From 19 beads we can have two sets of nine beads and 1 more bead. Removing 12 beads means removing one set of nine beads and then three more beads. Thus, from the two sets of nine beads from 19, we can remove one set of nine beads. Now, we have one set of nine beads and a single bead from the 19 beads. To take away three more beads we can first pick up the single bead from the 19 beads leaving with us only one set of nine beads. Still we have to take away two beads of the total 12 we wanted to remove from the 19. To do that we have obviously to use the single set of nine beads from the 19 beads! From these nine beads, when we remove the two beads, we will be left with seven beads! This is of course, the remainder of nine after subtraction.
How the process works with these numbers is shown in figure 1. In following this procedure of casting out nines, remember that the remainder of nine for the number nine is zero. Hence, you can ignore the digit 9 in the numbers while calculating remainder of nine. In the example 1, 2 or 3 in figure 3 of the previous article, we could have ignored the digit 9 in the minuend / subtrahend. Also remember that, finally, we are interested only in the remainder of nine and not in how many sets of nine are there in the minuend or the subtrahend or the remainder. However, number of sets of nine are calculated here just to emphasize the point that we can always borrow one nine from the minuend to subtract the extra remainder of nine of the subtrahend, if required.
The same process is made clear in Figure 2. Here, also, we could first have straightaway deleted one remainder of nines each from the minuend and the subtrahend, which is shown by using cancellation by green lines. Then, to delete the two remainders of nine of the subtrahend, two dots from a set of nine of the minuend and two dots from the remainders of the subtrahend are used.This is shown in the figure by using red lines for cancellation.
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