Famer Gray owns two rectangular fields of the same area. One is 700 yards longer then it is wide; the second is 450 yards shorter than the first; and 400 yards

Question

Famer Gray owns two rectangular fields of the same area. One is 700 yards longer then it is wide; the second is 450 yards shorter than the first; and 400 yards wider. What are the dimensions of each field?

Answer 1

Hi Zachary, let "x" be the width of the first field. Then the length of the first field is x + 700 yards. The second field has a length of the first field, x + 700 yards minus 450 yards. The width of the second field 400 more than the first field, "x".

 

The two fields have the same length, therefore the equation becomes:

 

A(rea) of field 1 = A(rea) of field 2

 

x * (x + 700) = (x + 400) * (x + 700 - 450)

 

X² + 700x = x² + 650x + 10000 (the right side is just multiplication of binomial terms)

 

The x² terms cancel out leaving:

 

700x = 650x + 100000

 

Solve for x gives x = 2000 yards.

 

Therefore, the width of the first field is 2000 yards. The length of the first field is 2000 + 700 or 2700 yards. 

 

The width of the second field is 2000 + 400 or 2400 yards. The length of the second field is 2000 + 700 - 450 or 2250 yards.

 

To check, place the values in the original equation and solve

 

(2000) * ( 2700) = (2400) * (2250)

 

5400000 = 5400000, the values check.

 

Answer 2

The area of a rectangle is length times width.

 

For one rectangle:

 

Let width = x

Let length = x + 700

 

 

For the second triangle:

 

Let width = x + 400

Let length = (x + 700) - 450 = x + 250

 

 

Since the areas are the same, we can write the equation.

 

x(x + 700) = (x + 400)(x + 250)

 

 

Solve for x.

 

x² + 700x = x² + 650x + 100000

 

 

Subtract x² on both sides of equation.

 

700x = 650x + 100000

 

 

Subtract 650x on both sides of equation.

 

50x = 100000

 

 

Divide both sides of equation by 50.

 

x = 2000

 

 

Now just plug in this value of x into the variables to get your dimensions of each rectangle.