CRAYON POW
Problem Statement Rachel enters a coloring marathon because she is skilled enough to color with both hands at the same time. She gets two crayons, one fat one that wears down to nothing after 15 hours and a skinny one that is used up after six hours. If she starts the marathon at 10:30am with both crayons at the same height, at what time will the skinny crayon be half the length of the fat crayon?

Process When I first saw this problem, my first thought was “there has to be some kind of algebraic way to figure this out algebraically”. The thing is, I like figuring things out without too much math to clear things out. I guessed the answer should be around 3-ish hours as I read the question. The way I did it was cut the crayons into sections that disappear per hour. For the skinny crayon, the dimensions are 6 x 1 because one section disappears per hour and the whole crayon will be gone in six. Each hour, one of the six sections will be worn away. For the fat crayon, the dimensions were 2.5 x 6 units. The skinny crayon wears out after six hours and has one section each hour and the crayons are at the same starting length. The width is 2.5 because I divided 6 by 15 (hours until wear down). Each section is 0.4 units big because 2.5/6 = 0.4 so every hour, 0.4 units of the fat crayon will be gone. To figure out when the skinny crayon will be half the length of the fat, I set up a table based on the disappearing rates per hour.

Fat Skinny Time
6 6 0 hrs
5.6 5 1 hr
5.2 4 2 hrs
4.8 3 3 hrs
4.4 2 4 hrs

I started cutting the sections in half so now it was 0.2 per half hour for the fat and 0.5 for the skinny crayon.

Fat Skinny Time
4.8 3 3 hrs
4.6 2.5 3 hrs 30 min
4.4 2 4hrs

I cut the sections in half again so it was actually into fourths now. It was 0.1 per 15 minutes for the fat crayon and 0.25 for the skinny.

Fat Skinny Time
4.6 2.5 3hrs 30 min
4.5 2.25 3hrs 45 min

I got 2.25 as half of 4.5 when I checked so the final time is 3 hours and 45 minutes. The starting time was 10:30 am so add 3hrs and 45 min to that and got 2:15pm as the final time that the skinny will be half the height of the fat crayon. I figured the problem out this way because it