What powerful mathematical tool do you use everyday? If you have ever considered how much food you can eat and still leave room for dessert, or made a rough calculation of how long it will take to travel to school, you are using estimates to help guide and inform your decisions

Frolking, Steve

Rosentrater, Lynn

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University of New HampshireUniversity of New Hampshire Scholars' RepositoryEarth Sciences Scholarship Earth Sciences4-25-1999Two Answers are Better than OneSteve FrolkingUniversity of New Hampshire - Main Campus, steve.frolking@unh.eduLynn RosentraterUniversity of New Hampshire - Main CampusFollow this and additional works at: https://scholars.unh.edu/earthsci_facpubThis Article is brought to you for free and open access by the Earth Sciences at University of New Hampshire Scholars' Repository. It has been acceptedfor inclusion in Earth Sciences Scholarship by an authorized administrator of University of New Hampshire Scholars' Repository. For moreinformation, please contact nicole.hentz@unh.edu.Recommended CitationFrolking S, Rosentrater L. 1999. Two answers are better than one. PUMAS (Practical Uses of Math and Science). The On-line Journalof Math and Science Examples for Pre-College Education, http://pumas.jpl.nasa.gov.Document ID: 04_25_99_2Date Received: 1999-04-25   Date Revised: 1999-07-17   Date Accepted: 1999-07-17Curriculum Topic Benchmarks: S.15.4.6, M.1.4.8, M.3.4.4, M.3.4.4Grade Level: [9-12] High SchoolSubject Keywords: estimating, back-of-the-envelope calculation, significant figures, checkingyour workRating:  moderate  Two Answers are Better than OneBy: Steve Frolking and Lynn Rosentrater, Institute for the Study of Earth, Oceans, and Space,University of New Hampshire       e-mail:  steve.frolking@unh.eduFrom:  The PUMAS Collection http://pumas.jpl.nasa.gov©1999, California Institute of Technology. ALL RIGHTS RESERVED. Based on U.S. Gov't sponsored research.What powerful mathematical tool do you use everyday? If you have ever considered how muchfood you can eat and still leave room for dessert, or made a rough calculation of how long it willtake to travel to school, you are using estimates to help guide and inform your decisions.Estimating means pproximating your answer rather than making a precise calculation. Making anestimate begins with a specific question, like how long will you need to save in order to buysomething that costs about $100.  You earn about $20 per week from allowance, babysitting orother chores, and can usually save about half of that, then it will take you about 10 weeks to savethe total amount:$100 / $10 per week = 10 weeksAs you save, you may have to revise your estimate based on your actual earnings and savings eachweek.Suppose your class is having a party at school, and you want to serve pizza, chips, juice and sodapop, and ice cream.  About how much money will you need to raise to buy enough food for 100people?  You need to estimate how much food you will need, and the costs per item of the food.One pizza and one bottle of juice or soda pop will feed about 4 people, so you need 100 people/4people per pizza or soda pop ~ 20-30 pizzas and 20-30 soda pops.  A bag of chips and a half-gallon of ice cream are each enough for about 10 people, so you ne d ten of each of these.Suppose each pizza costs $7, a bag of chips is $2, a half-gallon of ice cream costs $4, and drinkscosts $1 per 2 liter bottle.  Then you need about:30 x $7 + 10 x $2 + 10 x $4 + 30 x $1 = $300.00As another check on your answer, express it in a different way: $300.00 for the party for 100people is $3.00 per person, which seems pretty reasonable.  Also, if you only ordered 25 pizzas,you would have some money left (how much?) for cups, paper plates, and napkins.Scientists use estimates as a check on more precise measurements; estimation is an easy way toquantify early hypotheses that can be tested or checked later in formal analysis. This is where twoanswers are better than one. An initial estimated answer can be used to help interpret a secondcalculated answer, or identify problems with a methodology when the two don’t match. Estimatingis also useful when you have a problem with several components. You can estimate the importanceof each component and concentrate your research and thinking on the most important part of theproblem.When working with complex problems, especially those involving large numbers, it is useful writeyour numbers with only one significant figure, that is to say, your number will have only onenumeral that is not a zero.  