Find the value of $(0.34\overline{68}+0.17\overline{32})$.
To show the mechanism of solution, form repeating sequence for each repeating digit pair,
$0.3 4 6 8 6 8 6 8\overline{68}......$
$0.1 7 3 2 3 2 3 2\overline{32}......$
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$0.5 2 0 1 0 1 0 0.......$
The third pair of 68 and 32 from the left add up to 100, with a carry over of 1. This carry over flows through the two more pairs of repeating digits on the left into the non-repeating digit area, and also results in $01$ as a seemingly repeating pattern.
In other words, first pair of repeating digits from the right add up to 100, while all the repeating digit pairs on its left add up to 101, consisting of carry over of 1 and a two digit sequence of 01.
This is clear, but what the final result will be is not so clear.
In a non-terminating repeating digit sequence, the repeating pattern of say, $68$ continues to the right endlessly to infinity. The same is true for the second repeating digit pattern of 32.
In the first instance we will assume infinity to end at a point where the absolute rightmost two pairs, $68$ and $32$ add up to 100 resulting in a two digit result of $00$ and a carry of 1. This is the only time a $00$ is produced and being the rightmost zeros these can be ignored.
Finite nature of the result of adding two infinite repeating sequences: However long the sequences of the repeating couple of digits are, addition of the two numbers will create the result of $00$ as the rightmost two digits and a carry of 1. In this case, the sum results in a finite result on adding two infinitely long sequences.
The carry of 1 will generate a sequence of $01$ all through to the left of infinity and finally will spill over into the non-repeating digit area.
Thus, the result of addition becomes, $0.52\overline{01}$. In this case, the infinity has its conventional definition of something with no-end point.
SSC CGL Number system Solution set 46: Learn to answer 10 dividend divisor quotient remainder questions in 12 mins. Basic to advanced concepts used.
