A large rectangle is divided into 4 non-overlapping smaller rectangles by two lines. One line is parallel to the base, and the other line is parallel to the height. If the areas of three of the small rectangles are 18,24, and 40, compute the only possible integer solution for the area of the fourth small rectangle. I'm not entirely sure where to start.
The original rectangle and the four smaller rectangles are related by the sides: side 1=a+b and side 2=c+d such that the areas of the small rectangles are:
ac, ad, bc, bd.
By factoring each area, and look for possible candidates for a,b,c and d to satisfy the above relations, we can find the sides of the fourth rectangle.
The first part is done for you below:
18=(2*3)*(3*3)
24=(2*2)*(2*3)
40=(2*2)*(2*5)
The fourth rectangle=?
18=(2*3)*(3*3)???
18=(2*3)*(3*3)???
This doesn't work
To solve this problem, let's start by labeling the four smaller rectangles as A, B, C, and D, with A representing the unknown rectangle. The areas of the small rectangles are given as 18, 24, and 40. Let's assign these areas to the corresponding rectangles.
Now, let's consider the relationship between the areas of these rectangles. Since the large rectangle is divided into four smaller non-overlapping rectangles, the total area of the large rectangle is the sum of the areas of the four smaller rectangles. Mathematically, we can express this relationship as:
Area of Large Rectangle = Area of Rectangle A + Area of Rectangle B + Area of Rectangle C + Area of Rectangle D
Since the area of Rectangle A is unknown, we can write this equation as:
Area of Large Rectangle = Unknown Area (A) + 18 + 24 + 40
Let's say that the area of Rectangle A is represented by 'x'. Now we can rewrite the equation as:
Area of Large Rectangle = x + 18 + 24 + 40
The area of the large rectangle is not given in the problem. However, we do know that the large rectangle is formed by dividing a larger rectangle into four smaller rectangles. Since both lines are parallel to the sides of the large rectangle, the two lines divide the large rectangle into three equal sections.
This means that the total area of the large rectangle is three times the area of any of the smaller rectangles. So, we have:
Area of Large Rectangle = 3 * Area of Rectangle A
Substituting the expression for the area of the large rectangle into our equation, we get:
3 * Area of Rectangle A = x + 18 + 24 + 40
Simplifying this equation, we have:
3x = x + 82
On solving this equation, we find that x = 41.
Therefore, the only possible integer solution for the area of the fourth small rectangle (Rectangle A) is 41.