Fifth. Consider the series of last figures which have been successively cut off from the prepared dividend and from the modified dividends as constituting a number, the figure first cut off being in the units' place, the next in the tens' place, and so on. Call this the first infinite number, because its left-hand portion consists of a series of figures repeating itself indefinitely toward the left. Imagine another infinite number, identical with the first in the repeating part of the latter, but differing from this in that the same series is repeated uninterruptedly and indefinitely toward the right into the decimal places.
Subtract the first infinite number from the second, and shift the decimal point as many places to the left as there were zeros dropped in the process of obtaining the prepared divisor.
The result is the quotient sought.
Examples.
1. The following is taken at random. Divide 1883 by 365.
First. The divisor, since it ends in 5, must be multiplied by 2, giving 730. Dropping the O, we have 73 for the prepared divisor.
Second. The last figure of the prepared divisor being 3, this is the extraneous multiplier. Multiplying the truncated divisor, 7, by the extraneous multiplier, 3, and adding 1, we have 22 for the current multiplier.
Third. The dividend, 1883, has now to be multiplied by the product of 3, the extraneous multiplier, and 2, the multiplier used in preparing the divisor. The product, 11298, is the prepared dividend.
Fourth. From the prepared dividend, 11298, we cut off the last figure 8, and multiply this by the current multiplier, 22. The product, 176, is added to the truncated dividend, 1129, and gives 1305 for the first modified divisor. The whole operation is shown thus:
1 8 8 3
6
-------
1 1 2 9|8
1 7 6 -
-----
1 3 0|5
1 1 0 -
-----
2|4 0
8 8 ---
---
|9 0
-----
1 9|8
1 7 6 -
-----
1 9|5
1 1 0 -
-----
1 2|9
1 9 8 -
-----
2|1 0
2 2 ---
2 4