Take the following quiz:
1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? ____cents
2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? _____minutes
3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? _____days
Each of these questions has an obvious answer that is wrong. However, if you take one minute to think about the questions--BEFORE YOU ANSWER--you will come to the correct, less obvious answer. Over the years, I have found that those who can answer these questions tend to do very well in my economics classes.
MORAL: before answering the question, spend a minute or two thinking about what the question is.
=================ANSWERS=================
Question 1: Though the quick intuitive answer is 10 cents, a moment's reflection leads to the realization that 10 cents is not a full dollar less than $1.00. (If you're sleepy: 10 cents is 90 cents less than $1.00.) The accurate solution can be reached through a little 8th grade algebra:
If the cost of the ball is x, the cost of the bat is x + 1.
x + (x+1) = 1.10
2x +1 = 1.10
2x = 0.10
x = 0.05, or 5 cents
Answer: the ball costs 5 cents and the bat costs $1.05, for a total of $1.10.
Question 2: Your brain screams at you that the answer must be 100, because your intuitive side sees the 5-5-5 pattern in the first example, and 100-100-100 just looks right. But if it takes 5 minutes for 5 machines to make 5 widgets, it doesn't take 20 times as long for 20 times as many machines to make 20 times the widgets. It will take the same 5 minutes for 100 machines to make 100 widgets, and it will take 5 minutes for 1000 machines to make 1000 widgets, and so on, because each machine spits out one widget every five minutes. That is the rate of widget production for the machines, and it doesn't change no matter how many machines you are running at once.
Answer: it would take 5 minutes for 100 machines to make 100 widgets.
Question 3: This trick here is that the lily pads grow at an exponential rate, not an arithmetic rate. On the day before the 48th day, the pond was only half-covered in lily pads; the day before that, one-quarter covered; the day before that, one-eighth covered; the day before that, one-sixteenth covered. Go back two weeks from the 48th day (day 34) and you will be hard-pressed to find any lily pads on the lake. It will be only 1/16,384 covered on that day. This means only .006% of the surface will be covered by a lily pad. Imagine how powerful a microscope you'd need to detect any lily paddage at all on day 1.
Answer: the pond will be half-covered in lily pads on the 47th day.
HT: Big Thinking
The first problem looks very easy and straight forward, but if you try to solve it in your mind you will get wrong answer: $1.10-$1=$.10. That is what I would get if I try to solve it without suing paper and pen. I always recommend my kids not to solve problems in their mind and write all the steps on the paper. This helps to have a visionary idea of problem solving, also, the teacher will see how problem was solved. In case if the answer is wrong, the student will get partial credit for showing the steps and detail.
ReplyDeleteFor the second problem, ratio won’t solve the problem. This is not as easy as the first problem. If the first sentence of the problem says that each machine needs 5 minutes to make a widget, then you can make any number of widgets by using the same number of machines in 5 minutes. How about if each of these five machines makes a part of a widget? What if this is a real-life problem? Does a company scarify its resources by using 100 machines to make 100 widgets in 5 minutes? Is it beneficial?
To solve the last problem, you have to look at it from different angle. It is easy to solve this problem back ward. The patch covered the whole lake in day 48. Yesterday, day 47, the size of the patch was half of today’s. Therefore, on day 47, the patch covered only half of the lake.
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ReplyDeleteThe seeming simplicity of the problems can easily lead a fast thinking mind to overlook the correct answer in favor of the obvious answer that tends to jump out at one when first reading problem. Had I not been forewarned in advance to pause and take a minute to think about the questions, I can see myself choosing the wrong answer for one or more of the problems as I skimmed over the problem in a rush to solve and move on.
ReplyDeleteThe first few years after founding and operating a chocolate products manufacturing business for me were full of errors made when working on projects. The most obvious mistakes that I made had to do with fixed, variable costs, economic profit and pricing decisions. Some decisions were made with disregard for opportunity costs and others without consideration for fixed costs. Although I have been able to sustain a steady rate of growth and expanded the company, looking back I can see a few hundred thousand dollars’ worth of mistakes that would have made a difference between at times taking a paycheck, or investing in marketing or equipment.
