Category Archives: Quant

 Math Payoffs

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“When am I every going to use this?”

 

It’s a familiar refrain from math students everywhere, so I’ll take this opportunity to show you exactly where you can use your math training. It’s a great example of how preparing for your test can also help you in the real world.

 

My wife was recently at the store to take advantage of two promotions. First, she had a 20% off coupon. Second, the store was offering a $10 gift card for all $50 purchases (with certain products excluded). Due to cashier error, the $10 gift card didn’t come up automatically when she checked out. That’s where the math comes in.

 

She needed to be able to calculate the total amount of her purchases before tax, subtract out the purchases of items that weren’t eligible and then subtract 20% of that total to make sure that it all came out to over $50.

 

Both the manager and the cashier would have preferred to rely on the machine. Their position was that if the magic box doesn’t spit out a code for a gift card, the customer probably shouldn’t get one. A customer who was less confident in her math would have relented and walked out without the $10, but my wife was perfectly comfortable walking through each step of the process to make sure she got what she deserved.

 

It’s only $10, and maybe that doesn’t seem like adequate payoff for the tedium of math classes and test prep, but that’s one example. Every time you decide whether to buy your flight, hotel and rental car separately or bundle them, that’s a math payoff. When you decide whether to buy the giant package of toilet paper at Costco, that’s a math payoff. And when you decide whether to invest money toward retirement or pay off student loans given the interest rates, that’s a math payoff.

 

Sitting there poring over numbers and remembering how to calculate percent discounts and the like may not be your idea of a good time, but it pays off in the end.

 In Someone Else’s Shoes

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Many of my friends have moved beyond the college and graduate level of standardized tests to the professional licensing tests of bar exams and medical boards. However, the same disdain that they showed for standardized tests the first time around has magically reappeared. It seems that you can’t escape from tests, but the same concepts that worked then work now.

 

As I discussed the upcoming boards with a doctor friend of mine he mentioned that he’s just trying to get through a lot of questions in order to learn the tricks. The fact that he called them tricks tipped me off to a fundamental problem in the way that he was approaching the test. Tests don’t have tricks, they have cleverly designed wrong answers.

 

In order to see the difference, you’ve got to reverse your perspective. Instead of viewing everything as the student, think about the question from the question-writer’s perspective. Let’s look at an example.

 

A stew mixture consists of beef, potatoes and carrots in the ratio of 5:4:3 respectively by weight. If 60 ounces of the mixture is prepared, the mixture includes how many more ounces of beef than carrots?

 

If I’ve written this question, my goal isn’t to trick anyone, but it is to separate those people who understand this ratio concept from those who do not. The correct answer to this question is 10, but I still need to come up with four incorrect answers to include. If I indiscriminately pick numbers, I run the risk of picking poorly and allowing someone who doesn’t understand the ratios concept I’m trying to test to get this question correct. While that might not sound like such a terrible thing, if everyone gets all the questions correct I haven’t done my job of separating the very best students from the merely average ones.

 

So how do I pick those incorrect answers? Well, I want to make sure that students are taking that last step of taking the difference between beef and carrots, so I’ll include answer choices that have the number of ounces of beef (25) and carrots (15) individually. I also want to make sure students understand that to find the amounts in our mixture they have to multiply that ratio by a constant. If they merely subtract they’ll get another one of our wrong answers (2). Finally, I want to make sure that students can isolate the correct two parts of a three-part ratio, so I’ll include what they would get if they subtracted beef minus potatoes (5).

 

As you can see right there, the point of that process is not to trick you, but to make sure that you understand the concept being tested. Rather than cursing the ugly tricks that the test-maker has come up with, try to understand the test from a different perspective and you’ll not only stress less, you’ll be able to spot the source of many of those wrong answer choices and be able to avoid them!

 Strategy Sunday: Overlapping Sets

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John Venn had an awesome beard.

His legacy, however, is a pictorial representation of overlapping sets that was named after him: the Venn diagram.

Look at that beard one more time.

Go ahead, I’ll wait.

Now if a guy has a beard that good and is remembered for something other than his beard, that other thing must be pretty special.

A Venn diagram is simply two overlapping circles, however, those overlapping circles create four separate regions. The red region represents the items that are part of Group 1, but not part of Group 2. The blue region represents the items that are part of Group 2, but not part of Group 1. The purple region represents the items that are members of both sets, and the white region represents the items that are members of neither set.

In equation form, this is written as follows:

Total = G1 + G2 – Both + Neither

However, in my experience, the graphical version of this analysis is much more helpful in allowing you the pick out the piece of information that is needed without any unnecessary confusion. The best strategy when you see overlapping sets is to begin by drawing a Venn diagram and filing in the known information. Consider the following example:

Ninety-two sports agents gathered for a conference on trends in free agency. Fifty-six of the agents had been in the industry more than ten years, but did not represent players in multiple sports. The number of agents who represented players in multiple sports was twice as many as the number of single sport agents who had been in the industry less than ten years. What is the maximum possible number of agents who represent players in multiple sports and have been in the industry more than ten years?

Trying to drop this paragraph of information neatly into a formula is asking for trouble, however, if your first reaction is to see two overlapping sets and draw a Venn diagram, you’re on the right track.

The next step is to add in the information that the question provided.

Make note of the fact that the 2X covering the multi-sport agents encompasses that entire circle, including the overlap. Using this graph you can now clearly see that the three regions we have values for cover all 92 conference attendees, so 56+2X+X=92. Solving this, we get that X=12.

To finish our problem, in order to get the greatest possible number of agents into that central overlapping region, we’d simply take all 24 of our multi-sport agents and put them in the middle, yielding our answer of 24.

When you face overlapping sets, remember to use a Venn diagram to represent your information because it’s an incredibly useful tool, and not just because he had an awesome beard.

 Answers Don’t Crawl Out of The Rubble

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If you haven’t seen it yet, take a moment to watch this video which shows one of the more feel-good stories to come out of the horrific damage caused by the tornadoes in Oklahoma.

This woman goes into the storm with a plan. Granted, it’s not the best plan, but it’s a plan nonetheless. Then the storm comes and throws everything around and all that’s left is a pile of rubble. And somehow, under a pile of debris, appears the dog that was feared to be lost. It’s a tremendous story, but it’s not a model for you to take in your test prep.

Some students have what I call “Mad Scientist Disease”. They imagine that upon seeing numbers and equations they should immediately start performing calculations, scribbling notes so quickly that smoke rises from the paper as the sweat pours down the intensely furrowed brow. After several moments of this process they imagine that an answer will reveal itself as correct. They create rubble and hope that the thing they hope for will crawl out from under it.

Math problems are not best solved by hoping for the miraculous. Unlike the tornadoes in Oklahoma, students who go down this road are creating the disaster rather than dealing with a force beyond their control. Rather than jumping directly into calculations you should begin at the end. Figure out what it is that you need to solve for in order to find your answer and then figure out what information you need in order to get that intermediate information. Once you’ve plotted out your steps, THEN it’s safe to begin the process of doing calculations.

Planning your approach before the danger comes is the best way to minimize the danger of a natural disaster, and the best way to survive your test unscathed.