Checking of Calculations: Casting Out Nines

While doing arithmetic calculations, we should normally check our calculation. But the checking should not be as tedious as the original problem. To solve this problem I am explaining below a very frequently used method which is discussed in Vedic Mathematics as well as by many other mathematicians.

Vedic Sutra: Vedic Mathematics Technique

Beejank: The Sum of the digits of a number is called Beejank. If the addition is a two digit number, then these two digits are also to be added up to get a single digit.

Find the Beejank of 632174.

As above we have to follow

632174  –> 6 + 3 + 2 + 1 + 7 + 4 –> 23 –> 2 + 3 –> 5

But a quick look gives 6 & 3 ; 2 & 7 are to be ignored because 6+3=9,2+7=9.

Hence remaining 1 + 4 –> 5 is the beejank of 632174.

Checking of Addition

Thumb Rule: Whatever we do to the number, we also do to their digit sum: then the result                 we get from the digit sum of the number must be equal to the digit sum of the answer.

For example: The number: 12+45+96+75+25 =253

The digit sum = 3+9+6+3+7 =28=10=1

Answer’s digit sum: 2+5+3 =10=1 (verified)

Another example:  3.5+23.4+17.5 = 44.4

The digit sum: 8+9+13=8+9+4=21=3

Answer’s digit sum: 12=3 (verified)

Casting Out Nines

This method is also known as “casting-out-nines“. The method involves converting each number into its “casting-out-nines” equivalent, and then redoing the arithmetic. The casting-out-nines answer should equal the casting-out-nines version of the original answer. Below are examples for using casting out nines to check addition.

We get the casting-out-nines equivalent of a number by adding up its digits, and then adding up those digits, until you get a one digit number. If our answer is 9, then that becomes 0. As a short cut, we don’t have to add in any of the 9’s in our work, as these are the equivalent of 0. We can just “cast out” those 9’s. For example, 19 becomes 1, without even adding 1 and 9 and getting 10, and then adding 1 and 0 and getting 1. As a further short cut, we can group numbers together which add up to 9, and replace them with 0. 2974 becomes 4, because we can cast out the 9 and the 2+7 (which is also 9 or 0). Well, let’s try an arithmetic problem:

137892     3

+ 92743   + 7

——    —

230635     1

3+7=10, casting out 9 we get 1.

This rule is also applicable to subtraction, multiplication and up to some extent to division also

In the next post I will explain the use of this method for all of them.

Concept: CHECKING OF CALCULATIONS
Beejank: The Sum of the digits of a number is called Beejank. If the addition is a two digit number, then these two digits are also to be added up to get a single digit.
Find the Beejank of 632174.
As above we have to follow
632174  –> 6 + 3 + 2 + 1 + 7 + 4 –> 23 –> 2 + 3 –> 5
But a quick look gives 6 & 3 ; 2 & 7 are to be ignored because 6+3=9,2+7=9.
Hence remaining 1 + 4 –> 5 is the beejank of 632174.
Checking of Addition
Thumb Rule: Whatever we do to the number, we also do to their digit sum: then the result                 we get from the digit sum of the number must be equal to the digit sum of the answer.
For example: The number: 12+45+96+75+25 =253
The digit sum = 3+9+6+3+7 =28=10=1
Answer’s digit sum: 2+5+3 =10=1 (verified)
Another example:  3.5+23.4+17.5 = 44.4
The digit sum: 8+9+13=8+9+4=21=3
Answer’s digit sum: 12=3 (verified)
This method is also known as “casting-out-nines”. The method involves converting each number into its “casting-out-nines” equivalent, and then redoing the arithmetic. The casting-out-nines answer should equal the casting-out-nines version of the original answer. Below are examples for using casting out nines to check addition.
We get the casting-out-nines equivalent of a number by adding up its digits, and then adding up those digits, until you get a one digit number. If our answer is 9, then that becomes 0. As a short cut, we don’t have to add in any of the 9’s in our work, as these are the equivalent of 0. We can just “cast out” those 9’s. For example, 19 becomes 1, without even adding 1 and 9 and getting 10, and then adding 1 and 0 and getting 1. As a further short cut, we can group numbers together which add up to 9, and replace them with 0. 2974 becomes 4, because we can cast out the 9 and the 2+7 (which is also 9 or 0). Well, let’s try an arithmetic problem:
137892     3
+ 92743   + 7
——    —
230635     1
3+7=10, casting out 9 we get 1.
This rule is also applicable to subtraction, multiplication and up to some extent to division also
In the next post I will explain the use of this method for all of them.
Concept: CHECKING OF CALCULATIONS
Beejank: The Sum of the digits of a number is called Beejank. If the addition is a two digit number, then these two digits are also to be added up to get a single digit.
Find the Beejank of 632174.
As above we have to follow
632174  –> 6 + 3 + 2 + 1 + 7 + 4 –> 23 –> 2 + 3 –> 5
But a quick look gives 6 & 3 ; 2 & 7 are to be ignored because 6+3=9,2+7=9.
Hence remaining 1 + 4 –> 5 is the beejank of 632174.
Checking of Addition
Thumb Rule: Whatever we do to the number, we also do to their digit sum: then the result                 we get from the digit sum of the number must be equal to the digit sum of the answer.
For example: The number: 12+45+96+75+25 =253
The digit sum = 3+9+6+3+7 =28=10=1
Answer’s digit sum: 2+5+3 =10=1 (verified)
Another example:  3.5+23.4+17.5 = 44.4
The digit sum: 8+9+13=8+9+4=21=3
Answer’s digit sum: 12=3 (verified)
This method is also known as “casting-out-nines”. The method involves converting each number into its “casting-out-nines” equivalent, and then redoing the arithmetic. The casting-out-nines answer should equal the casting-out-nines version of the original answer. Below are examples for using casting out nines to check addition.
We get the casting-out-nines equivalent of a number by adding up its digits, and then adding up those digits, until you get a one digit number. If our answer is 9, then that becomes 0. As a short cut, we don’t have to add in any of the 9’s in our work, as these are the equivalent of 0. We can just “cast out” those 9’s. For example, 19 becomes 1, without even adding 1 and 9 and getting 10, and then adding 1 and 0 and getting 1. As a further short cut, we can group numbers together which add up to 9, and replace them with 0. 2974 becomes 4, because we can cast out the 9 and the 2+7 (which is also 9 or 0). Well, let’s try an arithmetic problem:
137892     3
+ 92743   + 7
——    —
230635     1
3+7=10, casting out 9 we get 1.
This rule is also applicable to subtraction, multiplication and up to some extent to division also
In the next post I will explain the use of this method for all of them.

Vineet Patawari

Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!

7 thoughts to “Checking of Calculations: Casting Out Nines”

  1. Gm. Casting Out Nines is very good method to check the Addition , Subtraction , Multiplication quickly ,accurately ! This method mostly can be used for Competitive Exams .

  2. You need to add when checking subtraction! Remember that subtraction is nothing more than addition in REVERSE.

    So to check subtraction, we take the digit sum of the answer,(difference), and add it to the digit sum of the number we subtracted, (subtrahend). If the digit sum of this sum is equal to the digit sum of the original number, (augend), then the subtraction problem is correct.

    Ex: 319 – 278 = 41

    (Think: digit sum of 41 is 5. The digit sum of 278 is 8. Add these to get 13 whose digit sum is 4. The digit sum of 319 is also 4, so the problem is correct.)

    1. sir good things u talked …… when we got number like -5 at that time we have to do 9-5 =4 one thing we have to keep in mind that we have to consider + digital roots

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