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Posted on
21
Jun 2019

Number Theory Problem Form – Wedding Guest

Today we’re going to be looking at what at first seems to be an allocation problem. But on further reflection, actually turns out to be a much simpler number theory problem.

If we take a look at the problem stem these numbers 143 and 77. They stick out to us and they stick out not just because they don’t seem to have any relative association but also because they’re sort of odd-looking numbers, they don’t look like most the numbers were used to seeing. Say 48 or 24 or 36 something easily divisible clearly breakable into factors. Here were given these two disparate numbers and we’re being asked to formulate not what the tables are made up of but how many tables there are.

Solving the Problem

So we look at these two numbers and we examine first the 77 because it’s a simpler lower number. 77 breaks into factors of 7 and 11. This clues us in as to what to look for out of the 143. 143 must have a factor of 7 or 11. And in fact 143 is evenly divisible by 11 and it gives us 13.

Which means that the maximum number of tables is 11. Each one has 13 people from the bride’s side and 7 people from the groom’s side. 13 plus 7 there’s 20 people at each table. Times the 11 tables is 220.

And we can back check our math, 143 plus 77 is 220. We don’t need to go that far but that might help deliver some comfort to this method. So in reality this is a very creative clever way the GMAT is asking us for a greatest common factor.

Graphical Solution Path

Another way to think about this is that we need an equal number of groups from the bride and groom side. The number of people on the bride’s side doesn’t have to equal the number of people on the groom’s side. We just need them broke in into the same number of equal groups. Graphically, the illustration shows us how a certain number of different sized groups combined into this common table. So 13 and 7 and we have 11 groups of each.

Problem Forming

This is a great problem to problem form. It will give you some additional mental math or common result experience by forcing you to figure out numbers that you can present that at first don’t look like they match but in fact do have a common factor. You’ll notice that if they had given you 16 people and 36 people finding that common factor might be easier.

So as you problem form this try and do it in a way that sort of obscures the common factor. Either try it maybe where they have multiple common factors and you tweak things like what is the greatest number what is the fewest number of tables. Or even do a perspective shift and take a look at say how many guests are represented at each table. Or on the bride’s side or on the groom’s side. Of course there are conceptual shifts to this and you can make this story about anything. Once you control the story or rather the structure of the story this problem becomes very very straightforward.

Hope this helped, I look forward to seeing you guys soon.

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Posted on
12
Feb 2019

If you’re doing math on the GMAT, watch this.

I want to discuss one of the core tenents of Apex’s quantitative philosophy on the GMAT. “If you’re doing math, you’re doing something wrong.” Meaning, if you find yourself doing math, that’s a signal from the exam that you’re using a sub-optimal solution path. By math I don’t mean any calculation whatsoever, but any calculations that aren’t reasonable. That don’t come out easily, neatly and cleanly, once you’re well practiced with mental math. So it’s not that we’ll never do a calculation, but every calculation we do should be deliberate and smooth.

The Most Overused Solution Path

Let’s go a little deeper into this, because it’s a really important concept. Many, many people preparing for the GMAT spend way too much time worrying about the math. Being freaked out about the math and on the exam doing the math. The applied mathematical solution path is the most over used solution path on the quantitative side of the GMAT. Particularity among engineers, and with people who do a lot of self-prepping. They look to the back of the book or look to previous experience as students. And get caught up in the idea that their answer needs to be precise. This gets in the way of using our estimation solution path or other higher solution paths, which can get us to the correct answer much more quickly.

The GMAT isn’t Testing Your Math Skills

How do we know that math is not what the GMAT wants us to do? It’s quite simple. If the GMAT was the referendum on how well you can do mental math, then the scores would reflect your ability to do so. MBA programs at top business schools would be filled with people with extraordinary, almost savant like mental math abilities. We know this isn’t the case.

Actually, as we improve on our mental math, we get diminishing returns with it. So we see a lot of clients getting up to the 70th, 80th, or 90th percent level even, on the quantitative side of things. Then, all of a sudden they plateau; they can’t get any higher. The reason is they are so focused on the math. They are missing the bigger logical reasoning picture or the structure of quantitative problems that doesn’t rely on doing math that allows both quick and accurate solutions.

Key Things to Avoid

While math has its place, we want to be sure that we’re not putting it on a pedestal. And that when we’re performing computations, we’re doing so with great deliberation, intentionality, and that we have a good reason for doing any computation we’re doing. If you find yourself diving into the equation or doing a lot of processing, stop. Say “Wait a minute, there must be a better way to do this.”

Another option is that sometimes you make a basic error early on and that leads to ugly numbers and math. But you should never, never, never be multiplying decimals out to the fourth decimal. That sort of math is the true trigger, the true signal, that there’s a better way to solve the problem. When you’re self prepping, this is what you want to look for.

So by the time you get to the exam, you’re not catching yourself doing math, but you’ve already incorporated it into your process, the fact that math shouldn’t be your default.

So, remember, guys, if you’re doing math, you’re doing something wrong and you can take this one to the bank.

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Posted on
12
Feb 2019

Profit & Loss Problem Form

The profit and loss problem form that this problem fits into is one that has strong DSM’s into mathematics. Here we are tempted to do the math in part because that’s so easy. It’s so available to us.

This is characteristic of a mid-level arithmetic problem where there’s some shifts and shimmies but overall it’s a fairly straightforward problem that utilizes no more than the four basic operations. So on the one hand this profit and loss problem is pre-algebra or even sort of grade school math. On the other this makes the solution path much more elusive.

Solving the Problem Using Math

So of course we can follow the math. We can add up all the costs, five thousand plus two dollars, times twenty thousand. Then contrast that with the revenue that comes in which is 12×20,000. But then we’re left with the ugly division problem that brings us to the profit per t-shirt. This is where the GMAT sticks us.

Instead of handling this in aggregate it’s strongly preferable to handle it with a higher level solution path. Let’s take a look at a few:

Higher Level Solution Path: Distribution

One way to do this is to distribute the fixed cost over the cost per t-shirt. This is actually a lot easier than it seems. Twenty thousand t-shirts, five thousand dollars, five over twenty is one-quarter.

Therefore, it costs one-quarter per t-shirt in addition to the two dollars in variable cost. So twelve minus two is equal to ten dollars, minus one quarter is equal to nine dollars and seventy-five cents.

Higher Level Solution Path: Graphical Equalization

We can also use a graphic equalization method in order to get to the same conclusion. If the numbers were more complicated, understanding that that shift is one-quarter down. That is the fixed cost is one-quarter down.

Then we know we’re looking for something that ends in a seventy-five cents. That allows us to eliminate all the answer choices that don’t end in 0.75. Then we can use scale to determine that 9.75 is the correct answer.

Practice Problems

There are more complicated versions of this problem form. In particular, I’d encourage you to explore being told that the t-shirt company is breaking even. Then determining the amount of variable costs or fixed cost that’s there or even the production run. Similarly, you can be given a target profit or loss, the break-even just being the zero, so it’s a bit easier and have to reverse engineer the relationships.

Once again, this doesn’t have to be done algebraically. As you begin to appreciate the subtlety of the ratio between costs production run and total P&L all of these problems should be simplified and should be very straightforward.

Continue your GMAT practice with the Wedding Guest GMAT problem.

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