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All Vedic Math tricks for multiplication

Introduction

Vedic Math is an ancient and effective system of mathematics derived from the Vedic texts of India. This technique helps simplify and speed up mathematical calculations, making it very useful for students and educators today. Many people in Pakistan also learn and use Vedic Math to improve their confidence and skills in mathematics. Knowledge has no nationality, and any method that helps enhance learning should be embraced and shared. Vedic Maths is a super-fast way of making all mathematical calculations easy and gives accurate results. Vedic Math has given us various tricks to solve any type of mathematical problem in just a few minutes.

These formulas are in the form of sutras; under each sutra, we can see the mathematical operations hidden. Starting from addition, subtraction, multiplication, division, decimals, fractions, HCF/LCM, squares, cubes—their roots—along with algebra, trigonometry, geometry, calculus, etc. When compared to our conventional method, the steps might be a little lengthier, but here, the tricks can find solutions in a very simple way. That’s the reason why most of the students are falling for the Vedic Maths Method.

The Vedic Maths

Multiplication is nothing but when two numbers are multiplied, one of them is the multiplicand and the other is the multiplier. Students generally learn multiplication by the conventional method, i.e., we just have the multiplicand and the multiplier and need to multiply given numbers and write the final answer, but under the Vedic Method, for every type of number or question, we have a method to calculate.

Excited to know what they are???

Don’t worry, I will explain it in detail.

All Vedic Math tricks for multiplication

Multiply by 11 – Ekadhikena Purvena(“One more than the previous one.”)   

Steps:

1. Add each pair of adjacent digits and write the sum in between them.
2. If the sum is more than 9, carry over the extra digit.

Example 1: 45 × 11
Add each pair of adjacent digits and write the sum in between them.

=4 + 5 = 9
=495 Ans

Example 2: 72 × 11

Add each pair of adjacent digits and write the sum in between them.
→ 7 + 2 = 9
 = 792 Ans    

Example 3:  Find 345 ×11

Step 1:  Add the digits consecutively: (3+4=7) and (4+5=9).

Step 2:  Place the numbers obtained in step 2 in the middle with the digits

So, the answer is 3795.

Example 3:  Find 345 ×11

Step 1:  Add the digits consecutively: (7+4=11  write 1, carry 1 to 7  7 + 1 = 8)

and (4+5=9).

Step 2:  Place the numbers obtained in step 2 in the middle with the digits

So, the answer is 8195.

In short:

To multiply by 11, just add each digit to its neighbor — quick and easy

General multiplying two-digit numbers – Urdhva-Tiryagbhyam 

General multiplying two-digit numbers – Urdhva-Tiryagbhyam The Vedic Maths: Multiplication Mastery Urdhva-Tiryagbhyam is a vertically and crosswise method to find the product of two numbers. The number of steps required to get the product is one less than the total number of digits of both numbers. We have some problems understanding the method.

1. Multiplication of Two 2-Digit Numbers

To multiply a 2-digit number by another 2-digit number, we need 3 steps.
The following pattern helps to remember the vertical and crosswise method.

Example 1: Find 53 × 47

Step 1: Multiply the digits in the ones’ place (vertically).

3×7=21

Write 1, carry 2 to the next step.

Step 2: Multiply crosswise and add the products.

(5×7)+(3×4)=35+12=47

Add the carry 2 → 47 + 2 = 49
Write 9, carry 4 to the next step.

Step: :3 Multiply the digits in the ten’s place (vertically).

5×4=20

Add the carry 4 → 20 + 4 = 24

Now write all parts together from left to right:

53×47=2491

Example Find 63 × 57

Step 1: Multiply the digits in the place (vertically).

3×7=21

Write 1, carry 2 to the next step.

Step 2: Multiply crosswise and add the products.

(6×7)+(3×5)=42+15=57

Add the carry 2 → 57 + 2 = 59
Write 9, carry 5 to the next step.

Step 3: Multiply the digits in the ten’s place (vertically).

6×5=30

Add the carry 5 → 30 + 5 = 35

Now write all parts together from left to right:

53×47=3591

 2. Multiplication of Two 3-Digit Numbers

To multiply a 3-digit number by another 3-digit number, we need 5 steps.

Example: Find 123 × 214

Step 1: Multiply the digits in the ones’ place (vertically).

  3×4=12

Write 2, carry 1 to the next step.

