Question

a farmer has one square mile of land.

① if he divides his land into square fields that are \\(\frac{1}{2}\\) mile long and \\(\frac{1}{2}\\) mile wide, how many fields will he have?
\\(\underline{\quad 4 \quad}\\) fields

② if he divides his land into square fields that are \\(\frac{1}{3}\\) mile long and \\(\frac{1}{3}\\) mile wide, how many fields will he have?
\\(\underline{\quad \quad}\\) fields

③ if he divides his land into square fields that are \\(\frac{1}{4}\\) mile long and \\(\frac{1}{4}\\) mile wide, how many fields will he have?
\\(\underline{\quad \quad}\\) fields

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?
\\(\underline{\quad \quad}\\) fields

b. what is the total area of his land in square miles?
\\(\underline{\quad \quad}\\) square miles

Explanation:

Problem ②

Step1: Find area of each small field

The area of a square is side length squared. Each small field has side length $\frac{1}{3}$ mile, so its area is $(\frac{1}{3})^2=\frac{1}{9}$ square miles.

Step2: Find total number of fields

The total area of the land is 1 square mile. To find the number of fields, divide the total area by the area of one field: $1\div\frac{1}{9}=1\times9 = 9$.

Step1: Find area of each small field

Each small field has side length $\frac{1}{4}$ mile, so its area is $(\frac{1}{4})^2=\frac{1}{16}$ square miles.

Step2: Find total number of fields

Divide the total area (1 square mile) by the area of one field: $1\div\frac{1}{16}=1\times16 = 16$.

Step1: Find total area of land

The farmer originally has 1 square mile and buys another $\frac{1}{2}$ square mile, so total area is $1+\frac{1}{2}=\frac{3}{2}$ square miles.

Step2: Find area of each small field

Each small field has side length $\frac{1}{4}$ mile, so its area is $(\frac{1}{4})^2=\frac{1}{16}$ square miles.

Step3: Find total number of fields

Divide total area by area of one field: $\frac{3}{2}\div\frac{1}{16}=\frac{3}{2}\times16 = 24$.

Answer:

9

Problem ③