Percentages are simple problems involving percentage calculations: "What is 75% of 40?" belongs in this mold. They have a very practical real-life application in bank lending and interest rates, occasionally make guest appearances in Accounting problems, and tend to show up on project status reports. As a result, I felt that a refresher on these would not be amiss.
Averages are similarly easy, although I wanted to approach them from a roundabout direction. Most problems in this area involve calculating the average of a set of values; I wanted to turn this on its head and ask a question where you needed to get a specific value in a series, given the average as one of your clues.
Multiple-variable problems, on the other hand, were certainties. I had given a friend some help on one of these for her MBA class before, so if there were any problems that were likely to show up, it would be these. You've probably seen them before; these are the tedious word problems where you have to figure out the values of more than one item.
Logical reasoning is harder to describe. This is not a subset of problems, mind you, as much as it is the ability to organize given information, identify an unknown value, and then use one to work towards the other. It's obviously used extensively in business, although in a less mathematical sense. But I'm convinced that it's the mathematical training that helps us apply it to non-math outlets.
Given this outline, all that I had left to do was to write the questions:
1. Three chickens can lay three eggs in three days. In how many days can you expect 18 chickens to lay 18 eggs?
This was a question of logical reasoning, plain and simple. I see it appear in a lot of IQ tests and other cognitive exams, and I've seen some very smart people give some very wrong answers as well. Most people, in fact, will trust in their faulty pattern-recognition senses and say "18 days!" right off the bat.
The answer is a lot more mundane. If three chickens lay three eggs, with all other factors being equal, then it stands to reason that each chicken laid one egg. If these three chickens laid those three eggs in a matter of three days, then it follows that each chicken needs three days to lay one egg. Therefore eighteen chickens would come up with eighteen eggs in those same three days.
2. You have exactly Php 35,000.00 in a bank account that gains 2% compound interest per annum. Assuming that you neither deposit nor withdraw any money from that account, how much will the account contain after two years?
I wanted to throw in at least question that implied a real-world application of a mathematical principle, and this was that question. The easy way to solve this is to just get ((35,000 x 102%) x 102%) for a total of Php 36,414.00. That said, I only realized afterwards that banks hardly use the term "compound interest" anymore, which puts up the wall between theory and practice again.
3. I need an average score of 93 among my exams in order to pass one of my courses. So far, the grades that I got in five earlier exams were 90, 97, 87, 100, and 86. What is the minimum grade that I should get on the sixth (and final) exam in order to pass?
This is a reverse-average problem that is commonly known as "the Student's Dillemma", and I'm sure that a lot of people out there learned to put these calculations together at some point in their academic lives. I won't cover the answer here, as it's really just a throwback to what we were all probably doing around our final exam weeks.
4. A bicyclist travels at a steady rate of 8 kilometers per hour. She leaves her house at 2:00pm and rides her bike to the supermarket. Halfway there, she realizes that she's forgotten her shopping list and returns home to get it, then sets out for the supermarket again. She arrives there at 4:30pm. What is the distance from her house to the supermarket?
I put this problem here for one basic purpose: It encourages the solver to draw a chart. I feel that visualization is an important part of logical reasoning — if you can envision the cyclist's journey in your mind, then so much the better, but if not, you can just doodle something that lets you conclude that she travelled a total of twice her original intended distance from 2:00 to 4:30. That means that she normally travels the path to the supermarket in 75 minutes at 8 km/h... which makes the distance 10 kilometers.
5. A 200-liter mixture is comprised of 20% water, 30% salt, 10% sugar, 15% sand, and 25% gold. This mixture is left out in the sun for a few hours, after which all the water is found to have evaporated. What percentage of the resulting mixture is made up of gold?
And now the problems get a whole lot harder. I set up this percentage problem to illustrate the fact that percentages are non-constant values, and that they change with the introduction or removal of new factors. The easiest way to get the answer here is to realize that you're just looking for an equivalent of 25 parts out of the remaining 80 units, which makes 31.25%.
One interesting quirk about this problem was the fact that you technically don't need the volume of the original mixture to solve it. In fact, you can give the original mixture and quantity you want and the answer will still be the same... but I wasn't about to introduce that to people who had spent years away from their high school math classes.
