Introduction:
In this problem, we are given two individuals, PpQq and PPQq, who are heterozygous for two different genes P and Q. We are asked to determine the fraction of their progeny that will show at least one homozygous recessive character. We are also given that the alleles assort independently.

Independent Assortment:
Independent assortment refers to the random distribution of alleles during gamete formation. It occurs when genes are located on different chromosomes or when they are far apart on the same chromosome. In our problem, since the genes P and Q are located on different chromosomes or are far apart on the same chromosome, we can assume that they assort independently.

Genotypes of the parents:
The genotype of the first parent is PpQq, which means it has one dominant allele (P) and one recessive allele (p) for gene P, and one dominant allele (Q) and one recessive allele (q) for gene Q. The genotype of the second parent is PPQq, which means it has two dominant alleles (P) and one recessive allele (q) for gene Q.

Calculating the possible genotypes of the progeny:
To determine the possible genotypes of the progeny, we need to consider the different combinations of alleles that can be inherited from the parents. Since the alleles assort independently, we can consider each gene separately.

For gene P:
- The first parent can pass on either the dominant allele (P) or the recessive allele (p).
- The second parent can only pass on the dominant allele (P).

For gene Q:
- The first parent can pass on either the dominant allele (Q) or the recessive allele (q).
- The second parent can pass on either the dominant allele (Q) or the recessive allele (q).

Determining the fraction of progeny with at least one homozygous recessive character:
To determine the fraction of progeny with at least one homozygous recessive character, we need to consider the different possible genotypes of the progeny.

Possible genotypes for gene P:
- PP (25% chance): This genotype can only occur if the first parent passes on the dominant allele (P) and the second parent also passes on the dominant allele (P).
- Pp (50% chance): This genotype can occur if either parent passes on the dominant allele (P) and the other parent passes on the recessive allele (p).
- pp (25% chance): This genotype can only occur if the first parent passes on the recessive allele (p) and the second parent also passes on the recessive allele (p).

Possible genotypes for gene Q:
- QQ (25% chance): This genotype can only occur if the first parent passes on the dominant allele (Q) and the second parent also passes on the dominant allele (Q).
- Qq (50% chance): This genotype can occur if either parent passes on the dominant allele (Q) and the other parent passes on the recessive allele (q).
- qq (25% chance): This genotype can only occur if the first parent passes on the recessive allele (q) and the second parent also passes on the recessive allele (q).

To calculate the fraction of progeny with at least one homozygous recessive