A farmer has two rectangular fields. The larger field has twice the length and four times the width of the smaller field. If the smaller field has area K, then the area of the largest field is greater than the area of the smaller field by what amount?

• Length of the smaller field = L
• Width of the smaller field = W

According to the information given, the larger field has twice the length and four times the width of the smaller field. Therefore:

• Length of the larger field = 2L
• Width of the larger field = 4W

Now, we can calculate the areas of both fields:

Area of the smaller field = L * W = K (given)

Area of the larger field = (2L) * (4W) = 8LW

To find the difference in the areas of the larger and smaller fields, subtract the area of the smaller field from the area of the larger field:

Difference in areas = Area of the larger field - Area of the smaller field Difference in areas = 8LW - K

Now, we can express this difference in terms of the dimensions of the smaller field:

Difference in areas = 8(LW) - K

Since we are given that the area of the smaller field is K, we can substitute this value:

Difference in areas = 8(K) - K

Now, Difference in areas = 8K - K Difference in areas = 7K

So, the area of the largest field is greater than the area of the smaller field by 7K.