Using Least Common Multiple to find the number of rectangle pieces of a given size to fit a square surface

Question

I would be grateful if someone could help me with this question. I have calculated the answer to be 24 but the answer in the book says 12.

DANCE FLOORS A dance floor is to be made from rectangular pieces of plywood that are 6 feet by 8 feet. What is the minimum number of pieces of plywood that are needed to make a square dance floor?

I calculated the Least Common Multiple for the two values 6 and 8 which came out to be 24.I presume this is the number of pieces of plywood required to cover the area.

Prime factors for 6 : 2 and 3. Prime factors for 8 : 2, 2, 2.

The Prime factors selected on basis of greatest number of times of appearance : 2, 2, 2, 3 when multiplied it comes to 24.

Even using a factor tree to calculate the LCM gives 24.

I do not know where I am making the error. Thanks in advance.

Answer 1

The question asks us to use rectangle shaped pieces of plywood to make a square. The two sides are 6 feet and 8 feet. The aim is to have enough pieces to make all sides equal. LCM(6, 8) is 24. At this length both the sides will become equal (but use different number of rectangles as shown in the picture below). The longer side uses 3 pieces and the shorter side uses 4 pieces to come to the same length. The LCM is 24. So Area of the square will be 24 X 24 = 576. If 'x' is the number of pieces needed to reach this square of 576.

      Then (6 * 8) * x = Total Area of the square 
            48 * x = 576.
                 x = 576 / 48
                   = 12 pieces.

This image shows 12 rectangles of 6 feet X 8 feet dimensions used to make a square

This is the explanation for the point that Mr Omielan has stated.

The value of 48 * x has to be a perfect square. Calculating the values of all combinations of 48 * x where x is a number starting from 1 to 12. 48 * 1 = 48 . square root of 48 does not yield a whole number hence it is not a perfect square. Similarly 48 * 2 = 96 whose square root again does not yield a whole number. But 48 * 3 gives 144 whose square root is 12. This makes 144 a possible choice for the area of the square. Raising x to 4, 5, 6, 7, 8, 9, 10, 11 will not give a number whose square root is a whole number. The next value which gives a whole number is 12.

So we now have two numbers, 3 and 12.

Taking the value 3 as the number of plywood pieces

 

We can take two pieces and get the length 12 as seen in the picture below.

This image shows that two pieces have been used to fill a possible square. There is still the area of 4 X 12 remaining if the other side has to be 12 also to make a square.

The remaining area of 4 X 12 can only be filled if we split a piece to 6 X 4 which will then make a perfect square as seen in the next image.

The final square with two full pieces and one split piece

Answer 2

Since these sheets have a common factor we should divide by that factor before computing LCM. $$(6,8)=2\quad \dfrac{(6,8)}{2}=(3,4) \quad (3,4)=12.\quad$$ It takes only 3 sheets in one direction and 4 in another to make a 24×24 square.