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सूत्र १४. एकन्यूनेन पूर्वेण

(Sūtra 14. ekanyūnena pūrvena) - By one less than the one before.
 

The Sūtra: ekanyūnena pūrvena (By one less than the one before) is used to multiply (x × y) a number with repititions of 9 (999...).
 
As an illustration, let us use this Sūtra for:
15 × 99
  Steps 15 × 99
1. Subtract one from the number. This is the first part of the answer. In this case, we get:
15 - 1 = 14
2. Subtract the number from the nearest power of ten (10n), as the second part of the answer. In this case, the nearest 10n to 99 is 100
And, 100 - 15 = 85
3. Join the results to get the answer. In this case, Joining 14 and 85, we get 1485, which is the answer!
So, for a practitioner of Vedic Mathematics, for something like:
24 × 99
  24 - 1 = 23
And, 100 - 24 = 76
 
Thus, 24 × 99 = 2,376
Again, for something like:
356 × 999
  356 - 1 = 355
And, 1000 - 356 = 644
 
Thus, 356 × 999 = 355,644
And, for something like:
878 × 9999
  878 - 1 = 877
And, 10000 - 878 = 9122
 
Thus, 878 × 9999 = 8,779,122
But, why does it work? For this Sūtra (ekanyūnena pūrvena), let us consider the following:
  For any number, x × 99...
= x(10n - 1), where 10n is nearest
= 10nx - x
= 10nx - 10n + 10n - x
= 10n(x - 1) + (10n - x)
 
Note that, since the results are 'joined', 10n is always adjusted as the sum.
 
Lastly, please remember that, as in any other form of mathematics, the mastery of Vedic Mathematics require practice and the judgement of applying the optimal method for a given scenario - a guideline of which, is presented in Applications »
 
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