I don't get this problem, at all.

Assume that two pigments, red and blue, mix to give the normal purple color of petunia petal. Separate biochemical pathways synthesize the two pigments, as shown in the top two rows of the accompanying diagram. "White" refers to compounds that are not pigments. (Total lack of pigment results in a white petal.) Red pigment forms from a yellow intermediate that is normally at a concentration too low to color petals.
Pathway I ... → white1 →E blue
Pathway II ... → white2→A yellow→B red
C↑
Pathway III ... →white3 →D white4
( the letters should be on top of the arrows, and the numbers are subscripts)
A third pathway, whose compounds do not contribute pigment to petals, normally does not affect the blue and red pathways, but, if one of its intermediates(white) should build up in concentration, it can be converted into the yellow intermediate of the red pathway. In the diagram, the letters A through E represent enzymes; their corresponding genes, all of which are unlinked, may be symbolized by the same letters. Assume that wild-type alleles are dominant and encode enzyme function and that recessive alleles result in a lack of enzyme function. Deduce which combinations of true-breeding parental genotypes could be crossed to produce F2 progeny in the following ratios:
a. 9 purple: 3 green: 4 blue b. 9 purple: 3 red: 3 blue: 1 white c. 13 purple: 3 blue d. 9 purple: 3 red: 3 green: 1 yellow
(Note: No mutations, blue mixed with yellow makes green)

Please help!

To solve this problem, we need to understand the pathways involved in pigment synthesis and the genotypes of the parental organisms. Let's break it down step by step:

1. Start by analyzing the different pathways and their components. We have three pathways: Pathway I, Pathway II, and Pathway III. The pathways can be represented as follows:

Pathway I: ... → white1 → E (blue)
Pathway II: ... → white2 → A (yellow) → B (red)
Pathway III: ... → white3 → D → white4

2. Now, let's examine the phenotypes and genotypes associated with each pathway:
- Pathway I: Results in blue pigments. An active enzyme E is required, which is encoded by the gene E.
- Pathway II: Results in red pigments. It begins with a yellow intermediate and requires enzymes A and B, which are encoded by their respective genes A and B.
- Pathway III: Does not contribute to pigment synthesis. It includes enzymes D and white4, which are encoded by their respective genes D and white4.

3. Based on the provided ratios of F2 progeny, we need to identify the parental genotypes that could produce these ratios:

a. 9 purple: 3 green: 4 blue
This ratio suggests that two different pathways contribute to the purple color in the F2 progeny. From the given pathways, only Pathway I (blue) and Pathway III (white4) contribute to blue color. Therefore, the parental genotypes for this ratio must include at least one parent that is homozygous for the blue allele (EE) and another parent that is homozygous for the white allele (white4white4).

b. 9 purple: 3 red: 3 blue: 1 white
This ratio suggests that all three pathways contribute to the pigment synthesis. From the given pathways, Pathway I (blue), Pathway II (red), and Pathway III (white4) are involved. To produce this ratio, the parental genotypes should include at least one parent that is homozygous for the blue allele (EE), one parent that is homozygous for the red allele (AABB), and one parent that is homozygous for the white allele (white4white4).

c. 13 purple: 3 blue
This ratio suggests that only Pathway I (blue) is involved in pigment synthesis. To produce this ratio, the parental genotypes should include at least one parent that is homozygous for the blue allele (EE).

d. 9 purple: 3 red: 3 green: 1 yellow
This ratio suggests that Pathway I (blue), Pathway II (red), and Pathway III (white4) contribute to pigment synthesis. From the given pathways, we know that blue mixed with yellow results in green. Therefore, the parental genotypes should include at least one parent that is homozygous for the blue allele (EE), one parent that is homozygous for the red allele (AABB), one parent that is homozygous for the p white allele (white4white4), and one parent that is homozygous for the yellow allele (BB).

Remember, this is just an explanation of how to approach the problem. The specific genotypes of the parental organisms can only be deduced based on the given information and patterns of inheritance.