Chapter 10 – Combined Operations on C, D, CI, DI, CD, DF, CIF, A, B, BI, K and K’ Scales
In this unit we will show how the order of operations and the selection of scales can greatly reduce the moves required. The effect of this is to increase accuracy, speed and the range of problems you can handle on your Slide Rule.
Note: The A and B scales or the K and K’ scales (if you Slide Rule has a K’ Scale), can be used for multiplication and division. On the A and B scales the 1, 10 , 100 can be read as an index, while the K and K’ scales 1, 10, 100, or 1,000 can be used as an index in calculations. The BI scale can be used in conjunction the A and B scales in the same way as the CI with the C and D scales, or the CIF with the CF and DF scales.
Example 1:
(i.e. approx =
= (.03)2
= 0.0009)
Therefore the answer is 0.00114
Note: We find the reciprocal of 29.6 first and then the square. This is possible because .
If we were to use the reverse order of operations, it would necessitate reading the square of 29.6 off the A or B scale and transferring this value onto the C or CI scale to obtain the reciprocal, which of course is unsatisfactory.
Example 2:
Example 3:
Example 4:
Example 5:
Example 6:
Example 7: 29.64 = 768,000
OR
Example 8: 29.65 = 22,750,000
Example 9: 29.66 = 673,000,000
Example 10:
Note: The last two steps in the above example could be done as follows:
2. Set the hair line over 29.6 on the D scale.
Exercise 10(a)
In this section we will combine the basic methods of multiplication and division of Units 2, 3, 4 and 8. You may find it helpful to review these briefly before proceeding with this section.
Example 1: 12.4 x 8.4 x 0.157 = 16.35
Note: the progressive answer is marked on the D scale by the left index of the C scale.
Example 2: 2.32 x 60.5 ¸ 0.082 = 1710
Note: the progressive answer is marked on the D scale by the left index of the C scale.
Example 3: 7.5 ÷ 4.8 x 30.4 = 47.5
Note the progressive answer is marked on the D scale by the left index of the C scale.
Example 4:36.6 ÷ 0.71 ÷ 2.26 = 22.8
Note the progressive answer is marked on the D scale by the right index of the C scale.
Note: In combination multiplication and division rules are:
The reason for this is that when we multiply by the CI and D scales, or divide by the C and D scales, the answer is always marked on the D scale by the left or right index of the C scale, thus allowing a further multiplication or division without moving the slide. Hence, if the second operation is multiplication, we use the C and D scales, and if the second operation is a division we sue the CI and D scales. (Check these points carefully in the four examples above.)
Example 5: 3.35 x 47 ÷ 25.9 ÷ 41 x 8.85 = 1.312
(Note: in the above example we switched to the Folded scales for the last multiplication, as 8.85 on the C scale was off the end of the D scale. Otherwise it would have required the slide to have been moved.)
Exercise 10(b)
There are too many possible combinations for us to cover ever type, but the following table gives many possibilities. Note that often care must be taken in setting numbers on the A, B and K scales, as we will recall for example and are located at different positions on the A and B scale. Thus, numbers must be located on the A, B and K scales according to the rules given in Units 5 and 6. In some of the following, with certain numbers the answer may run off the end of the scale. In such cases it will be necessare to reset the slide unless the C and D scales are involved, when it will be possible to us the CF and DF scales and avoid a further movement of the slide. (In the following, ‘H.L." stands for Hair Line.)
Example |
Set HL Over |
Under HL Place |
Reset HL over |
Under HL answer |
a on A scale |
Index of B scale |
b on C scale |
on A scale |
|
a D |
Index C |
b B |
A |
|
a K |
Index B |
b C |
K |
|
a A b D |
b CI a BI |
b C b C |
A A |
|
a D a D |
Index C b CI |
b C Index C |
A A |
|
a D |
b CI |
b B |
A |
|
c A a D |
a CI c BI |
b C b C |
A A |
|
c K |
a CI |
b C |
K |
|
a A a A |
b C Index B |
Index C b CI |
A A |
|
a K |
b C |
Index C |
K |
|
a D a D |
b C b B |
b BI b CI |
A A |
|
a D |
b B |
Index B |
A |
|
a D a D |
b B b K’ |
a B index K’ |
A K |
|
a A |
a CI |
b CI |
A |
|
b D |
c C |
a B |
A |
|
a A |
b C |
c CI |
A |
|
a A |
b B |
c CI |
A |
|
Index D |
a C |
b CI |
D |
|
Index A |
a B |
b CI |
A |
|
Index D |
a C |
b CI |
A |
|
a K |
b C |
a CI |
K |
|
a A |
Index B |
b B |
D |
|
a A |
b BI |
c B |
D |
|
a D |
Index C |
b B |
D |
|
b A |
a CI |
a C |
D |
|
a A |
b B |
Index B |
D |
|
a A |
b C |
Index C |
D |
|
a D |
b B |
Index C |
D |
|
a A |
c B |
b B |
D |
|
a A |
b B |
c BI |
D |
|
a A |
b B |
c BI |
D |
|
a A |
b C |
c C |
D |
|
a D |
c B |
b C |
D |
|
c A |
b CI |
a C |
D |
|
b A |
a CI |
c B |
D |
|
b K |
a CI |
Index CI |
D |
|
a K |
b C |
Index C |
D |
|
b K |
a C |
Index D |
D |
|
b K |
c C |
a B |
D |
|
b K |
c B |
a C |
D |
|
Index A a A |
a B Index B |
b A b B |
CI DI |
|
b A Index D |
a CI a C |
Index D b BI |
C D |
|
b A |
a CI |
Index C |
DF |
|
a D |
b B |
Index C |
DF |
|
π A |
Index B |
a B |
D |
|
b A |
a CI |
Index CF |
D |
|
a D |
b B |
Index CF |
D |
|
b A |
π B |
a CF |
DF |
Exercise 10(c)
Exercise 10(d)