For example, if you know the length of something is 2350 meters youwould round that to 2000 meters (the ’2’ is the only significant figure). Now suppose you want toestimate one-seventh of that length to one significant figure, your answer will be 2000 Ö 7 ~ 300meters, though the precise answer would be 2350 Ö 7 = 335.71428571... meters.  Your answer of300 meters conveys the same level of precision (1 significant figure) as your initial estimate of2000 meters. Using significant figures requires that you round each number to the nearest ten, hundred,thousand, etc.  For example, both 8 and 12 are approximated by 10, and 75 is approximately 80; ayear has approximately 50 weeks.  There are specific rules for rounding numbers. However, youshould realize that if you were to round every number down, your estimate would be too low, andif you were to round every number up, your estimate would be too high. So try to make sure thatyou round some things up and some things down; this way your final answer will be a betterapproximation.Here are two more examples:1. You want to paint your room. The room is rectangular, about 10 feet long by 10 feet wideby 8 feet high.  There are two doors, 3 feet by 6 feet, and two windows, 2 ft by 3 ft. Your momsays, ‘Okay, you can paint the room, but you buy the paint and supplies.’  How much will it cost?Some useful information: paint is sold in gallons (1 gallon of paint covers about 450 square feet,and costs about $20) and quarts (1 quart covers about 100 square feet, and costs about $7). The area of the walls and ceiling is about:4 walls x (10 ft x 8 ft per wall) + 1 ceiling x (10 ft x 10 ft) ~ 400 ft2The area of the doors is about:2 doors x (3 ft x 6 ft per door) ~ 40 ft2The area of the window trim is determined by multiplying the trim width by the distance around thewindow (perimeter) and is about:2 windows x (0.5 ft wide x 10 ft perimeter) ~ 10 ft2So 1 gallon of paint should be enough for a single coat of paint on the walls and ceiling, and 1quart of paint should be enough for two coats of paint on the doors and trim.  The total cost wouldbe about $30 - $50, depending on whether the walls needed one or two coats of paint. You mightalso need some money ($10 - $20) for paint brushes and/or paint rollers, masking tape, and dropcloths.2. How much water flows out of the Mississippi River in a day? To estimate this, wehave to estimate several characteristics of the river.  Assume that near its mouth the river is about 1kilometer (km) wide and perhaps 10 meters (or 0.01 km) deep. We could estimate that the rate offlow was 8 km per hour (~200 km/day); this is equivalent to a very fast walk (5 mph) and seemsreasonable.  Then to estimate the flow we would multiply the width by the depth by the rate offlow:1 km x 1/100 km x 200 km/day » 2 km3/dayOur estimate shows that the Mississippi River discharges about 2 cubic kilometers per day.  But isit right? How would you check?  We could check with another estimate.  Almost all the water thatflows out of the Mississippi River fell as rain or snow (some was pumped from groundwateraquifers, but we can ignore this).  So the amount of rain and snow that falls on the area that isdrained by the Mississippi River (called the drainage basin) should be about this amount.(Actually, it should be about a factor 2 bigger, because typically more than half of the rain that fallsevaporates back into the atmosphere before getting to the river.) We can estimate this bymultiplying three numbers: the area of the drainage basin, the depth of water in a typical rain orsnow storm, and the typical storm frequency. What is the area of the Mississippi River drainage basin? To estimate this, picture a map of theUSA in your mind, or look at an atlas.  The country (except Hawaii and Alaska) is roughlyrectangular and is about 2000 km (north-south) by 5000 km (east-west), giving an area of about10,000,000 square kilometers.  The Mississippi drainage basin is roughly 1/4 of the country, orabout 3,000,000 km2.  It rains or snows about once a week, and here in New Hampshire we getabout 1 meter of precipitation (rain plus snow) per year.  If we assume the New Hampshire annualtotal is typical for the Mississippi basin (some areas will be wetter, some drier; what is the annualprecipitation in your area?) then a typical storm might produce 2 cm (= 1 m divided by 50 storms)or 0.00002 km of water.  So the average daily precipitation on the Mississippi River drainage basinis about: 3,000,000 km2 x 0.00002 km water/storm x (1 storm/7 days) »  8 km3/dayGiven all the estimating involved, these two results (2 and 8 km3/day) are close enough to beconsidered consistent with each other.  Neither is precisely right, but both are probably reasonableestimates.

Two Answers are Better than One

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Two Answers are Better than One

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