I recently came across this problem in a job assessment test. I can't go back and answer the question now, but not knowing the answer is bothering me. Any thoughts on the solution and the means of solving would be excellent.
Mary has three different colored boxes, blue, red, and green. She places 7 of these blue boxes on her desk. With these 7, she fills some and leaves some empty. In the boxes that she fills, she places 5 red boxes. In these red boxes, she fills some and leaves some empty. In the boxes she fills, she places 5 green boxes. Mary now has 87 empty boxes on her desk. How many total boxes does Mary have on her desk?
In regards to the answer, it was multiple choice. Three of the answers were below 87, so I discounted those as the answer was asking for the total of all the boxes, not just the remainder of the boxes beside those which were empty. The two answer I was left with were 107 and 207.
I'm unsure of what type of problem this exactly is, so I was unable to solve it. I went ahead and guessed 107 as my answer, It didn't seem like enough boxes were being filled to add up another 100, but I could certainly be wrong. I'll facepalm I'm sure if an answer is derived, but right now I'm just boggled.
Look forward to any thoughts,
Rodge
Best Answer
Let $x$ denote the number of blue boxes that get filled and $y$ the number of red boxes that get filled. Then there are $5x$ red boxes and $5y$ green boxes. Hence,
The total number of boxes is $$7+5x+5y=7+5\cdot(x+y)$$ and the number of empty boxes is $$\underbrace{7-x}_{\text{empty blue boxes}}+\underbrace{5x-y}_{\text{empty red boxes}}+\underbrace{5y}_{\text{empty green boxes}}=7+4\cdot(x+y)=87.$$
Hence, $x+y=20$ and thus there will be a total of $7+5\cdot 20=107$ boxes. So your guess was correct - congratulations!