Two squares have a combined ratio of 68cm^2. The side of one square is four times as long as a side of the other square. Find the perimeter of each square.

Question

This has to be solved with special factoring.

Answer 1

You definitely entered the question incorrectly, since "a combined ratio" doesn't make sense, and a ratio wouldn't have units, so the question must be "a combined area".

 

Normally, you should let your variable represent the thing you are trying to find, which is perimeter in this case. However, in this problem, we will let our variable represent the length of each side of one of the squares. Since the side of the larger square is stated relative to the smaller one, we will let x represent the length of a side of the smaller square.

 

Now, there are two squares and we need to know the perimeter of both, but we will not use a second variable, because we know a relationship between the two squares. The larger square's sides are 4 times as long as the shorter square's. Since x represents the length of a side of the smaller square, then  4x  has to represent the length of a side of the larger square. Make sure you are clear on x and 4x before you go on.

 

Now, we are given the combined area of the two squares. That means there areas added together. We know that the area of a square is the length of its side squared.

 

The side of the smaller square is x, so its area must be   x²

 

The side of the larger square is  4x, so its are must be  (4x)²    (and the parentheses are very important)

 

    Now, (4x)² = 16x², so that is the area of the larger square.

 

The combined area, then, is   x² + 16x² = 17x²     OK so far?

 

We are told that the combined area is  68 square centimeters, so  17x² = 68

 

Divide both sides by  17  to get  x² = 4, so  x  has to equal 2  (Mathematically, it could be -2, but that wouldn't make sense in the word problem)

 

The shorter square has sides of length 2 cm, and the larger one has sides of length 8 cm. Are we finished? No, because we haven't answered the question. We were asked to find the perimeters. The perimeter is 4 times the length of a side for a square so the perimeters are   8 cm   and   32 cm.