The problem was a simple one. A coworker wanted to use river rock to cover an area 6 feet by 20 feet to a depth of 3 inches, and wanted to know how much would be needed in cubic feet and cubic yards. The problem is straightforward one, and I took out my pocket calculator and did the following:
First I calculated the area in square feet: 6 x 20 = 120 square feet.
Next I converted it into square inches. Since a square foot is a square with 1 foot sides, one square foot is 12 x 12 = 144 square inches. So to convert square feet to square inches, just multiply by 144: 120 x 144 = 17,280 square inches.
Now that everything is in inches, all I had to do was to multiply by the height to get the cubic inches: 17,280 x 3 = 51,840 cubic inches.
That’s great, but we need cubic feet and cubic yards. A cubic foot is just a cube with all sides 12 inches, so a cubic foot = 12 x 12 x 12 = 1,728 cubic inches. So, to get the amount of needed rock in cubic foot, we divide 51,840 by 1,728: 51,840 ÷ 1,728 = 30 cubic feet. We get cubic yards the same way. A yard is three feet long, so a cube with sides of 1 yard = 3 x 3 x 3 = 27 cubic feet. So 30 ÷ 27 = 1 1/9 cubic yards. Call it 1.11 cubic yards and be done with it.
It turned out the coworker’s source for river rock sold it in bags of a half cubic feet, so cubic yards wasn’t needed at all. So it was: 30 ÷ ½ = 30 x 2 = 60 bags. Just as easy as it could be.
Or so I thought. It wasn’t until I was eating supper that I realized 3 inches is a quarter of a foot: 3 ÷ 12 = 3/12 = ¼. This meant each square foot covered to a depth of 3 inches is a quarter of a cubic foot. That meant that a 4 square foot area covered to a depth of 3 inches = 1 cubic feet. So, 6 x 20 = 120 square feet; then 120 ÷ 4 = 30 cubic feet. That’s simple enough that we can do the math in our head without using a calculator, or even pencil and paper.
It was then I felt very stupid. I should have realized that at the start. Instead, I took out my pocket calculator and muscled through the problem. I came up with the correct answer, but could have worked smarter, not harder, if I’d just thought about how inches relate to feet. Once we realize 3 inches is a quarter of a foot, we can exploit this for other depths. Let’s say we want 4 inches of river rock. 4 ÷ 12 = 4/12 = 1/3, so 3 square feet covered to a depth of 4 inches equals 1 cubic foot. That means 120 ÷ 3 = 40 cubic feet. If we bump it up to 6 inches, 6/12 = ½, so 120 ÷ 2 = 60 cubic feet. All very simple, and I completely missed it.
Unfortunately this isn’t the first time this has happened. I used to attribute this to using pocket calculators more than I had slide rules ( I was at the very end of the slide rule generation), and while this is partially true, it’s more because it’s easy to muscle your way through problems using a calculator, and we become accustomed to doing things this way.
A story has floated around for years about a light bulb company that supposedly gave new engineers the task of determining the volume of a light bulb. Very often, the new engineers would carefully measure the bulb and calculate the volume. More rarely, someone would simply fill the bulb with water and pour it out into a beaker. According to the story, that is what the company was really after: to teach their new engineers to think out a solution rather than plunging ahead by rote. Thinking ahead can sometimes result in simpler solutions.
If you notice a similarity between this and clumsy errors, you’re right. Both result when we act without thought.
I don’t know if it’s still done this way in school, but back in the day, we were taught to simply things as much as possible. That was an important skill when the most common calculating tools were a pencil and something to write on. It still is, even in this age of pocket calculators. But had wrote the calculation down, it might have leaped out at me that 3/12 = ¼. Might. I could have still muscled through it with a calculator, never once realizing there was a simpler way to solve the problem.
That’s the problem with habits: they’re so hard to break. But if we pause and think about a problem first, we might can do just that. Who knows? Maybe it won’t even require a calculator to solve it.