Question

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 1

There are 120 people in the marching band.

Explanation:

Let no. of people in the band be x.

In the first rectangular arrangement,

Width of rectangle (
`w_(1)`) = w people.

Length of rectangle (
`l_(1)`) = (w-7) people

Therefore,

no. of people in the band(x) = Area of the rectangle = Product of width and Length

x = width
`*` length

x =
`w* (w-7)` (equation 1);

In the second rectangular arrangement,

Width of rectangle (
`w_(2)`) = (w-3) people.

Length of rectangle (
`l_(2)`) = (w-5) people

Therefore,

no. of people in the band(x) = Area of the rectangle = Product of width and Length

x = width
`*` length

x =
`(w-3)*(w-5)` (equation 2);

Therefore, from equation 1 and equation 2 , we get,

`w*(w-7)=(w-3)*(w-5)`

`w^(2)-7w=w^(2) - 8w + 15`

Therefore,

w = 15 (equation 3)

From equation 1 and equation 3, we get

x =
`15*(15-7)`

x = 120.

Therefore,

no. of people in the marching band = 120 people