Mathematic problem

Question

I'm stuck on a problem that maybe most of you will be able to solve instantly. Here is the problem:

A publisher needs to bind $4500$ books. One print shop can bind these books in $30$ days, another shop can do it in $45$ days. How many days are necessary to bind all the books if both shops work in parallel?

The problem's answer is $18$ days, but I got $13$ days. Here is how I calculated this: First, I found the books per day for each shop: Shop 1 = $\frac{4500}{30} $ = $150$ books/day Shop 2 = $\frac{4500}{45}$ = $100$ books/day

Let $x$ be the number of days: $$150x + 100x = 4500$$ $$x = 4500 / 350 = 12.8$$ or $13$ days

What am I missing here? Thanks

Answer 1

You have somehow gotten $350$ from adding $100$ and $150$. It should be $250$ only.

Answer 2

A common way of solving these types of problems is figuring out the fraction of books bound in a day. One shop completes ${1}/{30}$ of the total books, and the other shop completes ${1}/{45}$ of the total books in a day. Add these fractions together, and you get ${1}/{30} + {1}/{45} = {3}/{90} + {2}/{90} = {5}/{90} = {1}/{18}$. That means ${1}/{18}$ of the books are bound in a day, so it takes $18$ days to complete them all. This method can be used for other problems of this form.