Vincent had three boxes, containing a total of 1100 styrofoam plates. The number of styrofoam plates in Box M to the total number of styrofoam plates was 3 : 11. He sold 462 styrofoam plates from Box N and sold ¹₃ of the styrofoam plates in Box P. The number of styrofoam plates left in Box N to the number of styrofoam plates left in Box P was 5 : 1. How many styrofoam plates were there in Box N at first?

	Box M	Box N	Box P	Total
Before	3 u	8 u (800)		11 u (1100)
		10 p + 462	3 p
Change		- 462	- 1 p
			2 p
After		5x2 = 10 p	1x2 = 2 p

11 u = 1100
1 u = 1100 ÷ 11 = 100

Number of styrofoam plates in Box N and Box P at first
= 8 u
= 8 x 100
= 800

The number of plates left in P is the repeated identity.
LCM of 1 and 2 = 2

10 p + 462 + 3 p = 800
10 p + 3 p = 800 - 462
13 p = 338

1 p = 338 ÷ 13 = 26

Number of plates in Box N at first
= 10 p + 462
= 10 x 26 + 462
= 260 + 462
= 722

Answer(s):722