The table below summarizes the results of an agricultural study comparing the germination rates of two seed varieties,…

To find x in terms of k, first determine the overall germination rate for Variety P. For Variety P, the total number of seeds planted is 400 + 600 = 1,000, and the total number of seeds that germinated is 320 + 420 = 740. Thus, the overall germination rate for Variety P is 740 / 1,000 = 0.74 (or 74%).

Because the overall germination rate for Variety Q equals that of Variety P, and the total number of Variety Q seeds planted is 300 + 700 = 1,000, the total number of Variety Q seeds that germinated must be 0.74 · 1,000 = 740.

Therefore, x + y = 740.

Next, express the germination rates for Variety Q under each treatment:

- Treatment 1 germination rate = x / 300

- Treatment 2 germination rate = y / 700

The problem states that the germination rate under Treatment 1 is k times the rate under Treatment 2:

x / 300 = k · (y / 700)

Solving this equation for y gives:

y = (700x) / (300k) = (7x) / (3k)

Substitute this expression for y into the total germinated equation x + y = 740:

x + (7x) / (3k) = 740

x · (1 + 7 / (3k)) = 740

x · ((3k + 7) / (3k)) = 740

x = (740 · 3k) / (3k + 7) = (2,220k) / (3k + 7)

Choice A is incorrect; it results from comparing raw counts of germinated seeds rather than germination rates (setting x = ky instead of x / 300 = k · (y / 700)).

Choice C is incorrect; it results from calculating the overall germination rate of Variety P as the simple, unweighted average of its two treatment rates ((320/400 + 420/600) / 2 = 0.75), yielding x + y = 750 instead of 740.

Choice D is incorrect; it results from inadvertently swapping the sample sizes for Treatment 1 and Treatment 2 when writing the rate relationship (setting x / 700 = k · (y / 300)).