Step 2: Multiply crosswise using the last two digits of both numbers.

(2×4)+(3×1)=8+3=11

Add carry 1 → 12
Write 2, carry 1 to the next step.

Step:  Multiply and add crosswise for three pairs (all digits interact):

(1×4)+(2×1)+(3×2)=4+2+6=12

Add carry 1 → 13
Write 3, carry 1 to the next step.

Step 4: Multiply the first two digits crosswise (left pair).

(1×1)+(2×2)=1+4=5

Add carry 1 → 6
Write 6.

Step 5: Multiply the hundreds digits vertically.

1×2=2

Now write all digits together (from left to right):

123×214=26322   Ans 

Example: Find 123 × 456

Step 1: Multiply the digits in the ones place (vertically).

  3×6=18

Write 8, carry 1 to the next step.

Step 2: Multiply crosswise using the last two digits of both numbers.

(2×6)+(3×5)=12+15=27

Add carry 1 → 28
Write 8, carry 2 to the next step.

Step:  Multiply and add crosswise for three pairs (all digits interact):

(1×6)+(2×5)+(3×4)=6+10+12=28

Add carry 2 → 30
Write 0, carry 3 to the next step.

Step 4: Multiply the first two digits crosswise (left pair).

(1×5)+(2×4)=5+8=13

Add carry 3 → 16
Write 6. Carry 1 to the next step.

Step 5: Multiply the hundreds digits vertically.

1×4=4

Add carry 1 → 5

Now write all digits together (from left to right):

123×456=56088

Multiplication of a 2-Digit Number by a 3-Digit Number

Example: Multiply 34 × 231

Step 1: Multiply the digits in the place (vertically)

4 × 1 = 4 → Write 4 (no carry)

Step 2: Multiply crosswise using tens and ones digits

(3 × 1) + (4 × 3) = 3 + 12 = 15
Write 5, carry 1 to the next step

Step 3: Multiply the hundreds and tens digits crosswise and add the products

(3 × 3) + (4 × 2) = 9 + 8 = 17
Add carry 1 → 18
Write 8, carry 1 to the next step

Step 4: Multiply the hundreds and tens digits vertically

3 × 2 = 6
Add carry 1 → 7

Step 5: Write all digits together from left to right

34 × 231 = 7854 Ans

Multiplication near base (100, 1000…) – Nikhilam  Base method

Nikhilam Sutra stipulates subtraction of a number from the nearest power of 10, i.e., 10, 100, 1000, etc. The powers of 10 from which the difference is calculated are called the base. These numbers are considered to be references to find out whether the given number is less than or greater than the base.

Nikhilam Sutra in Vedic Mathematics can be used as a shortcut to multiply and divide numbers in a faster approach. In English, it is translated as all from 9 and "last from 10," i.e., subtract the last digit from 10 and the rest of the digits from 9. Multiplication using Nikhilam Sutra is used when numbers are closer to a power of 10, i.e., 10, 100, 1000

Vedic Mathematics is a treasure trove of ancient Indian techniques that simplify complex mathematical operations. Among these, the Nikhilam Sutra, translated as “All from 9 and the Last from 10,” stands out for its elegance and efficiency in handling multiplications.

The number is below the base value
Example: What is 9 x 8?

Solution: Since we can see that the numbers are below the base value.

Base: 10

Deviations: −1 and −2

Step 1: Multiply deviations: (−1)×(−2)=2

Step 2: Add crosswise: 9 + (-2) = 7

Final Answer: 72

The number is above the base value
Example: What is 13 x 14?

Solution: Since we can see that the numbers are above the base value.

Deviations: +3  and +4

Step 1: Multiply deviations: (+3)×(+4)=12, [1 is carry]

Base: 10

Step 2: Add crosswise: 14 + 3 = 17

Add carry 1  17+1=18

Final Answer: 182

The number is above and below the base value
Example: Multiply 104 and 97:

Here you can see that one number is above and the other is below the base value:

Base: 100

Deviations: +4 and −3

Step 1: Multiply deviations: (+4)×(−3)=−12

Step 2: Add crosswise: 104−3=101

Adjust for the negative: We need to put 00  (since the base is 100) after Step 2 and subtract 12 from there. 

