Ratio and Proprotion
October 8, 2022Multiplication by 2 Digit Numbers – The Mental Math Approach
October 9, 2022In this lesson, you will learn the mental math approach for multiplication by 3 digit numbers. In the previous lesson on two digit multiplication, we looked into the vertical and cross wise technique or the Urdhva Tiryak technique from Vedic Maths. It is the same technique that we are going to extend for multiplication by 3 digits.
The aim of this lesson is to help you multiply 3 digit numbers in your head. Mental computation is feasible if you can multiply from left to right, which reduces the overhead involved due to carry. Since the approach to 3 digit multiplication technique explained in this section involves computing from left to right, it can make a significant difference to our mental computation skills. But remember that multiplication of 3 digit numbers from left to right is not fairly easy, as it requires considerable amount of practice. Please use this lesson as a guide to learn the approach and if you would like to practice, you can consider using our App on Math Tricks.
By the end of this post you will learn the following tricks
- 3 digit by 1 digit multiplication
- Multiplying 3 digits by 2 digits
- 3 digit by 3 digit multiplication trick
Multiplication by 3 digit numbers - Video
The approach involved in multiplication by 3 digit numbers is explained in the videos below with the help of two examples. The first example focuses on multiplying three digit numbers without using carry, while the second number deals with carry.
Multiplication by three digits
As explained in the video above, we are going to use the vertical and cross wise technique of Vedic maths to multiply three digit numbers. This technique helps us easily multiply 3 digit numbers mentally. Now let's look into this multiplication trick by categorizing the approach to multiplication by 3 digits into following sections:
- How to multiply two 3 digit numbers
- Multiplying 3 digits by 2 digits
- How to multiply 3 digits by 1 digit
How to multiply two 3 digit numbers
So now, given below is the three digit multiplication approach explained with the help of two examples. The approach illustrated in these examples can be used to multiply any three digit number.
Example 1: Product of 121 and 231
- In order to carry out the multiplication by 3 digits we will divide the solution into 5 parts. The number of parts here is obtained by 2x$n$-1, where $n$ represents the value in n-digit multiplication. In case of 3 digit multiplication n is 3. Hence,
121
x 231
------------------------
? | ? | ? | ? | ?
- Left most part is obtained by multiplying the left most digits 1 in 121 and 2 in 231 => 1x2 = 2.
121
x 231
------------------------
2 | ? | ? | ? | ?
- The second part is obtained by cross multiplying between the first two digits 1 and 3 & 2 and 2, then adding the respective products => 1x3 + 2x2 = 7
121
x 231
------------------------
2 | 7 | ? | ? | ?
- Middle part is obtained by doing a 3 digit cross wise multiplication of 1 and 1, 2 and 3, and 1 and 2, and then adding the respective products => 1x1 + 2x3 + 1x2 = 9
121
x 231
------------------------
2 | 7 | 9 | ? | ?
- The fourth part is obtained by cross multiplying between the last two digits 2 and 1 & 1 and 3, then adding the respective products => 2x1 + 1x3 = 5
121
x 231
------------------------
2 | 7 | 9 | 5 | ?
- Right most part is obtained by multiplying the right most digits in both the numbers => 1x1 = 1
121
x 231
------------------------
2 | 7 | 9 | 5 | 1
- Thus we have the result of multiplication of three digit numbers 121 and 231 is 27951.
Example 2: Product of 415 and 326
- Similar to example 1 lets find the product by dividing the solution into 5 parts:
415
x 326
------------------------
? | ? | ? | ? | ?
- The first part is obtained by taking the product of 4 and 3 => 4x3 = 12
415
x 326
------------------------
12 | ? | ? | ? | ?
- Now, the next part is obtained by doing a cross multiplication between the first two digits 4 and 2 & 1 and 3, then adding the respective products => 4x2 + 1x3 = 11
415
x 326
------------------------
12 | 11 | ? | ? | ?
Note: Except for the left most part, all the other parts must contain only one digit, the excess digit if any will be carried over to the left. Hence , the excess 1 in 11 will be carried over to the left part. Thus:
415
x 326
------------------------
13 | 1 | ? | ? | ?