I have been good with the sunken cost fallacy, abandoning unprofitable projects despite investments in packaging, yet at the same time, there were a few instances when I fell victim to the Hidden Cost Fallacy. The most significant was prior to opening my first location. I had a choice of two locations
Location 1 rent $ 1800
Renovation - $ 80,000- $ 100,000
Retail Value - very low - quiet block, low income area
Distance to travel to work 1 hr
Lease term 5 years
Location 2 Rent $ 4,000
Renovation $ 50,000 – $ 70,000
Retail Value – high, busy street, high income area, shopping destination
Distance to travel to work 15 minute walk
Lease Term 5 years
When choosing between locations I was concerned predominantly with the cost difference in rent, but did not consider the economic costs or the opportunity costs of travel time which amounted to up to 12 hours a week and over $ 100 weeklys in fuel and wear and tear costs (Gas was near record highs at the time). My inexperience also led me to ignore structural issues with the cheaper space that became obstacle to installing an energy efficient infrastructure. As a result, my electric and water costs were several times more than what they could have been. I had also dismissed the opportunity cost of better retail value that I was giving up with the more expensive location. Although it is difficult to put an exact number of economic loss or unrealized economic potential resulting from my choice to rent the cheaper space, the excesses in electric and water bill coupled with the lower costs of renovation place the actual cost of the cheaper space within a few hundred dollars of the rent for the more expensive location. These few hundred dollars would have been easily covered by the better retail opportunities of the more expensive location.
The first problem looks very easy and straight forward, but if you try to solve it in your mind you will get wrong answer: $1.10-$1=$.10. That is what I would get if I try to solve it without using paper and pen. As it mentioned in the book, Even though this kind of mistake is so obvious preventing it from occurrence doesn’t seems easy. I always recommend my kids not to solve problems, especially, math problems in their mind; put everything on the paper and write all the steps. This helps to have a visionary idea of the problem solving; also, the teacher will see how problem was solved. In case if the answer is wrong, the student will get partial credit for showing the steps and detail. Cost of deciding to spend a little bit of time and formulate even the simple look problem is to get a correct answer and earn full credit.
ReplyDeleteFor the second problem, ratio won’t solve the problem. This is not as easy as the first problem. If the first sentence of the problem says that each machine needs 5 minutes to make a widget, then you can make any number of widgets by using the same number of machines in 5 minutes. Using logic and analysis are very helpful to find an answer that makes more sense. But, how about if each of these five machines makes a part of a widget? They work in a series (sequentially) not parallel. What if this is a real-life problem? Does a company scarify its resources by using 100 machines to make 100 widgets in 5 minutes? Is it beneficial? What is the cost of decision?
I think we should read last problem up-side-down. It will save a lot of time, and be less confusing. Starting the problem from the beginning and thinking of algorithm makes the solution complicated. Hence, it is easy to solve this problem back ward. The patch covered the whole lake in day 48. Yesterday, day 47, the size of the patch was half of today. Therefore, on day 47, the patch covered only half of the lake. This problem can be solving in complicated way and using algorithms. The answer would be the same, but it cost forfeited times, and brain exhaustion or what you gave up to solve the problem.
In all three problems, decision and its consequences play important role. No matter what the decision is there is always consequence(s) that attaches to the decision. The consequence could be beneficial or comes at cost. The most important is that make a decision that is more beneficial.
If it weren’t for the first sentence in this blog, I would have gotten the wrong answers for the questions. When I first read through the questions I initially got the wrong answer because I didn’t take the time to actually think the question through and come to the right answer. When answering the question I was just using quick thinking and not using economics to solve the problem. When solving these problems, first figuring out what the question is asking or what is causing the problem, and how to solve it or fix it. The key to using economics to solve a problem is analyzing the question and working through the problem to find the right solution to fix it. Although the response that I as well as many others received intuitively springs to mind, it is incorrect. People often substitute difficult problems with simpler ones in order to quickly solve them. Realizing that I initially got the first two questions wrong shows that I am overconfident and placing too much faith in my intuition and don’t rely on my logical system. A number came to mind right away, and instead of doing the math, I chose the quick intuitive answer not using economics to solve the problems given. The trick in these questions seems to be, people who answer wrong trust their intuition more than using logics to find the answers.
ReplyDelete