paint was 2:3.

a) What fraction of the purple paint mixture was from the white paint?

b) He then used 3/8 of the purple paint to paint the wall and 2/3 of the remaining purple paint to paint the ceiling. He had 300 ml of purple paint left. What was the volume of the purple paint mixture at first?

Answer 1

a) Let's assume that the volume of red paint used is R, the volume of blue paint used is B, and the volume of white paint used is W.

Given that 1/4 of the purple paint mixture was made up of red paint, we can write an equation: R = 1/4(R + B + W) [Since the total volume of the purple paint mixture is R + B + W]

Simplifying this equation, we get: 4R = R + B + W

Since the ratio of the volume of blue paint to the volume of red paint is 2:3, we can write another equation: B/R = 2/3

Simplifying this equation, we get: 3B = 2R

We can now solve these two equations to find the values of R, B, and W.

From the second equation, we get: B = (2/3)R

Substituting this value of B in the first equation, we have: 4R = R + (2/3)R + W

Simplifying further, we get: 4R = (5/3)R + W

Multiplying through by 3, we have: 12R = 5R + 3W

Rearranging terms, we get: 7R = 3W

Therefore, we can conclude that 3W is divisible by 7. The only multiple of 3 that is divisible by 7 is 21. Therefore, W = 21/3 = 7.

So, 7R = 21, and R = 21/7 = 3.

Therefore, the fraction of the purple paint mixture that was from the white paint is 7/(R + B + W) = 7/(3 + (2/3) * 3 + 7) = 7/19.

b) Let's assume the volume of the purple paint mixture at first is P.

Given that the painter used 3/8 of the purple paint to paint the wall, the remaining volume of purple paint is (1 - 3/8) = 5/8 times the original volume.

So, the volume of purple paint remaining after painting the wall is (5/8)P. The painter then used 2/3 of this remaining purple paint to paint the ceiling.

So, the volume of purple paint used to paint the ceiling is (2/3)(5/8)P = (10/24)P = (5/12)P.

We are given that the volume of purple paint remaining is 300 ml. Therefore, we can write an equation: (5/12)P = 300

Multiplying through by 12/5, we get: P = 12/5 * 300 = 720

Therefore, the volume of the purple paint mixture at first was 720 ml.