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Division (Vedic Mathematics)

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1. Division by 9

Division by 9 is derived from the second sutra of Vedic Mathematics, known as the ‘Nikhilam sutra’.
Watch the following video to understand the method.
Solved Example
Example 1:
Divide 1241 by 9
Solution:

Q = 137, R = 8
In this case, the remainder is less than 9. But what will happen if the remainder is greater than 9?
To understand this case, watch the following video.
Solved Examples
Example 1:
Divide 28491 by 9
Solution:

Q = 3163
R = 24 = 6 + 2 × 9
∴Qnew = 3163 + 2 = 3165
Rnew = 6
Example 2:
Divide 92864 by 9
Solution:

Q = 10315
R = 29 = 2 + 3 × 9
∴Qnew = 10315 + 3 = 10318
Rnew = 2

2. Division Method - I

The contemporary method of division is a very long process, in which, there are chances of making errors and is also very time consuming. However, the Vedic method of division is very useful in dividing numbers.
Look at the following video to understand the method:
Solved Examples
Example 1:
Divide 10042 by 89
Solution:
Base = 100
Deficit = 100 − 89 = 11

Q = 112
R = 74
Example 2:
Divide 101010 by 8998
Solution:
Base = 10000
Deficit = 10000 − 8998 = 1002

Q = 11
R = 2032
Example 3:
Divide 10102 by 899
Solution:
Base = 1000
Deficit = 1000 − 899 = 101

Q = 11
R = 213
In all the above cases, the remainder was always less than the divisor. How to solve a problem, where the remainder turns out to be more than the divisor?
Look at the following video to understand the method:
Solved Examples
Example 1:
Divide 11007 by 88
Solution:
Base = 100
Deficit = 100 − 88 = 12

Q = 124
R = 95 = 7 + 1 × 88
∴Rnew = 7
Qnew = 124 + 1 = 125
Example 2:
Divide 15004 by 89
Solution:
Base = 100
Deficit = 100 − 89 = 11

Q = 167
R = 141 = 52 + 1 × 89
∴Rnew = 52
Qnew = 167 + 1 = 168
Note:
This method is very useful when the divisor is close to the base, that is, close to 10n.

3. Division Method - II

This method of division is derived from the 5th sutra of Vedic Mathematics, known as the “Parvartya sutra”, which means “transpose and apply”.
Transpose generally refers to the transpose of operators, i.e., “+” becomes “−”, “−” becomes “+”, “÷” becomes “×” and “×” becomes “÷”, and can be used in cases where the first digit of the divisor is 1. This Vedic method of division uses the rule of transpose of operators.
To learn this method, watch the following video.
Solved Examples
Example 1:
Divide 1232 by 110
Solution:

Q = 11
R = 22
Example 2:
Divide 1386 by 113
Solution:

Q = 12
R = 30
In the above examples, the remainder obtained after calculation came out to be less than the divisor. But what will happen if the remainder comes out to be greater than the divisor?
Let us understand this case with the help of following video.
Solved Examples
Example 1:
Divide 18428 by 161
Solution:

Q = = 11(10 − 9) = 111
R = 557 = 74 + 3 × 161
∴Qnew = 111 + 3 = 114
Rnew = 74
Example 2:
Divide 1221 by 192
Solution:

Q = = 0(10 − 7) = 3
R = 645 = 69 + 3 × 192
∴Qnew = 3 + 3 = 6
Rnew = 69
Let us now take another case. To get an insight into it, try dividing 13456 by 1123.

In this case, the quotient comes out to be 12, while the remainder comes out to be −20. However, can a remainder be negative? No. In fact, in such a case, one has to perform an additional step in.
Watch the following video to learn this case.
Solved Examples
Example 1:
Divide 13905 by 115
Solution:

Q = 122
R = = −125
Qnew = 122 − 1 = 121
Rnew = 115 − 125 = −10
Since Rnew is negative, the process is repeated once.
∴Qnew = 121 − 1 = 120
Rnew = 115 − 10 = 105
Example 2:
Divide 13456 by 1122
Solution:

Q = 12
R = = −10 + 2 = −8
∴ Qnew = 12 − 1 = 11
Rnew = 1122 − 8 = 1114
Till now, the concept involved division of a number where the first digit of the divisor was equal to 1. However, what will happen if the first digit of the divisor is not 1?
Look at the following video to understand the method.
Solved Examples
Example 1:
Divide 15224 by 224
Solution:

Q = = 13(10 − 4) = 136
R = 12 = −20 + 12 = −8
∴Qnew = 136 − 1 = 135
Rnew = 112 − 8 = 104
Since the leftmost digit of the divisor is 2, the quotient also needs to be divided by 2.
∴Qnew = 135 ÷ 2 = 67.5 and Rnew = 104
⇒ Qnew = 67.5 − 0.5 = 67 and Rnew = 104 + 0.5 × 224 = 104 + 112 = 216
Example 2:
Divide 19212 by 484
Solution:

Q =
R = 215 > 121
∴Rnew = 215 − 121 = 94
Qnew = 157 + 1 = 158
Since the leftmost digit of the divisor is 4, the quotient also needs to be divided by 4.
∴Qnew = 158 ÷ 4 = 39.5 and Rnew = 94
⇒ Qnew = 39.5 − 0.5 = 39 and Rnew = 94 + 0.5 × 484 = 94 + 242 = 336
Note:
This division method is very useful when the first digit of the divisor is 1.

