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- 1. TARUN GEHLOT How to Estimate a Math ProblemWhat happens whenever you encounter a problem you cant solve, or a question youcant answer? Rather than simply giving up, you try to estim ate a right answer or asolution; you try to come as close as you can to the result you are looking for. Thisprocess of estimating something you dont know can range from a simp le "educatedguess" to systematic theories developed in sciences like Statistics and NumericalAnalysis.Before beginning the understanding of how some estimation methods work, try to makea few "educated guesses" in order to answer the following questions (unless, of course,you already know the answers)! 1. How many water bottles do you need to fill-up your local swimming pool? 2. How many grains of brown sugar are there in a teaspoon? 3. What is the area under the graph of the curve y=x2 between the points 0 and 1? 4. How many letters are there in the novel you read last summer?Hopefully, your answers were not completely random, but they involved some sort ofintuition, or even systematic thinking. In the paragraphs that follow, I will try to helpyou organize your thoughts, so that the estimates you will be making from now on, willbe based on solid facts and arguments, rather than pure intuition and guesses.Think of what you knowThe first thing you need to do when encountering a problem you get stuck on, is tofigure out what you do know about the problem. It is very un likely that you will bepresented with a task you know absolutely nothing about. So, organize the things thatyou know and try to extract results from them.For example, what do you know about your local swimming pool? To calculate itsvolume, you need to know its length, its width and its depth. You definitely know itslength - it is 25 meters. As far as its depth is concerned, you are not sure - but you doknow that its deep end is 2 meters deep and that its shallow end is 1.20 meters deep.What about its width? You know the swimming pool has eight lanes. How wide is eachone of them?For another example, you dont know yet how to calculate areas of strange shapes.However, you must certainly know how to calculate the area under the curve y=x andalso the area under the curve y=12x. These are simp ly triangles. How do these relate toour original problem of finding the area under y=x2, though?Understand what you dont knowAll of the above problems involve at least one quantity which is unknown to you, andprevents you from finding your answer. In the swimming pool example, you dont knowthe width of each lane. In the example with the sugar, you dont know the volume of abrown sugar grain, even though you might remember that a teaspo on holds about 5ml.In the example with the book, you may remember that there are 300 pages, but youdont know how many letters there are in a page. TARUN GEHLOT (B.E, CIVIL, HONOURS)
- 2. However, once you decid e what a suitable value for all those unknown parametersmight be, you will be very close to finding a very sensible answer to your question.Approximating what you dont knowI will now give you a few ideas on what you could do, if you need to make a reasonableapproximation. Of course, every problem is different, and other approaches than thesemight work, but still they are a good way to start.1. Use inequalitiesEven if you dont know what the value of a particular quantity is, you might know that itcertainly will not be greater than a specific value nor will it be less than another value.For example, plot y=x, y=x 2 and y= 12x on the same axes. You should see that the areaunder the graph of y=x2 is certainly less than the area under the graph of y=x. But it isalso more than the area under the graph of y=12x. So, the area A that we are lookingfor, satisfies the inequality 14<A<12This is definitely a decent approximat ion, and a little mathemat ical intuition might leadus into picking the value inside the interval [ 14, 12], which corresponds to the rightresult...2. Work recursivelySometimes answering a hard question is only a matter of answering a series of easierquestions. The point is to identify exactly what you are looking for in each problem, sothat you know which are these easier questions.Your first hard question is: How many letters there are in a book? Since you know howmany pages there are in a book, you try to answer how many letters there are in onepage.But instead of answering that, try to find how many lines there are in one page. Thatshouldnt be hard. Guess about 25 (thats how many lines there are in a worddocument).So, how many letters are there in one line? Still a little hard - so think about how manywords there are in one line. About 10 sounds reasonable. Now, the average englishword has roughly 4.5 letters.Therefore, you have broken down calculat ing the number of letters in a book tocalculating the number of letters in a word, then a line, then a page and therefore thewhole book. This is how recursive thinking helps break down a complicated question toa few straightforward tasks.3. Compare to quantities you knowYou want to make an estimate about the volume of a grain of sugar. This is likely to bearound a milimetre, or even less in fact. To compare the size of a grain of sugar to amilimetre, all you need to do is draw a square with side 1mm, using your ruler. Doesthis look pretty much like the size of a grain? It really does, so this is your estimate.4. How many "units" fit in the wholeA typical idea in approximating is to see how many "small" things fit into a "larger".That way, given that we know the dimensions of the "small" thing, we can obtain anestimate for the dimensions of the larger one (and vice versa, of course). TARUN GEHLOT (B.E, CIVIL, HONOURS)
- 3. To estimate the width of the pool, which you dont know, how can you work? You canstart by estimating the width of a lane. Think that if there are four people resting at theend of the lane, they fit even if they are quite crammed. This means that since eachperson occupies 50 cm (in width), the lane needs to be about 2 meters wide. Therefore,a pool with eight lanes, should be about 16 meters wide.But you can also work the other way too. If somebody asks you, for example, howmuch a staple weighs, all you need to do is estimate how much a box (with 5000staples) weighs, and then divide!Get your answer!1. You found that the pool is 25 metres long, 16 metres wide and (on average) 1.6metres deep. So, its volume is V=25×16×1.6=640m 3So, if we use water bottles of 500 mL, we will be needing 1,280,000 of them in order tofill up our swimming pool!2. We estimate that a grain of sugar occupies a volume of 1 cubic milimetre. Now, ateaspoon has a capacity of 5mL, so it will contain about 5,000 grains of sugar.3. If you decide to stop with this approximation of the area under the curve, there isnot much more to do than simply guess a value between 14 and 12. Some might go forthe average of these two, that is 38. Somebody who is more lucky, might guess thecorrect value of 13 (which is not completely unlikely - it is just the second most naturalthought when somebody asks you to pick a number between a quarter and a half).4. A simple multip licat ion is all you need to do. In total, we estimateabout 300×25×10×4.5=337,500 letters. How many of these do you think are as?Think about your errorThis is probably the most important aspect of an approximat ion. In some cases, youmay be able to specify this precisely - for example in the calculat ion of the areaunder y=x2 you know that whatever valu e you choose in the interval [14, 12 ], your errorcannot be more than 14.In other cases, you need to consider how much each estimation you made affects yourfinal calculation. For example, if the volume of a grain of sugar is 1±0.2 cubicmilimetres, then your approximat ion for the number of the grains should be between4167 and 6250.Another way of describing an error, which is encountered in advanced statistics, isthinking about confidence intervals. So, you estimate that the quantity you are lookingfor has a value which lies with high probability in a specific interval.In general, the more precise you can be about the error of your approximat ion, themore sense this will make, and the more useful it will be. This is why there are entirebranches of mathematics are devoted to developing techniques for error control andoptimization of approximations. TARUN GEHLOT (B.E, CIVIL, HONOURS)
- 4. Final Thoughts...The point of this article was to show you how to make estimat ions of things younormally cant find. A good approximation is in many cases as useful as a propersolution, so you shouldnt underestimate its value.Most of the time, you will be encountering problems which, even though you might notknow how to solve, will have aspects which you could more or less estimate. And thebest way to estimate is not to guess, but to gradually analyze each problem anddiscover how you can make the most out of what you already know - in a s imilarfashion to what we did above.So, use the above methods, or improvise, and remember that if no one can find theanswer to a problem, your goal is to arrive as close to it as you can! Wednesday, March 06, 2013 TARUN GEHLOT (B.E, CIVIL, HONOURS)

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