Keywords: Vedic Mathematics Division Tricks, Fast Division Tricks, Shortcut for Division, Mental Math Division, Easy Math Tricks for Students

All Vedic Math tricks for division

Introduction 

Vedic Mathematics presents multiple interesting techniques that can revolutionize our approach to mathematical computations. This system integrates a range of sutras and sub-sutras designed to simplify and speed up arithmetic operations. From rapid multiplication and division to efficient squaring and subtraction, Vedic Mathematics offers methods that enhance mental calculation speed and accuracy.

The main advantage of these techniques is that they help the students spend less time on arithmetic calculations due to their clever tricks, which allows the students to focus more on the logical and reasoning parts of mathematics. 

All Vedic Math tricks for division

                                       

Nikhilam Navatashcaramam Dashatah

(“All from 9 and the last from 10.”)

Nikhilam, or base division, is a Vedic Math technique for quickly performing division when the divisor is slightly less than a power of 10. Instead of using the divisor with its full digits, its ten's complement is employed as a multiplier.

Example 1: Divide 1255 ÷ 99

Step 1: Nearest base = 100

Complement of divisor = 100 – 99 = 1

Step 2: Write the dividend in two parts (since the base has 2 zeros):

12 | 55

Step 3: Bring down the first part = 12; this is the quotient’s first part.

Step 4: Multiply by the complement = 12 × 1 = 12

Step 5: Add this to the second part: 55 + 12 = 67

Quotient = 12 and Remainder = 67

Example 2: Divide 1234 ÷ 98

Step 1: Nearest base = 100

Complement = 100 – 98 = 2

Step 2: Write the dividend in two parts → 12 | 34
Step 3: Bring down 12 (quotient part)
Step 4: Multiply by the complement.  12 × 2 = 24
Step 5: Add to the second part: 34 + 24 = 58

Quotient = 12, Remainder = 58

Example 3: Divide 1004 ÷ 999

Step 1: Nearest base = 1000

Complement = 1000 – 999 = 1

Step 2: Write the dividend in two parts (3 zeros in base): 1 | 004
Step 3: Bring down 1 (quotient part)
Step 4: Multiply 1 × 1 = 1
Step 5: Add to 004 → 004 + 1 = 005

Quotient = 1, Remainder = 5

Example 4: Divide 16256 ÷ 998

Step 1: Nearest base = 1000

Complement = 1000 – 998 = 2

Step 2: Write the dividend in two parts (3 zeros in base): 16 | 256
Step 3: Bring down 16 (quotient part)
Step 4: Multiply by the complement.  16 × 2 = 32
Step 5: Add to 256 → 256+32 = 288

Quotient = 16, Remainder = 288

Example 5: 23 ÷ 9

Step 1: Base = 10

Complement = 10 – 9 = 1
Step 2: Write the number in two parts (2 | 3)
Step 3: Bring down the first part → 2
Step 4: Multiply by complement: 2 × 1 = 2
Step 5: Add to the second part: 3 + 2 = 5

Quotient = 2, Remainder = 5

Example 6: 25 ÷ 11

Step 1: Base = 10

Complement = 10 – 11 = –1

Step 2: Write the number in two parts: 2 | 5
Step 3: Bring down 2 as a quotient.
Step 4: Multiply by the complement 2 × (–1) = –2
Step 5: Add to second part → 5 + (–2) = 3

Quotient = 2, Remainder = 3

Parāvartya Yojayet: (Transpose and Apply)

Paravartya Yojayet is one of the popular sutras of Vedic Maths that presents a new method of solving problems. Also referred to as the paravartya sutra or paravartya yojayet sutra, this technique means “Transpose and Apply.” It presents a special method of transposing one mathematical operation to another, tending to simplify complicated calculations.

What Does “Transpose and Apply” Mean?

In short, to transpose is to swap or change the locations of two things. In math, this might mean changing one operation into another—like changing subtraction into addition in some equations. This concept is behind the paravartya yojayet strategy and makes it a good technique to use when confronted with division and other operations.

Subtraction, for instance, can be transposed into addition in specific formulas.

Transpose and Apply are often utilized when dividing numbers whose divisor is a little more than any power of 10 and begins with 1.

Let us demonstrate this method through an example.

Example 1: Divide 785 by 24.

Solution:

Complement of 24: 9 – 2 = 7, and 9 – 4 = 5.

Therefore, the complement is 75.

Divide 785 by 24: Quotient = 32 (ignore the remainder).

Multiply the quotient by the complement: 32 x 75 = 2400.

Subtract 2400 from 785: 785 – 2400 = -1615 (negative value indicates remainder).

Since we have no more digits in the dividend, the final result is the quotient obtained: 32, and the remainder is -1615.

Therefore, the division of 785 by 24 using the Paravartya Yojayet method gives a quotient of 32 and a remainder of -1615.

Example 2: Divide 648 by 32.

Solution:

Complement of 32:
9 – 3 = 6, and 9 – 2 = 7.

Therefore, the complement is 67.

Divide 648 by 32: Quotient = 20 (ignore the remainder).

Multiply the quotient by the complement:
20 × 67 = 1340.

Subtract 1340 from 648:
648 – 1340 = –692 (negative value indicates remainder).

Since we have no more digits in the dividend, the final result is the quotient obtained: 20, and the remainder is –692.

Therefore, the division of 648 by 32 using the Parāvartya Yojayet method gives a quotient of 20 and a remainder of –692.

Example 3: Divide 972 by 36.

Solution:

Complement of 36:
9 – 3 = 6, and 9 – 6 = 3.

Therefore, the complement is 63.

