Reciprocal is also used in problems involvingtrigonometric functions of angles, as you will seelater in this chapter, in the solutions of problemscontaining identities.is formed. The proportion may be written in threedifferent ways, as in the following examples:RATIO AND PROPORTIONAlmost every computation you will make asan EA that involves determining an unknownvalue from given or measured values will involvethe solution of a proportional equation. Athorough understanding of ratio and proportionwill greatly help you in the solution of bothsurveying and drafting problems.The results of observation or measurementoften must be compared to some standard valuein order to have any meaning. For example, if themagnifying power of your telescope is 20diameters and you see a telescope in the marketthat says 50 diameter magnifying power, then onecan see that the latter has a greater magnifyingpower. How much more powerful? To find out,we will divide the second by the first number,which isThe magnifying power of the second telescope is2 1/2 times as powerful as the first. When therelationship between two numbers is shown thisway, the numbers are compared as a RATIO. Inmathematics, a ratio is a comparison of twoquantities. Comparison by means of a ratio islimited to quantities of the same kind, Forexample, in order to express the ratio between12 ft and 3 yd, both quantities must be writtenin terms of the same unit. Thus, the properform of this ratio is 4 yd:3 yd, not 12 ft:3 yd.When the parts of the ratio are expressed interms of the same unit, the units cancel eachother and the ratio consists simply of twonumbers. In this example, the final form of theratio is 4:3.Since a ratio is also a fraction, all the rulesthat govern fractions may be used in working withratios. Thus, the terms may be reduced, increased,simplified, and so forth, according to the rulesfor fractions.Closely allied with the study of ratio is thesubject of proportion. A PROPORTION isnothing more than an equation in which themembers are ratios. In other words, when tworatios are set equal to each other, a proportionThe last two forms are the most common. All ofthese forms are read, “15 is to 20 as 3 is to 4.”In other words, 15 has the same ratio to 20 as 3has to 4.The whole of chapter 13, NAVEDTRA10069-D1, is devoted to an explanation of ratioand proportion, the solution of proportionalequations, and the closely related subject ofvariation. In addition to gaining this knowledge,you should develop the ability to recognize acomputational situation as one that is availableto solution by proportional equation. A very largearea of surveying computations—the area thatinvolves triangle solutions—uses the proportionalequation as the principal key to the determinationof unknown values on the basis of known values.Practically any problem involving the conversionof measurement expressed in one unit to theequivalent in a different unit is solvable byproportional equation. Similarly, if you know thequantity of a certain material required to producea certain number of units of product, you candetermine by proportional equation the quantityrequired to produce any given number of units.In short, it is difficult to imagine anymathematical computation involving thedetermination of unknown values on the basis ofknown values that is not available to solution byproportional equation.Your knowledge of equations need not extendbeyond that required to solve linear equations;that is, equations in which the unknown appearswith no exponent higher than 1. The equationfor example, is a linear equation, because theunknown (technically known as the “variable’ ‘),x, appears to only the first power. The equationX2 + 2x = – 1, however, is a quadratic, not alinear, equation because the variable appears tothe second power.The whole of chapter 11 of NAVEDTRA10069-D1 is devoted to an explanation of linear1-7
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