So it will be like: 10100- 12 = 10088

Example: One Number Above and One Below
Example: Multiply 102 and 97:

Base: 100
Deviations: +2 and −3
Step 1: Multiply deviations: (+2)×(−3)=−6
Step 2: Add crosswise: 102−3=99
Adjust for the negative: Subtract 6 from 00 (since the base is 100), so it will be like: 9900- 6 = 9894

Example 4: Numbers Near 1000

Multiply 995 and 990:

Base: 1000
Deviations: −5 and −10
Step 1: Multiply deviations: (−5)×(−10)=50
Step 2: Add crosswise: 995−10=985
Final Answer: 985050

Use of proportional multiplication – Anurupyena

This sutra is used when numbers are not close to a simple base (like 10, 100, 1000)
But they are close to a multiple or sub-multiple of a base
(for example, 50 is half of 100, 200 is double of 100, etc.).

It means we adjust the base proportionally and then apply the Nikhilam Method using that new base.

Example 1: 48 × 42

Here, both numbers are near 50,
and 50 is half of 100, so base = 100, and proportion = ½.
48 is 2 less → (–2)
42 is 8 less → (–8)
We must divide the left part by 2.
Right part = 16

Step 1: Write deviations from 50:

Step 2: Cross subtract → 48 – 8 = 40 (or 42 – 2 = 40)

Step 3: Multiply deviations → (–2) × (–8) = +16

Step 4: Since the working base is 50 (which is ½ of 100),

40 ÷ 2 = 20 → Left part = 20

Combine: 20 | 16 → 2016

48 × 42 = 2016 Ans

Example 2: 12 × 13

Here, both are near 10, but closer to 10 × 10 = 100,
so we can take base = 10 and proportion = 10 × 10 automatically.

Step 1: Deviations
12 is +2, 13 is +3

Step 2: Cross add → 12 + 3 = 15
Step 3: Multiply deviations → 2 × 3 = 6
Step 4: Combine → 15 | 6 → 156

12 × 13 = 156

Example 3: 204 × 206

Here, both are near 200,  and 200 is double of 100, so the proportion = 2.

Step 1: Deviations
204 → +4
206 → +6

Step 2: Cross add → 204 + 6 = 210
Step 3: Multiply deviations → 4 × 6 = 24
Step 4: Since base = 200 = 2 × 100,
Multiply the left part (210) by 2 → 420

Combine → 420 | 24 → 42024

204 × 206 = 42024

Multiplication of the given number by 5ⁿ. (5, 25, 125, …)

Step 1: Add as many zeroes at the end of the given number as there are of 5

Step 2: Divide the resultant number by 2^{(Power of 5)}, to get the result.

Example:  Multiply 94 by 125

Solution:

Given: 94 ×125

Here 125 = 5³. The power of 5 is 3.

Step 1: Now, add 3 zeros at the end of 94, and hence it becomes 94000

Step: Now, divide 94000 by 2³. Hence, it becomes 

= 94000/8

= 11750

Therefore, 94 ×125 is 11750

Example:  Multiply 456 by 25

Solution:

Given: 456 ×25

Here 25 = 5². The power of 5 is 2

Step 1: Now, add 2 zeros at the end of 456, and hence it becomes 45600

Step 2: Now, divide 45600 by 2². Hence, it becomes 

= 45600/4

= 11400

Therefore, 456 ×25 is 11400

Multiply by 99,999,9999, etc. – (complement)   

 To multiply given numbersby 9999s..., we have an interesting formula,

Step-1:- By one less than one before

Step-2:- All from nine and last from ten

Example: 67 × 99

Base = 100
Step 1: Subtract 1 from the number → 67 – 1 = 66

Step 2: Subtract the number from 100 → 100 – 67 = 33

Step 3: Write both results together → 66 | 33

67 × 99 = 6633

Example: 254 × 999

Base = 1000

Step 1: Subtract 1 from the number → 254 – 1 = 253
Step 2: Subtract the number from 1000 → 1000 – 254 = 746
Step 3: Write both parts together → 253 | 746

254 × 999 = 253746

Example: 87 × 9999

Base = 10000

Step 1: Subtract 1 from the number → 87 – 1 = 86
Step 2: Subtract the number from 10000 → 10000 – 87 = 9913
Step 3: Combine → 86 | 9913

87 × 9999 = 869913

Conclusion:

We have seen a few of the mathematical techniques available in Vedic Math in this article. There are many more techniques that are helpful in quickly solving 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.