The area of a rectangle is 2.6 square meters, and that of a square is two and twenty-five fourteenth square meters. Which figure has the largest area?

The area of a rectangle is 2.6 square meters, and that of a square is two and twenty-five fourteenth square meters. Which figure has the largest area?

The area of a square is two and twenty-five fourths of a square meter

The area of a rectangle is 2.6 square meters, while the area of a square is 14 / 25 square meters?

14 / 25 = 2.56 ＜ 2.6
So the rectangle has a large area

Draw a rectangle one meter wide from one side of a square. The area of the remaining rectangle is 15.75 square meters

Let the side length of the square be x, then the length of the remaining rectangle is x, and the width is X-1
X (x-1) = 15.75
The solution is: x = 4.5
So the area of the rectangle is 4.5 * 1 = 4.5 (square meters)

The length of the rectangle is increased by 3 meters and the width by 5 meters. The area of the rectangle is 65 square meters larger than that of the original rectangle

Let the side length of the square be a and the area of the rectangle be X,
So the rectangle is A-3 in length and a-5 in width
The area of rectangle is (A-3) (a-5)
It can be seen from the title
a*a=（a-3）(a-5)+65
a=10.
So the area of the rectangle is (A-3) (a-5)
=7×5
=35
Why add the increased area to the area of the shadow and divide it by the area of the shadow?
Your misunderstanding should be "the increased area is added to the area of the shadow part in the figure, and then divided by (the increased length by 3 meters + the increased width by 5 meters)"
It can be understood as (65 + 3 * 5) / (3 + 5) in this way:
The length of the rectangle with height of (3 + 5) and area of 65 + 3 * 5 is the edge length of the new square
Because the small rectangle in the lower right corner is transverse to the bottom of the rectangle in the upper left corner, to buy a rectangle with an area of 65 + 3 * 5 and a height of (3 + 5), there is a lack of a shadow area, so it needs to add another shadow area
Similarly, if the small rectangle in the upper left corner is vertical to the right side of the rectangle in the lower right corner, it is necessary to buy a rectangle with an area of 65 + 3 * 5 and a length of (3 + 5). At this time, it lacks the area of a shadow part, so it is necessary to add an area of a shadow part. In this way, the height of the rectangle, that is, the edge length of the new square, can be obtained
(here, the horizontal direction of the rectangle is called long, and the vertical direction is called high.)