- Next the third part is obtained by doing a 3 digit cross wise multiplication of 4 and 6, 1 and 2, and 5 and 3, and adding the respective products => 4x6 + 1x2 + 5x3 = 41
415
x 326
------------------------
13 | 1 | 41 | ? | ?
=> 13 | 5 | 1 | ? | ?
- Now, the fourth part is calculated by cross multiplying between the last two digits 1 and 6 & 5 and 2, then adding the respective products => 1x6 + 5x2 = 16
415
x 326
------------------------
13 | 1 | 41 | 16 | ?
=> 13 | 5 | 2 | 6 | ?
- Last part is obtained by multiplying the right most digits in both the numbers => 5x6 = 30
416
x 326
------------------------
13 | 5 | 2 | 6 | 30
= 13 | 5 | 2 | 9 | 0
- Thus we have calculated the 3 digit multiplication of 416 and 326 using the vertical and cross wise technique from left to right without the use of calculator.
Multiplying Three digits by Two digits
So now, given below is the three digit multiplication approach explained with the help of two examples. The approach illustrated in these examples can be used to multiply any three digit number.
Example 1: Product of 121 and 231
- In order to carry out the multiplication by 3 digits we will divide the solution into 5 parts. The number of parts here is obtained by 2x$n$-1, where $n$ represents the value in n-digit multiplication. In case of 3 digit multiplication n is 3. Hence,
121
x 231
------------------------
? | ? | ? | ? | ?
- Left most part is obtained by multiplying the left most digits 1 in 121 and 2 in 231 => 1x2 = 2.
121
x 231
------------------------
2 | ? | ? | ? | ?
- The second part is obtained by cross multiplying between the first two digits 1 and 3 & 2 and 2, then adding the respective products => 1x3 + 2x2 = 7
121
x 231
------------------------
2 | 7 | ? | ? | ?
- Middle part is obtained by doing a 3 digit cross wise multiplication of 1 and 1, 2 and 3, and 1 and 2, and then adding the respective products => 1x1 + 2x3 + 1x2 = 9
121
x 231
------------------------
2 | 7 | 9 | ? | ?
- The fourth part is obtained by cross multiplying between the last two digits 2 and 1 & 1 and 3, then adding the respective products => 2x1 + 1x3 = 5
121
x 231
------------------------
2 | 7 | 9 | 5 | ?
- Right most part is obtained by multiplying the right most digits in both the numbers => 1x1 = 1
121
x 231
------------------------
2 | 7 | 9 | 5 | 1
- Thus we have the result of multiplication of three digit numbers 121 and 231 is 27951.
Example 2: Product of 415 and 326
- Similar to example 1 lets find the product by dividing the solution into 5 parts:
415
x 326
------------------------
? | ? | ? | ? | ?
- The first part is obtained by taking the product of 4 and 3 => 4x3 = 12
415
x 326
------------------------
12 | ? | ? | ? | ?
- Now, the next part is obtained by doing a cross multiplication between the first two digits 4 and 2 & 1 and 3, then adding the respective products => 4x2 + 1x3 = 11
415
x 326
------------------------
12 | 11 | ? | ? | ?
Note: Except for the left most part, all the other parts must contain only one digit, the excess digit if any will be carried over to the left. Hence , the excess 1 in 11 will be carried over to the left part. Thus:
415
x 326
------------------------
13 | 1 | ? | ? | ?
- Next the third part is obtained by doing a 3 digit cross wise multiplication of 4 and 6, 1 and 2, and 5 and 3, and adding the respective products => 4x6 + 1x2 + 5x3 = 41
415
x 326
------------------------
13 | 1 | 41 | ? | ?
=> 13 | 5 | 1 | ? | ?
- Now, the fourth part is calculated by cross multiplying between the last two digits 1 and 6 & 5 and 2, then adding the respective products => 1x6 + 5x2 = 16
415
x 326
------------------------
13 | 1 | 41 | 16 | ?
=> 13 | 5 | 2 | 6 | ?
