All from 9 and the Last from 10: This is one of the 16 sutras, Nikhilam Navatascaramam Dasatah.
This is a very important formula that can be used for subtraction from base numbers like 10, 100, 1000 and so on, i.e. power of 10.
Here, all the digits of 2nd figure (the number which is subtracted) are subtracted from 9 and the last digit of 2nd figure is subtracted from 10.
For example:
2 is subtracted from 9 (= 7), 7 is subtracted from 9 (= 2) and the last digit 6 is subtracted from 10 (= 4).
There are some special cases of this formula.
Special Case 1:
If the number ends in zero, use the last non-zero number as the last number.
Here, you have to ignore the 0 in 4250 and use this formula for 425, i.e. subtract 4 from 9 (= 5), subtract 2 from 9 (= 7) and subtract 5 from 10 ( = 5).
For the last digit 0, 0 – 0 = 0.
Special Case 2:
In all of the above sums, you may have noticed that the number of zeros in the first number is the same as the number of digits in the number being subtracted.
For example 1000–481 has three zeros and 481 has three digits which is same as the number of zeroes in 1000.
But how to subtract 10000 – 425, where 2nd figure has less digits than number of zeroes in 10000?
In such cases, append zeroes to the start of 2nd figure and then apply this formula.
Special Case 3:
When subtracting from a multiple of a power of 10, just decrement the first digit by 1, then subtract remaining digits.
For example,
Special Case 4:
When you have a problem like 8000 – 4222 where both numbers have the same number of digits, reduce the first digit of the first number by one more than the first digit of the second number to get the first digit of the answer.
And apply the formula to the remaining figures.
Money
A great application of “All from 9 and last from 10” is money. Change can be calculated by applying this sutra mentally!
It is often the case the payment is made with bills only, these are multiples of “10”.
Maths is Fun and Easy 