A high school marching band can be arranged in a rectangular formation with exactly three boys in each row and exactly five girls in each column. There are several sizes of marching band for which this is possible. What is the sum of all such possible sizes?
Let the number of rows and columns be r and c respectively. Although both r and c can be any quantity, we know that each row must contain exactly 3 boys and each column exactly 5 girls. Therefore, we get 3r+5c=rc. Rearranging, rc-3r-5c=0. Then
r(c-3)-5c=0
r(c-3)-5c+15=15
r(c-3)-5(c-3)=15
(r-5)(c-3)=15
So now we can substitute two integers that multiply to 15 to find (r-5) and (c-3).
r-5=1, r=6 and c-3=15, c=18
r-5=3, r=8 and c-3=5, c=8
r-5=5, r=10 and c-3=3, c=6
r-5=15, r=20 and c-3=1, c=4
With the values of r and c, multiply them to find the possible sizes and add to find the answer.
6 x 18 = 104
8 x 8 = 64
10 x 6 = 60
20 x 4 = 80
Hence, the sum of all possible sizes is 104+64+60+80=312
312
To find the sum of all possible sizes of the marching band, we need to determine the number of rows and columns that satisfy the given conditions.
Let's consider the number of rows first. Since each row has exactly three boys, the number of rows must be a multiple of 3. Let's use the variable "r" to represent the number of rows.
Similarly, since each column has exactly five girls, the number of columns must be a multiple of 5. Let's use the variable "c" to represent the number of columns.
From the given information, we can conclude:
- Number of boys in each row = 3
- Number of girls in each column = 5
- Number of rows = r
- Number of columns = c
Therefore, the total number of boys is 3 * r, and the total number of girls is 5 * c.
We know that the total number of students in the marching band should be the same regardless of how we calculate it. Therefore, we can write the equation:
Total number of boys = Total number of girls
3 * r = 5 * c
To determine the possible sizes, we need to find the values of "r" and "c" that satisfy this equation.
One way to find the solutions is to enumerate the possible values. Let's start with small values and incrementally increase them, checking if there exists a solution for each combination.
By observation, we can determine that if "r" is a multiple of 5 and "c" is a multiple of 3, there will be a solution to the equation.
Keeping this in mind, let's find all possible values for "r" and "c" that satisfy the equation.
For "r" = 5 and "c" = 3, the equation becomes:
3 * 5 = 5 * 3
15 = 15 (valid)
For "r" = 10 and "c" = 6, the equation becomes:
3 * 10 = 5 * 6
30 = 30 (valid)
For "r" = 15 and "c" = 9, the equation becomes:
3 * 15 = 5 * 9
45 = 45 (valid)
We can continue this pattern and find that for "r" = 20 and "c" = 12, and "r" = 25 and "c" = 15, the equation holds true as well.
Therefore, the possible sizes of the marching band are 15, 30, 45, 60, and 75.
Now, we can calculate the sum of these possible sizes:
15 + 30 + 45 + 60 + 75 = 225
Hence, the sum of all possible sizes of the marching band is 225.