At the beginning of a program, the 105 members of a marching band stand in a rectangular formation named Formation A. All of the band members then move into Formation B, which is a different rectangular formation with six more rows, but with two fewer band members per row. How many rows are in Formation A?
OK, this is what I think:
Since 105 =3 x 5 x 7 =15 x 7.
So, you would have 15 rows of 7 kids per row in Formation A.
In Formation B, you would have =15 + 6 = 21 rows with 5 kids per row. So that:
15 x 7 = 21 x 5
+128460 Let the number of rows of the original formation = R
And let the number of columns = C
So R * C = 105 ⇒ C = 105/R (1)
Now...in the new formation.....the number of rows is ( R + 6) and the number of columns = (C - 2)
So we have that
(R + 6) (C - 2) = 105 .... expand...
R*C + 6C - 2R - 12 = 105
105 + 6C - 2R - 12 = 105 ...simplify...
6C - 2R = 12 sub (1) into this
6(105/R) - 2R = 12 multiply through by R
630 - 2R^2 = 12R rearrange
2R^2 + 12R - 630 = 0 divide through y 2
R^2 + 6R - 315 = 0 factor
(R + 21) ( R - 15) = 0
Set both factors to 0 and solve for R
R = -21 reject
R = 15 accept
So....there were 15 rows in Formation A
