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The recombinant frequency is the proportion of recombination offspring produced in a cross. It represents the frequency of crossing over that occurs between the genes present on a chromosome.

The recombinant frequency can be used to determine the sequence of genes on the chromosome. The following formula is used to calculate the recombinant frequency of a cross:

\({\rm{Recombination frequency}}\,{\rm{ = }}\frac{{{\rm{Number of recombinants}}}}{{{\rm{Total number of offsprings}}}}\,{\rm{ \times 100}}\)

Four types of offspring were obtained from the cross: wild-type, black purple, gray purple, and black red. Wild-type (721) and black purple (751) are wild-type offspring, while gray purple (49) and black red (45) are recombinants obtained from the cross.

The total number of offspring produced in the cross is \(721\, + 751 + 49 + 45 = 1566\).

The total number of recombinants is \(49 + 45 = 94\).

The recombination frequency is calculated by substituting the values into the following equation:

\(\begin{aligned}{l}{\rm{Recombination frequency}}\,{\rm{ = }}\frac{{94}}{{1566}}\,{\rm{ \times 100}}\\{\rm{ = 6\% }}\end{aligned}\)

Thus, the recombination frequency between the genes for the body color and eye color is 6%.

In problem 3, the recombination frequency between the body color and wing size obtained was 17%. In this cross, the recombination frequency between the body color and eye color genes is 6%.

Thus, the recombinant frequency between the body color and wing size and between body color and eye color is known. Therefore, the recombinant frequency between eye color and wing size is required to determine the order of genes for the body color, eye color, and wing size.

In order to determine the recombinant frequency between eye color and wing size, a testcross needs to be performed using heterozygous for normal wings and red eyes with vestigial wings with purple eyes.

Let B be the allele for the gray body color and b for the black body color, W be the allele for the normal wings and w for the vestigial wings, and Ebe the allele for the red eyes and e for the purple eyes.

The genotype of heterozygous fruit fly with normal wings and red eyes would be WeEe, and the genotype of recessive fruit fly with vestigial wings and purple eyes would be wwee.

Thus, the cross between fruit fly with normal wings and red eyes (WeEe)and vestigial wings and purple eyes (wwee) needs to be performed to know the recombinant frequency between the genes for eye color and wing size.

As a result, recombinant frequencies of all the genes would help determine the sequences of the three genes on the chromosome.