All from 9 and the Last from 10

The ten’s complement of a number tells us how far it is from the next higher power of 10.  In our example, this is obtained by subtracting 847,536 from 1,000,000.

Quickly finding this 10’s complement is important in many arithmetic computations especially in “completing the whole” which is the subject of our Math Tip last week.

This is done by using the Vedic Math sutra Nikhilam Navatashcaramam Dashatah meaning All from 9 and the Last from 10. Instead of using the traditional method of putting the subtrahend 847,536 below the minuend 1,000,000 and subtracting from right to left which would require 6 “borrowings” or regroupings, we can proceed from left to right by subtraction every digit of the subtrahend from 9 except the last digit which we will subtract from 10.

Using this method, we can immediately announce the ten’s complement of any number without using pen and paper or calculators. Just by looking at the figure 847,536, we can say that the correct answer is 4) 152, 464.

Here is the step by step mental process:

1. Look at subtrahend: it is eight hundred forty seven thousand, five hundred thirty six.
2. The difference is 8 from 9 or one hundred.
3. 4 from 9 or fifty
4. 7 from 9 or two thousand,
5. 5 from 9 or four hundred
6. 3 from 9 or sixty
7. the last, 6 from 10 or four.

This Nikhilam method is also very useful in addition and subtraction of decimals, including money calculations and in base multiplication and division.

Perform the following subtraction using “All from 9 and the last from 10”

1. 1,000 – 456 =
2. 10,000 – 3,859 =
3. 100,000 – 83,942 =
4. 1,000,000 – 598,274 =
5. 1 – 0.24 =
6. 1 – 0.789 =
7. 1 – 0.4587 =
8. 1 – 0.68942 =
9. P100 – 24.75 =
10. P1,000 – 786.38 =

Answers will be published in the next post.

Here are the answers for last week’s exercises in Completing the whole:

1. 8 + 9 + 2 = (8 +2) + 9 = 19
2. 6 + 7 + 4 + 9 + 3 = (6 + 4) + (7 + 3) + 9 = 29
3. 11 + 32 + 89 = (11 + 89) + 32 = 132
4. 24 + 38 + 50 + 226 = (24 + 226) + 50 + 38 = 338
5. 17 + 38 + 79 + 45 = [(17 + 38) + 45] + 79 = 179
6. 3/5 + 3/4 + 2/5 = (3/5 + 2/5) + 3/ 4 = 1 3/ 4
7. 58 min + 57 min + 56 Min = (58 + 2) + (57 + 3) + (56 – 2 – 3) = 2 hours 51 mins
8. 3 ft 5 in + 4 ft 6 in + 5 ft 7 in = (3 ft 5 in + 5 ft 7 in) + 4 ft 6 in = 13 ft 6 in.
9. 9999 + 999 + 99 + 9 + 5 = (9999 + 1) + (999+1) + (99 + 1) + (9 + 1) + (5 – 4) = 11,111
10. 1 3/4 + 2 4/5 + 3 2/8 = (1 3/ 4 + 3 2/8) + 2 4/5 = 7 4/5