Divide 972 by 36: Quotient = 27 (ignore the remainder).

Multiply the quotient by the complement:
27 × 63 = 1701.

Subtract 1701 from 972:
972 – 1701 = –729 (negative value indicates remainder).

Since we have no more digits in the dividend, the final result is the quotient obtained: 27, and the remainder is –729.

Therefore, the division of 972 by 36 using the Parāvartya Yojayet method gives a quotient of 27 and a remainder of –729.

Dhvajanka (Flag Digit Method)

Big numbers can be divided fast and easily using the Dhvajanka Sutra. In this method, the divisor is split into two parts:

1. Mukhyanka (Main Part): All digits of the divisor except the last one.

2. Dhvajanka (Flag Digit): The last digit of the divisor.

Step-by-Step Method

Identify Mukhyanka and Dhvajanka.

Example: Divisor = 23 → Mukhyanka = 2, Dhvajanka = 3

Start division using Dhvajanka
Divide the dividend from left to right using the flag digit (Dhvajanka).

Adjust with Mukhyanka
Multiply the quotient by Mukhyanka and subtract from the next part of the dividend.

Repeat
Repeat steps 2 and 3 until all digits of the dividend are used.

Write the final answer

Combine all quotient digits for the final result.

If there is any remainder, write it down.

Example: Divide 3456 ÷ 23

Solution: Identify parts of the divisor:

Divisor = 23

Mukhyanka (Main Part) = 2

Dhvajanka (Flag Digit) = 3

Use Dhvajanka to divide the first part of the dividend:

Dividend first digit/group: 3

Divide by Dhvajanka = 3

Quotient digit = 1

Adjust with Mukhyanka:

1 × 2 = 2

Subtract from first group: 3 − 2 = 1

Bring down the next digit of the dividend → 14

Repeat division:

14 ÷ 3 ≈ 4 (next quotient digit)

Adjust with Mukhyanka: 4 × 2 = 8

Subtract: 14 − 8 = 6

Bring down next digit → 65

Repeat:

65 ÷ 3 ≈ 21 (next quotient digits)

Adjust with Mukhyanka: 21 × 2 = 42

Subtract: 65 − 42 = 23

Final quotient and remainder:

Quotient = 150

Remainder = 6

Vinculum Process

Vinculum means a bar (line) present over the symbol/digit.

English Translation: Complement of a number.

The vinculum process, or vinculum numbers, is the very basis of Vedic mathematics. Vinculum numbers are a concept used in Vedic mathematics and are those numbers that have at least 1 negative digit (having a bar over them). Also called bar numbers.

The Vinculum Process is a method in Vedic Mathematics where we convert numbers into their bar (negative) forms to simplify division. This reduces the steps and makes calculations faster.

Divide 3456 ÷ 23 

Solution: Convert Divisor to Vinculum Form

Divisor = 23

Last digit = 3 → Complement = 10 – 3 = 7

Put a bar over it → 2 7̅

So, the divisor in Vinculum form is 27̅.

Start Division

Take the first part of the dividend = 34

Divide by the main part of the divisor (2) → 34 ÷ 2 = 17

Multiply by the Vinculum digit

17 × 7̅ = 119̅

Subtract 119̅ from 34 using bar subtraction → New remainder = 6

Bring down the next digit

Bring down 5 → 65

Divide 65 by 2 → 65 ÷ 2 = 32

Multiply 32 × 7̅ = 224̅

Adjust using bar subtraction → New remainder = 6

Bring down the last digit

Bring down 6 → 66

Divide 66 by 2 → 66 ÷ 2 = 33

Multiply 33 × 7̅ = 231̅

Adjust using bar subtraction → Final remainder = 6

Combine Quotient

Quotient = 150

Remainder = 6

Shunyam Samyasamuccaye 

(If the sum is the same, it is zero)

This sutra is used in algebraic simplifications and solving equations, especially in cases where terms cancel each other out. "Shunyam" means Zero, "Samya" means Same / Equa,l, and "Samuccaye" means in combination/in sum. So, it essentially tells us: If two equal terms appear on opposite sides or in a sum, their net effect is zero.

Example: Simple numbers

Consider: (x − 5) + (5 − x)
Here, (x − 5) and (5 − x) are opposites.
Their sum: (x − 3) + (3 − x) = 0 
This is Shunyam Samyasamuccaye in action.

Anurupyena (Proportionately adjust dividend/divisor)

In division, Anurupyena helps to simplify calculations by making the divisor a convenient number, usually a power of 10 or an easy factor, and then adjusting the dividend proportionally. 

Basically: Change the divisor to an easier number, and adjust the dividend in the same proportion.

Example 1: Divide 432 ÷ 8

Step 1: Make divisor 8 → 80 (×10)

Step 2: Multiply dividend 432 → 4320 (×10)

Step 3: Divide 4320 ÷ 80 = 54 

Example 2: Divide 225 ÷ 25

Step 1: Make divisor 25 → 100 (×4)

Step 2: Multiply dividend 225 → 900 (×4)

Step 3: Divide 900 ÷ 100 = 9 

Example 3: Divide 360 ÷ 12

Step 1: Make divisor 12 → 120 (×10)

Step 2: Multiply dividend 360 → 3600 (×10)

Step 3: Divide 3600 ÷ 120 = 30 

Conclusion

We have seen a few of the mathematical techniques available in Vedic Math in this article. Many more techniques help quickly solve polynomial equations, differential calculus, etc.  The main advantage of these techniques is that they help the students spend less time on arithmetic calculations due to their clever tricks, which allows the students to focus more on the logical and reasoning parts of mathematics.