N4081 Use of Proportions

11. Use of Proportions. If the slide is drawn to any position, the ratio of any number on the D scale to its opposite on the C scale is, in accordance with the setting for division, equal to the number on the D scale opposite the index on the C scale. In other words, when the slide is set in any position, the ratio of any number on the D scale to its opposite on the C scale is the same as the ratio of any other number on the D scale to its opposite on the C scale. For example

FIG. 1.
draw 1 of C opposite 2 on D (see Fig. 1) and find the opposites indicated in the following table:

C (or CF)	1	1.5	2.5	3	4	5	x
D (or DF)	2	3	5	6	8	10	2 × x

and draw 2 of C over 1 on D and read the same opposites. The same statement is true if in it we replace C scale by CF scale and D scale by DF scale. Hence, if both numerator n and denominator d of a ratio in a given proportion are known, we can set n of the C scale opposite d on the D scale and then read, for an equal ratio having one part known, its unknown part opposite the known part. We could also begin by setting d on the C scale opposite n on the D scale. It is important to observe that all the numerators of a series of equal ratios must appear on one scale and the denominators on the other. For example, let it be required to find the value of x satisfying

——

—

Here the known ratio is 9/7. Hence
                              push hairline to 7 on D,
                              draw 9 of C under the hairline,
                              push hairline to 56 on D,
                              at the hairline read 72 on C.
or
                              push hairline to 9 on D,
                              draw 7 of C under the hairline,
                              push hairline to 56 on C,
                              at the hairline read 72 on D.

The CF and DF scales could have been used to obtain exactly the same settings and results. Figure 2 indicates the setting.

FIG. 2.
To find the values of x, y, and z defined by the equations

—

3.15

———

5.29

———

4.35

57.6

———

————

183.4

note that C and D indicate the respective scales for the numerators and the denominators, observe that 3.15/5.29 is the known ratio, and

                              push hairline to 529 on D,
                              draw 315 of C under the hairline;
                              opposite 435 on D, read x = 2.59 on C;
                              opposite 576 on C, read y = 96.7 on D;
                              opposite 1834 on D, read z = 109.2 on C.

The positions of the decimal points were determined by noticing that each denominator had to be somewhat less than twice its associated numerator because 5.29 is somewhat less than twice 3.15.

When an answer cannot be read, apply the italicized rule of §6. Thus to find the values of x and y satisfying

—

———

587

14.56

————

97.6

5.78

————

to 976 on D set 1456 of C; then, since the answers cannot be read, push the hairline to the index on C, draw the right index of C under the hairline and

opposite 587 on D, read x = 87.6 on C;
opposite 578 on C, read y = 38.75 on D.

Here the positions of the decimal points were determined by observing that each denominator had to be about six times the associated numerator.

When a result cannot be read on the C scale nor on the D scale it may be possible to read it on the CF scale or on the DF scale. Thus, to find x and y satisfying the equations

C (or CF)

—————————

D (or DF)

4.92

———

————

3.23

————

13.08

                              to 323 on D set left index of C;
                              opposite 492 on CF, read   x = 15.89 on DF;
                              opposite 1308 on DF, read   y = 4.05 on CF.

A slight inspection of the scales will show the value of the statement: If the difference of the first digits of the two numbers of the known ratio is small use the C and D scales for the initial setting; if the difference is large use the CF and DF scales. Since in the next to the last example, the difference between the first digits was great, the CF and DF scales should have been used for the initial setting. This would have eliminated the necessity for shifting the slide.

EXERCISES
Find, in each of the following equations, the values of the unknowns.