Division in Vedic Mathematics

There are so many shortcuts for multiplication but hardly any shortcuts for division. Nandeesh has translated a Sanskrit Sutra to reduce long division to one line short-cut. Join me in thanking him for his great efforts.

Long Division reduced to one-line shortcut

Example 1:  716769 ÷ 54.

Reduce the divisor 54 to 5 pushing the remaining digit 4 “on top of the flag” (Dhvajanka so to say).

Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.

Let us walk through the steps of this example:
716769 ÷ 54 = 13273.5


  1. 7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2
  2. The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 – 4 x 1 = 1717 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2
  3. Again subtract the product of the flagged number (4) and the previous quotient (3), 26 – 4 x 3 = 1414 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 4
  4. 47 – 4 x 2 = 3939 ÷ 5 = 7 remainder 4. Put down the 7 and carry over the 4
  5. 46 – 4 x 7 = 1818 ÷ 5 = 3 remainder 3. Put down the 3 and carry over the 3
  6. 39 – 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 2
  7. 20 – 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the division

Example 2:  45026 ÷ 47

Reduce the divisor 47 to 4 pushing the remaining digit 7 “on top of the flag” (Dhvajanka so to say).

Corresponding to the number of digits flagged on top (in this case, one), the rightmost part of the number to be divided is split to mark the placeholder of the decimal point or the remainder portion.

Let us walk through the steps of this example:

45026 ÷ 47 = 958.0

4975
4745026
0 9 5 8 0
  1. 4 ÷ 4 = 0 remainder 4. Put the quotient 0, the first digit of the solution, in the first box of the bottom row and carry over the remainder 4
  2. The product of the flagged number (7) and the previous quotient (0) must be subtracted from the next number (45) before the division can proceed. 45 – 7 x 0 = 4545 ÷ 4 = 9 remainder 9. Put down the quotient 9 and carry over the remainder 9.
  3. Again subtract the product of the flagged number (7) and the previous quotient (9), 90 – 7 x 9 = 2727 ÷ 4 = 5 remainder 7. Put down the quotient 5 and carry over the remainder 7.
  4. 72 – 7 x 5 = 3737 ÷ 4 = 8 remainder 5. Put down the quotient 8 and carry over the remainder 5.
  5. 56 – 7 x 8 = 0there is nothing left to divide, so this cleanly completes the division.

In reply to the query of Pratiush

To divide 716769 by 156:  Split divisor as 15 and 6

———————     11        17       12             12      15        9

15 6               71         6          7         6               9        0        0

—————– 4         5          9         4               6        7        3

Answer:  4594.673

Remarks:  In the first step we have written that 4 ÷ 4 = 0 remainder 4 instead of 4 ÷ 4 = 1 remainder 0. Otherwise, in the following step, we would have to subtract 7*1=7 from 05 which is not possible.

This post is contributed by Nandeesh Nagarajaia. He is a Chemical Engineer who did his B.Tech from NIT Suratkal.  He is now in IT field as Assistant General Manager(Systems) in Hindustan Copper Limited.

Other guest posts by Nandeesh –

  1. Quick Method to evaluate polynomials – Horner’s Method
  2. Quick calculations for extremely large numbers
  3. Shortcut multiplication for approximate numbers
  4. Heron’s method of finding roots

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