Chapter 5.1, Problem 45MS

Here we celebrate the power of algebra as a powerful way of finding unknown quantities by naming them, of expressing infinitely many relationships and connections clearly and succinctly, and of uncovering pattern and structure.

Tea time. Your grandmother has a very large collection of tea cups, sugar bowls, and mugs. Some have a one-holed handle, some have two holes, some have none. She wants you to organize them onto three shelves so that each shelf contains only vessels that are equivalent by distortion. She has ${\text{n}}^{\text{2}}$ cups with a one-holed handle, $\text{40-n}$ with two holes, and $\text{30-n}$ with no holes. If she has 150 items altogether, determine how many she has of each type.