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A Nonlinearity Brainteaser

From Cold Spring Shops:

Here's a pop quiz the Berkeley Energy Institute gave to readers.

Which of the following two choices leads to more fuel savings:

(A) buying a 15 mpg Cadillac Escalade instead of a 12 mpg Chevrolet Suburban, or

(B) buying a 50 mpg Toyota Prius instead of a 29 mpg Toyota Corolla?

Read it carefully: it's decision making at the margin. Ceteris paribus conditions are provided for the readers. "Assume you would drive the same distances and speeds no matter which car you have." Thus the total fuel use will be the least with the Prius and the greatest with the Suburban.

Click through to the post "Rectangular Hyperbolas" if you want to see the answer.

I care about this because the misinterpretation of nonlinearity in this example is very close to the misinterpretation of nonlinearity in casual analysis of macroeconomic policy and events. For example, it's really easy for dim bulbs in the legacy media to claim the the Great Recession was comparable to the Great Depression if they don't log the real GDP data first.

Usefulness of Basic Math

In the kitchen this morning …

Mix # 1 for waffles required 3/2 cups of mix, 3/4 cup of water, 3 tablespoons of water, and 1 egg.

But, I have only 3/4 cups of the mix, so the egg is going to be a problem.

Mix # 2 from a second box required 2 cups of mix, 3/4 cup of water, and 1/3 cup of oil, but no eggs.

I don’t want to try and come up with half an egg, and it’s clear from the first recipe that it’s substituting an egg for some oil. But how much?

A quick trip to the internet shows that 1/4 cup of oil can be substituted for 1 egg.

So, I measure out a half batch of Mix # 1: 3/4 cup of mix, 3/8 cup of water, 3/2 tablespoons of oil, and 1 egg.

Knowing there’s 1/2 egg extra in there, I substitute the extra 1/2 egg for 1/8 cup of oil in the second recipe. I’m making a half-batch of that too, so I use: 1 cup of mix, 3/8 cup of water, and 1/6 cup of oil minus the 1/8 cup of oil I’m getting from the egg.

To simplify, you note that the denominators 6 and 8 are common factors of 24. Then 1/6 cup is the same as 4/24 cup, and 1/8 cup is the same as 3/24 cup. So what I needed was 4/24 minus 3/24 equals 1/24 cup of oil.

A cup has 8 ounces. There’s two tablespoons in an ounce, so that’s 16 tablespoons in a cup. There’s 3 teaspoons in a tablespoon, so that’s 48 teaspoons in a cup. And 1/24 of a cup is then 48/24, or exactly 2 teaspoons. That’s how much extra oil I needed to add.*

I added that oil, whisked both half-batches together, started pouring onto the new waffle iron I got for Christmas, and the kids were happy with the waffles.

Tim Worstall is probably going to comment back that he just would’ve gone out for breakfast, but he’s not as geeky as me.

* The beauty of the customary system of weights and measures is that it is based on multiples of 2’s and 3’s, making the sort of arithmetic I used much more straightforward for someone with a basic grasp of fractions than using a calculator.

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Wall Street Interview Question

It is not widely known that in interviews for finance positions on Wall Street that it is typical to ask "brainteasers".

The goal is to see how you react to an unusual technical situation when under pressure.

Here is one that was directed last month at an SUU alumnus I mentor:

Suppose you had one million cans lined up with a 7 digit integer number under each can. What is a good method to find if the can numbered 0047000 is in the lineup? The only information you were told was that the numbers 1) start at 0000001, 2) the cans are ordered from smallest to largest number, but 3) numbers may be skipped. (Hint: a 7 digit number can go much higher than one million.)

(I cleaned this up a bit so that I could use it as an exam question for a senior level class). Post answers to the comments.

FWIW: He got the offer.

I'll be posting more of these in the future, but you can also go and buy this book for a collection of somewhat older questions: