Reciprocal is also used in problems involving trigonometric functions of angles, as you will see later in this chapter, in the solutions of problems containing identities. is  formed.  The  proportion  may  be  written  in  three different  ways,  as  in  the  following  examples: RATIO  AND  PROPORTION Almost  every  computation  you  will  make  as an  EA  that  involves  determining  an  unknown value from given or measured values will involve the  solution  of  a  proportional  equation.  A thorough  understanding  of  ratio  and  proportion will  greatly  help  you  in  the  solution  of  both surveying  and  drafting  problems. The  results  of  observation  or  measurement often must be compared to some standard value in order to have any meaning. For example, if the magnifying   power   of   your   telescope   is   20 diameters and you see a telescope in the market that says 50 diameter magnifying power, then one can see that the latter has a greater magnifying power.  How  much  more  powerful?  To  find  out, we  will  divide  the  second  by  the  first  number, which is The magnifying power of the second telescope is 2  1/2  times  as  powerful  as  the  first.  When  the relationship between two numbers is shown this way, the numbers are compared as a RATIO. In mathematics,   a   ratio   is   a   comparison   of   two quantities.  Comparison  by  means  of  a  ratio  is limited  to  quantities  of  the  same  kind,  For example,  in  order  to  express  the  ratio  between 12 ft and 3 yd, both quantities must be written in  terms  of  the  same  unit.  Thus,  the  proper form  of  this  ratio  is  4  yd:3  yd,  not  12  ft:3  yd. When  the  parts  of  the  ratio  are  expressed  in terms  of  the  same  unit,  the  units  cancel  each other   and   the   ratio   consists   simply   of   two numbers.  In  this  example,  the  final  form  of  the ratio  is  4:3. Since  a  ratio  is  also  a  fraction,  all  the  rules that govern fractions may be used in working with ratios. Thus, the terms may be reduced, increased, simplified,  and  so  forth,  according  to  the  rules for  fractions. Closely  allied  with  the  study  of  ratio  is  the subject   of   proportion.   A   PROPORTION   is nothing  more  than  an  equation  in  which  the members  are  ratios.  In  other  words,  when  two ratios  are  set  equal  to  each  other,  a  proportion The last two forms are the most common. All of these forms are read,  “15  is  to  20  as  3  is  to  4.” In other words, 15 has the same ratio to 20 as 3 has to 4. The   whole   of   chapter   13,   NAVEDTRA 10069-D1,  is  devoted  to  an  explanation  of  ratio and  proportion,  the  solution  of  proportional equations,   and   the   closely   related   subject   of variation. In addition to gaining this knowledge, you  should  develop  the  ability  to  recognize  a computational situation as one that is available to solution by proportional equation. A very large area  of  surveying  computations—the  area  that involves  triangle  solutions—uses  the  proportional equation  as  the  principal  key  to  the  determination of unknown values on the basis of known values. Practically any problem involving the conversion of  measurement  expressed  in  one  unit  to  the equivalent   in   a   different   unit   is   solvable   by proportional equation. Similarly, if you know the quantity  of  a  certain  material  required  to  produce a  certain  number  of  units  of  product,  you  can determine by proportional equation the quantity required  to  produce  any  given  number  of  units. In  short,  it  is  difficult  to  imagine  any mathematical   computation   involving   the determination of unknown values on the basis of known values that is not available to solution by proportional   equation. Your knowledge of equations need not extend beyond  that  required  to  solve  linear  equations; that is, equations in which the unknown appears with  no  exponent  higher  than  1.  The  equation for  example,  is  a  linear  equation,  because  the unknown (technically known as the “variable’ ‘), x, appears to only the first power. The equation X² +  2x  =  –  1,  however,  is  a  quadratic,  not  a linear, equation because the variable appears to the  second  power. The  whole  of  chapter  11  of  NAVEDTRA 10069-D1 is devoted to an explanation of linear 1-7