A marching band is performing in a rectangular arrangement, with a width of $w$ people. The length consists of seven less people, or $w - 7$. In the second stage of their performance, the band re-arranged to form a rectangle with a width of $w-3$ people, and with a length of $w - 5$ people. How many people are in the marching band?

Answer:There are 120 people in the marching band.Step-by-step explanation:Let no. of people in the band be x.In the first rectangular arrangement,Width of rectangle ([tex]w_{1}[/tex]) = w people.Length of rectangle ([tex]l_{1}[/tex]) = (w-7) peopleTherefore, no. of people in the band(x) = Area of the rectangle = Product of width and Lengthx = width [tex]\times[/tex] lengthx = [tex]w\times (w-7)[/tex]  (equation 1);In the second rectangular arrangement,Width of rectangle ([tex]w_{2}[/tex]) = (w-3) people.Length of rectangle ([tex]l_{2}[/tex]) = (w-5) peopleTherefore, no. of people in the band(x) = Area of the rectangle = Product of width and Lengthx = width [tex]\times[/tex] lengthx = [tex](w-3)\times(w-5)[/tex]  (equation 2);Therefore, from equation 1 and equation 2 , we get,[tex]w\times(w-7)=(w-3)\times(w-5)[/tex][tex]w^{2}-7w=w^{2} - 8w + 15[/tex]Therefore,w = 15  (equation 3)From equation 1 and equation 3, we getx = [tex]15\times(15-7)[/tex]x = 120.Therefore,no. of people in the marching band = 120 people