Question

④ a. suppose the farmer buys another \\(\frac{1}{2}\\) square mile of land and divides all his land into square fields \\(\frac{1}{4}\\) mile long and \\(\frac{1}{4}\\) mile wide. how many fields will he have? ______ fields

Explanation:

Step1: Find the original area of the land

The original land is a rectangle with length \(1\frac{1}{2}\) miles (which is \(\frac{3}{2}\) miles) and width 1 mile. The area of a rectangle is length times width, so the original area \(A_{original}=\frac{3}{2}\times1 = \frac{3}{2}\) square miles.

Step2: Find the new total area of the land

The farmer buys another \(\frac{1}{2}\) square mile of land. So the new total area \(A_{total}=A_{original}+\frac{1}{2}=\frac{3}{2}+\frac{1}{2}=\frac{3 + 1}{2}=\frac{4}{2}=2\) square miles.

Step3: Find the area of each square field

Each square field has length \(\frac{1}{4}\) mile and width \(\frac{1}{4}\) mile. The area of a square is side times side, so the area of each field \(A_{field}=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\) square miles.

Step4: Find the number of fields

To find the number of fields, we divide the total area by the area of each field. So the number of fields \(n=\frac{A_{total}}{A_{field}}=\frac{2}{\frac{1}{16}}\). When dividing by a fraction, we multiply by its reciprocal, so \(n = 2\times16=32\).

Answer:

32