# Combining Partial Sums

If the carries are combined in 2/33/44, the result is 574. What would be the result of combining carries in  23/34/56/15”

This question was given in several age groups in the recently held 4th MATH-Inic Vedic Mathematics National Challenge.

In the example given: if the carries are combined in 2/33/44,  the result was obtained by adding 2 to the first 3 in 33 to get 5, then the second 3 is added to the first 4 to get the middle digit and the last 4 is the last digit of the answer. This can be simplified as 2/33/44 = 2 + 3| 3 + 4 | 4 = 574.

Using this example, by combining carries in 23/34/56/15, the result would be

2 | 3 + 3 | 4 + 5 | 6 + 1 | or 26975

Now how do you evaluate this: 16/56/97/108/78/36/9 ?

The easiest way is to separate the tens digits and the units digit in the partial sum and then, starting from the left, add the ten’s digits to the unit’s digit of the next sum to the right. Repeat until a single digit remains in each partition.

16/56/97/108/78/36/09 = 1|6 + 5|6 + 9|7 + 10|8 + 7|8 + 3|6 + 0 | 9

= 1 | 11| 15| 17| 15| 11 | 06 |9

= 1 + 1| 1 + 1 | 5 +1|7 + 1|5 + 1| 1 + 0| 6 | 9

= 2|2|6|8|6|1|6|9

= 22686169