A rectangular quilt has 96 squares of the same size. How many shapes are there in which the quilt can be arranged?
How many factors are there? 12.
1,2,3,4,6,8,12,16,24,32,48,96
So there are 6 pairs of factors to choose
Well, let's quilt our way through this problem! Since the quilt is rectangular, it implies that it has both a length and a width. Now, we need to find the factors of 96 and pair them up to find all the possible dimensions of the quilt. Breaking out my juggling balls, let's get started!
The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. Now, let's put on a little math-hat-juggling-show and pair them up: (1, 96), (2, 48), (3, 32), (4, 24), (6, 16), and (8, 12).
This means we've got six pairs, and for each pair, we can arrange the quilt's dimensions in two different ways, either flipping it horizontally or vertically. So, all you math-fun-loving folks, get ready for this!
Multiply the number of pairs (6) by 2, and voila! We end up with a grand total of 12 possible shapes in which the quilt can be arranged.
Now, that's a lot of quilting fun! Feel free to add some colorful costumes and juggling balls to your quilt-making experience. Just try to make sure the quilt doesn't end up as a clown costume!
To calculate the number of possible arrangements, we need to find all the possible factors of 96 and then consider the arrangements of these factors.
First, let's find the factors of 96:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Now, let's consider the possible arrangements using these factors:
For each factor, we can consider it as the length of one side of the rectangle, and the other side would be the total number of squares divided by this factor.
For example, if we take the factor 2, the length of one side of the rectangle would be 2, and the other side would be 96/2 = 48. So, one possible arrangement is a rectangle of size 2 by 48.
Similarly, we can calculate the other possible arrangements for each factor:
2 x 48 = 96 (rectangle)
3 x 32 = 96 (rectangle)
4 x 24 = 96 (rectangle)
6 x 16 = 96 (rectangle)
8 x 12 = 96 (rectangle)
12 x 8 = 96 (rectangle)
16 x 6 = 96 (rectangle)
24 x 4 = 96 (rectangle)
32 x 3 = 96 (rectangle)
48 x 2 = 96 (rectangle)
96 x 1 = 96 (rectangle)
So, there are 11 possible arrangements for the quilt.
To determine how many different arrangements there are for the quilt, we need to calculate the number of possible ways to arrange the squares.
Since the quilt is rectangular, it consists of rows and columns. The total number of squares (96) can be distributed among the rows and columns in various combinations.
Let's start by finding all the factors of 96. The factors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96.
Each factor represents a possible number of rows or columns for the quilt. For example, if we have 1 row and 96 columns, it would be one arrangement. Similarly, if we have 96 rows and 1 column, it would be another arrangement.
To find all possible arrangements, we pair up the factors. For instance, if we have 2 rows and 48 columns, it is one arrangement, and if we have 48 rows and 2 columns, it is another arrangement. We continue this pairing process until we have exhausted all the factors.
Let's go through the pairs of factors:
- (1, 96) - 1 row, 96 columns
- (2, 48) - 2 rows, 48 columns
- (3, 32) - 3 rows, 32 columns
- (4, 24) - 4 rows, 24 columns
- (6, 16) - 6 rows, 16 columns
- (8, 12) - 8 rows, 12 columns
For each pair, we consider both possible arrangements: rows × columns and columns × rows. So, we multiply the number of rows with the number of columns for each arrangement.
(1, 96): 1 row × 96 columns = 96 arrangements
(96, 1): 96 rows × 1 column = 96 arrangements
(2, 48): 2 rows × 48 columns = 96 arrangements
(48, 2): 48 rows × 2 columns = 96 arrangements
(3, 32): 3 rows × 32 columns = 96 arrangements
(32, 3): 32 rows × 3 columns = 96 arrangements
(4, 24): 4 rows × 24 columns = 96 arrangements
(24, 4): 24 rows × 4 columns = 96 arrangements
(6, 16): 6 rows × 16 columns = 96 arrangements
(16, 6): 16 rows × 6 columns = 96 arrangements
(8, 12): 8 rows × 12 columns = 96 arrangements
(12, 8): 12 rows × 8 columns = 96 arrangements
Summing up all the arrangements for each pair of factors, we find that there are 12 + 12 + 12 + 12 + 12 + 12 = 72 different arrangements for the quilt.
Therefore, there are 72 different shapes in which the rectangular quilt can be arranged.