6. A motorboat needs three hours to travel upstream, but it only needs one hour to travel downstream. When there is no current, the motorboat moves at a constant four kilometers per hour. What is the rate at which the river's current flows?
This took things a little further; It's actually rather difficult to solve if you slept through most of your math classes. In fact, it's another problem that encourages you to draw... although a chart instead of a diagram is needed in this case.
Rate-Time-Distance problems like these usually need a bit of background. You need to know that Rate x Time = Distance, of course, but you also need to know that an opposing force will lower an object's effective rate of travel (and vice-versa). Ergo, the river will slow you down by its own rate when you go upstream, but it'll make you go faster by the same rate when you go downstream.
Assuming that the river's rate is R, we get:
(4 + R) x 1 = distance travelled downstream = distance travelled upstream = (4 - R) x 3
From there, it just boils down to:
(4 + R) = (4 - R) x 3
4 + R = 12 - 3R
4R = 8
R = 2 km/h
7. Three bowling balls and four frying pans weigh 54 pounds in total. Four bowling balls and one telephone weigh 54 pounds in total. Three telephones and eight frying pans also weigh 54 pounds in total. What is the total weight of one bowling ball, one frying pan, and one telephone?
This is the classic three-variable problem: Three unknown quantities, and three equations. I chose the objects completely at random only because I like choosing objects completely at random.
The interesting part is that I deliberately screwed around with the numbers here — while a bowling ball weighs 12 pounds and a telephone weighs 6 pounds, a frying pan weighs 4.5 pounds. I find that some solvers normally get thrown off by the decimal value for some reason, perhaps because it makes them think that they're on the wrong track.
8. Anthony, Beatrice and Charles win the lottery on a single ticket. They decide that they will each take 30% of the total, and then set aside the remaining 10% for future needs. After the money is deposited in their bank, however, each of the three friends arrives separately to claim their share. Anthony arrives first and withdraws 30% of the money. Beatrice arrives a few hours later, and withdraws 30% of what's left. Finally, Charles arrives some time later and withdraws 30% of what's left. At this point, only Php 205,800.00 is left in the account. How much did the three friends originally win in the lottery?
This is actually a problem that gets featured in a lot of puzzle books. While it's possible to solve this by means of basic algebra, the circumstances of the problem tend to leave people confused on where to start. Yes, this actually centers more on logical reasoning than percentages. Yes, I threw the two of them together to try and confuse my solvers further.
Problems like these encourage logical thinking — they force the solver to stop, think, and determine their battle plan before trying to tackle the problem. In this case, the way the logic should go is that that 205,800 represents 70% of the money that Charles saw (before he took his 30%). This amount represents 70% of the money that Beatrice originally found in the account, which is 70% of the money that Anthony found in the unblemished account. Therefore:
205,800 is 70% of the money that Charles saw...
— Charles originally saw 294,000 in the account.
294,400 is 70% of the money that Beatrice saw...
— Beatrice originally saw 420,000 in the account.
420,000 is 70% of the money that Anthony saw...
— Anthony originally saw 600,000 in the account.
And now I must admit that it all turned out to be for nothing, because we both ended up so busy the night before the exam that we never got around to the problems. I managed to fire off a quick question about chickens and eggs (which my friend got wrong), but we otherwise weren't able to go through the eight items above.
The next morning, my friend called to tell me that the exam was much easier than he expected, and that math only played a very small role in this regard. There were plenty of real-world logical scenarios and a few questions of general knowledge, but nothing beyond the simplified "What is 75% of 40?" percentage problems that I mentioned at the start of this article.
I suppose it's too much to ask. Math is more a tool for mental stimulation in schools, something that acts as a precursor to the logical thought that we use when we're older. You can't expect business professionals to maintain passion (much less practical use) for these theoretical concepts... especially when it comes to a post-graduate program that concentrates on cooperation and networking.
The test still sits on my desk, however, waiting for the next time that another MBA applicant asks for my assistance. On top of that, I have more word problems where that came from. It's only a matter of time before somebody asks me for another favor...