- Last part is obtained by multiplying the right most digits in both the numbers => 5x6 = 30
416
x 326
------------------------
13 | 5 | 2 | 6 | 30
= 13 | 5 | 2 | 9 | 0
- Thus we have calculated the 3 digit multiplication of 416 and 326 using the vertical and cross wise technique from left to right without the use of calculator.
Three digit by One digit Multiplication Trick
So now, given below is the three digit multiplication approach explained with the help of two examples. The approach illustrated in these examples can be used to multiply any three digit number.
Example 1: Product of 121 and 231
- In order to carry out the multiplication by 3 digits we will divide the solution into 5 parts. The number of parts here is obtained by 2x$n$-1, where $n$ represents the value in n-digit multiplication. In case of 3 digit multiplication n is 3. Hence,
121
x 231
------------------------
? | ? | ? | ? | ?
- Left most part is obtained by multiplying the left most digits 1 in 121 and 2 in 231 => 1x2 = 2.
121
x 231
------------------------
2 | ? | ? | ? | ?
- The second part is obtained by cross multiplying between the first two digits 1 and 3 & 2 and 2, then adding the respective products => 1x3 + 2x2 = 7
121
x 231
------------------------
2 | 7 | ? | ? | ?
- Middle part is obtained by doing a 3 digit cross wise multiplication of 1 and 1, 2 and 3, and 1 and 2, and then adding the respective products => 1x1 + 2x3 + 1x2 = 9
121
x 231
------------------------
2 | 7 | 9 | ? | ?
- The fourth part is obtained by cross multiplying between the last two digits 2 and 1 & 1 and 3, then adding the respective products => 2x1 + 1x3 = 5
121
x 231
------------------------
2 | 7 | 9 | 5 | ?
- Right most part is obtained by multiplying the right most digits in both the numbers => 1x1 = 1
121
x 231
------------------------
2 | 7 | 9 | 5 | 1
- Thus we have the result of multiplication of three digit numbers 121 and 231 is 27951.
Example 2: Product of 415 and 326
- Similar to example 1 lets find the product by dividing the solution into 5 parts:
415
x 326
------------------------
? | ? | ? | ? | ?
- The first part is obtained by taking the product of 4 and 3 => 4x3 = 12
415
x 326
------------------------
12 | ? | ? | ? | ?
- Now, the next part is obtained by doing a cross multiplication between the first two digits 4 and 2 & 1 and 3, then adding the respective products => 4x2 + 1x3 = 11
415
x 326
------------------------
12 | 11 | ? | ? | ?
Note: Except for the left most part, all the other parts must contain only one digit, the excess digit if any will be carried over to the left. Hence , the excess 1 in 11 will be carried over to the left part. Thus:
415
x 326
------------------------
13 | 1 | ? | ? | ?
- Next the third part is obtained by doing a 3 digit cross wise multiplication of 4 and 6, 1 and 2, and 5 and 3, and adding the respective products => 4x6 + 1x2 + 5x3 = 41
415
x 326
------------------------
13 | 1 | 41 | ? | ?
=> 13 | 5 | 1 | ? | ?
- Now, the fourth part is calculated by cross multiplying between the last two digits 1 and 6 & 5 and 2, then adding the respective products => 1x6 + 5x2 = 16
415
x 326
------------------------
13 | 1 | 41 | 16 | ?
=> 13 | 5 | 2 | 6 | ?
- Last part is obtained by multiplying the right most digits in both the numbers => 5x6 = 30
416
x 326
------------------------
13 | 5 | 2 | 6 | 30
= 13 | 5 | 2 | 9 | 0
- Thus we have calculated the 3 digit multiplication of 416 and 326 using the vertical and cross wise technique from left to right without the use of calculator.
What Next?
As you can see in this lesson, a three digit multiplication is a simple extension of the two digit multiplication trick. This trick requires a considerable amount of practice. But once mastered, you would be amazed with how good a mental workout it would turn out to be. For practise, you can start with 3 digit by 1 digit multiplication from left to right which will help you practise calculating carry mentally and then start with 3 digit by 3 digit multiplication.
Remember, you won't be able to master left to multiplication over night but by putting in a considerable amount of effort you will definitely fall in love with this technique. Now take the quiz below and the exercises associated with this lesson to get started.
