Algebra Question

Question

 Suppose a flat rectangular board is built by gluing together a number of square pieces of the same size.

1. If twenty squares are glued together to make a 4x5 rectangular board, how many of these squares are
2. completely surrounded by other squares?
3. If the dimensions of the board are a x b, then how many squares (in terms of a and b) are completely
4. surrounded by other squares?

Answer 1

First given the situation with 4x5 squares for the rectangle. If you draw this situation out as a grid, the only squares that are completely surrounded by other squares are those in between the first and last columns and between the top and bottom rows. The number of squares completely surrounded is 6.

Thinking about this situation... if we have 4 columns across and 5 rows down, we can get our value of 6 by taking the number of columns and subtracting 2 (for the first and last column) and taking the number of rows and subtracting 2 (for the top and bottom rows).

(4-2)x(5-2)

2x3=6

Now let's put this in terms of dimensions pxq

(p-2)(q-2) will give you the number of squares completely surrounded by other squares FINAL ANSWER

Answer 2

a. If you glue together square pieces creating a 4x5 rectangle, then the board contains 20 squares. On the 2 long sides there are 5 squares each and on the 2 short sides there are 2 squares each for a total of 14 squares making up the perimeter. That leaves 6 squares contained within the perimeter.

You may ask why the 2 short sides have 2 squares each instead of 4. The reason is thet the 2 end squares on each long set of squares makes up a corner and those corners belong to the short sides as well.

b. Now suppose you glue squares forming an axb rectangle. If a is the long side and b is the short side, then there are 2a + 2 (b-2) = 2(a+b) - 4 squares forming the perimeter. Since the rectangle has axb squares, the number of squares completely inside the perimeter are ab - 2(a+b) + 4.

Check:

In part a., a=5 and b=4. 2(a+b) - 4 = 2(5+4) - 4 = 18-4 = 14 squares make up the perimeter.

ab - 2(a+b) + 4 = 5*4 - 2(5+4) + 4 = 20 - 18 + 4 = 6 squares are contained within the perimeter.