15. Proportions involving the CI scale. The CI scale may be used in connection with proportions containing reciprocals. Since any number a = 1 ÷ 1/a and since 1/a = 1/a ÷ 1, we have

Rule (C). The value of any ratio is not changed if any factor of its numerator be replaced by 1 and its reciprocal be written in the denominator, or if any factor of its denominator be replaced by 1 and its reciprocal be written in the numerator. Thus a/b = a (1/b) = 1/[b (1/a)]. Hence b b (I /a) x b if - = be, we may write - - = - ; if ax be, we a a (I /C) (1 lb)
The reciprocal of a number is obtained by dividing 1 by the number. Thus, 1/2 is the reciprocal of 2, 2/3 ( = 1 ÷ 3/2) is the reciprocal of 3/2, and 1/a is the reciprocal of a.

The reciprocal scales CI and CIF are marked and numbered like the C, and CF scales respectively but in the reverse (or inverted) order; that is, the numbers represented by the marks on these scales increase from right to left. The red numbers associated with the reciprocal scales enable the operator to recognize these scales. A very important consideration may be stated as follows: When the hairline is set to a number on the C scale, the reciprocal (or Inverse) of the number is at the hairline on the Cl scale; conversely, when the hairline is set to a number on the CI scale, its reciprocal is at the hairline on the C scale. A similar relation exists between the CF and CIF scales. If the operator will close his rule, he can read the opposites indicated in the diagram.
CI 1 2 4 5 8 9 x
C or D 1 0.5
(= 1/2)		 0.25
(= 1/4)		 0.2
(= 1/5)		 0.125
(=1/8)		 0.1111
(= 1/9)		 1/x

By using the facts just mentioned, we can multiply a number or divide it by the reciprocal of another number. Thus to find 28/7, we may think of it as 28 × 1/7 and

                              to 28 on D set index of C;
                              opposite 7 on CI read 4 on D.

Again to find 12 × 3, we may think of it as 12 ÷ 1/3 and

                              push hairline to 12 on D,
                              draw 3 of CI under the hairline;
                              opposite index of C, read 36 on D.

When the CI scale is used in multiplication and division, the position of the decimal point is determined in the usual way.

The DF and CIF scales may be used to perform multiplications and divisions in the same manner as the D and CI scales; thus to multiply 40.3 by 1/9.04,

                              to 403 on DF set index of CF;
                              opposite 904 on CIF, read 4.46 on DF.

Again to multiply 40.3 by 1/0.207,

                              to 403 on D set left index of C;
                              opposite 207 on CIF read 194.7 on DF.

EXERCISES
 1. Use the CI scale to find the reciprocals of
  0.0625 16,
  0.00385 260,
  1.389 0.72,
  15.38 0.065,
  0.0575 17.4,
  0.0541 18.5,
  0.01490 67.1.
 2. Using the D scale and the CI scale,
  2 multiply 18 by 1/9 and
  162 divide 18 by 1/9.
 3. Using the D scale and the CI scale
  74.0 multiply 28.5 by 1/0.385 and
  10.97 divide 28.5 by 1/0.385.
Also find
  74.0 28.5/0.385 and
  10.97 28.5 × 0.385
by using the C scale and the D scale.
 4. Using the D scale and the CI scale
  199.5 multiply 41.3 by 1/0.207 and
  8.55 divide 41.3 by 1/0.207.
 5. Perform the operations of Exercises 2, 3, and 4 by using the CIF scale and the DF scale.
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