One table at a bake sale has 75 cookies.Another table has 60 cupcakes.Witch table allows for more rectangular arrangements when all the cookies and cupcakes are displayed?Explain
prime factors of 75 ... 5, 5, 3 ... 25 x 3, 15 x 5
prime factors of 60 ... 5, 3, 2, 2 ... 30 x 2, 20 x 3, 15 x 4, 12 x 5, 10 x 6
To determine which table allows for more rectangular arrangements, we need to compare the factors of the number of cookies and cupcakes on each table. The number with more factors will allow for more rectangular arrangements.
First, let's find the factors of 75 (number of cookies):
Factors of 75: 1, 3, 5, 15, 25, 75
Next, let's find the factors of 60 (number of cupcakes):
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
If we compare the number of factors of 75 and 60, we can see that 60 has more factors. This means that the table with 60 cupcakes allows for more rectangular arrangements.
To explain further, the factors of a number represent the different ways that number can be divided evenly. When arranging rectangular objects, the number of factors determines the different lengths and widths that can be used to create rectangles. The greater the number of factors, the more possible rectangular arrangements there can be.
In this case, 60 has more factors (12 in total) compared to 75 (6 in total), which means there are more possibilities for arranging cupcakes on a rectangular table compared to arranging cookies.