4. Straight Division

Straight division is the most general method of division provided by Vedic Mathematics. It is based on the “Urdhva-tiryak” sutra, which translates into English as “vertical and cross-wise”.
This method can be applied to all cases of division, which are difficult to solve by any other method.
This method can be better understood by dividing a number by a 2-digit divisor.
Look at the following video to understand the method of straight division:
Solved Examples
Example 1:
Divide 1942 by 42
Solution:

Step I:
1 ÷ 4 gives Q = 0, R = 1
Step II:
Gross dividend = 19
Actual dividend = 19 − 2 × 0 = 19 − 0 = 19
19 ÷ 4 gives Q = 4, R = 3
Step III:
Gross dividend = 34
Actual dividend = 34 − 2 × 4 = 34 − 8 = 26
26 ÷ 4 gives Q = 6, R = 2
Step IV:
Gross dividend = 22
Actual dividend = 22 − 2 × 6 = 22 − 12 = 10
This actual dividend is the remainder.
⇒ R = 10
Q = 46
The above method can be applied for dividing any number by a 2-digit divisor. Often, we need to calculate division up to the required decimal places. This is a major application of division under Vedic Mathematics, and hence, calculating division up to the required decimal place becomes as easy as calculating division normally.
Watch the following video to learn the method.
Solved Example
Example 1:
Divide 7453 by 79 up to two decimal places.
Solution:

Step I:
7 ÷ 7 gives Q = 1, R = 0
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 0, R = 7 is taken.
Step II:
Gross dividend = 74
Actual dividend = 74 − 9 × 0 = 74 − 0 = 74
74 ÷ 9 gives Q = 10, R = 4
However, if R = 4, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 9, R = 11 is taken.
Step III:
Gross dividend = 115
Actual dividend = 115 − 9 × 9 = 115 − 81 = 34
34 ÷ 7 gives Q = 4, R = 6
Step IV:
Gross dividend = 63
Actual dividend = 63 − 9 × 4 = 63 − 36 = 27
27 ÷ 7 gives Q = 3, R = 6
Step V:
Gross dividend = 60
Actual dividend = 60 − 9 × 3 = 60 − 27 = 33
33 ÷ 7 gives Q = 4, R = 5
∴7453 ÷ 79 = 94.34
The division method for divisors with more than two digits can now be generalized. This involves the application of the “Urdhva-tiryak” sutra, which means “vertical and cross-wise”.
Go through this video to understand this concept.
Solved Example
Example 1:
Divide 15824 by 331
Solution:

Step I:
Gross dividend = Actual dividend = 15
15 ÷ 3 gives Q = 5 and R = 0
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 4, R = 3 is taken.
Step II:
Gross dividend = 38
Cross-product of 04 and 31 = 0 × 1 + 3 × 4 = 0 + 12 = 12
Actual dividend = 38 − 12 = 26
26 ÷ 3 gives Q = 8 and R = 2
However, if R = 2, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 7, R = 5 is taken.
Step III:
Gross dividend = 52
Cross-product of 47 and 31 = 4 × 1 + 3 × 7 = 4 + 21 = 25
Actual dividend = 52 − 25 = 27
27 ÷ 3 gives Q = 9 and R = 0
However, if R = 0, then in the next step, the actual dividend comes out to be negative.
Hence, Q = 8, R = 3 is taken.
Step IV:
Gross dividend = 34
Cross-product of 031 and 478 = 0 × 8 + 4 × 1 + 7 × 3 = 0 + 4 + 21 = 25
Actual dividend = 34 − 25 = 9
9 ÷ 3 gives Q = 3 and R = 0
Steps to find remainder:
The last two digits of the quotient and the flag are 47 and 31 respectively.
Cross-product of 47 and 31 = 4 × 1 + 3 × 7 = 4 + 21 = 25
Product of cross-product and 10 = 25 × 10 = 250 … (1)
The last digits of the quotient and the flag are 7 and 1 respectively.
Product of these last digits = 7 × 1 = 7 … (2)
(1) + (2) = 250 + 7 = 257 … (3)

⇒ R = 524 − 257 = 267
∴Q = 47, R = 267
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