chap mathematical preliminaries derive an equation that relates the parameters to the problem select values for the parametersand apply the equation to yield an estimated solution when doing estimationsa good way to reassure yourself that the estimate is reasonable is to do it in two different ways in generalif you want to know what comes out of systemyou can either try to estimate that directlyor you can estimate what goes into the system (assuming that what goes in must later come outif both approaches (independentlygive similar answersthen this should build confidence in the estimate when calculatingbe sure that your units match for exampledo not add feet and pounds verify that the result is in the correct units always keep in mind that the output of calculation is only as good as its input the more uncertain your valuation for the input parameters in step the more uncertain the output value howeverback of the envelope calculations are often meant only to get an answer within an order of magnitudeor perhaps within factor of two before doing an estimateyou should decide on acceptable error boundssuch as within %within factor of twoand so forth once you are confident that an estimate falls within your error boundsleave it alonedo not try to get more precise estimate than necessary for your purpose example how many library bookcases does it take to store books containing one million pagesi estimate that -page book requires one inch on the library shelf (for examplelook at the size of this book)yielding about feet of shelf space for one million pages if shelf is feet widethen shelves are required if bookcase contains shelvesthis yields about library bookcases to reach this conclusioni estimated the number of pages per inchthe width of shelfand the number of shelves in bookcase none of my estimates are likely to be precisebut feel confident that my answer is correct to within factor of two (after writing thisi went to virginia tech' library and looked at some real bookcases they were only about feet widebut typically had shelves for total of shelf-feet so was correct to within on bookcase capacityfar better than expected or needed one of my selected values was too highand the other too lowwhich canceled out the errors example is it more economical to buy car that gets miles per gallonor one that gets miles per gallon but costs $ morethe