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?
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.
To solve this problem, let's assign some variables:
Let's say:
- Vr is the volume of red paint
- Vb is the volume of blue paint
- Vw is the volume of white paint
Now let's solve the problem step-by-step:
a) What fraction of the purple paint mixture was from the white paint?
1. We know that 1/4 of the purple paint was made up of red paint.
2. We also know that the ratio of the volume of blue paint to the volume of red paint was 2:3.
Based on this, we can set up the following equations:
Vr = (1/4) * (Vr + Vb + Vw) (Equation 1)
Vb/Vr = 2/3 (Equation 2)
We'll solve these equations to find the values of Vr and Vb, and then calculate the fraction of white paint.
3. Rearrange Equation 2 to get Vb = (2/3) * Vr.
4. Substitute this value of Vb in Equation 1:
Vr = (1/4) * (Vr + (2/3) * Vr + Vw)
5. Simplify the equation:
Vr = (1/4) * (5/3) * Vr + (1/4) * Vw
6. Multiply both sides by 12 to get rid of the fractions:
12Vr = 20/3 * Vr + 3Vw
7. Rearrange the equation:
9Vr - (20/3) * Vr = 3Vw
8. Simplify the equation further:
27Vr - 20Vr = 9Vw
9. Solve for Vw:
7Vr = 9Vw
10. Divide both sides by 9V:
Vw/Vr = 7/9
Therefore, the fraction of the purple paint mixture that was from the white paint is 7/9.
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?
1. Let's say the initial volume of the purple paint mixture was V. We know that he used 3/8 of V to paint the wall. So the remaining volume of the purple paint mixture is (1 - 3/8) * V = 5/8 * V.
2. He used 2/3 of the remaining volume (5/8 * V) to paint the ceiling. So the volume of purple paint left is (1 - 2/3) * (5/8 * V) = 1/3 * (5/8 * V) = 5/24 * V.
3. We are given that there were 300 ml of purple paint left, so 5/24 * V = 300.
4. Multiply both sides of the equation by (24/5):
V = 300 * (24/5) = 1440 ml.
Therefore, the volume of the purple paint mixture at first was 1440 ml.