Casting Out Nines
When you solve a sum whether it is
addition, subtraction, multiplication or division, a doubt constantly haunts
you. To avoid any mistake you begin to check every step and thus lose your
precious time. Don’t you think that there should be a single method which is
equally beneficial in all the basic mathematical operations? The answer to your
question is YES, there is a method to check all the four fundamental
mathematical operation i.e. addition, subtraction, multiplication or division
and that too in few seconds. This wonder method is called Casting out Nines or Nines-
Remainder Method.
How
does this method work?
This is a big question. But you will
be surprised to know that this method is so simple that even a primary school
going kid can understand and check his/her calculation.
Casting
out Nines literally means to throw nines. Now let us focus on its working.
v Add the digits
of a number across, dropping out 9, to get a single figure. If it is not a
single figure, add the digits obtained so as to get a single figure
between 0 to 8.
v 9 is not taken
into account in this process, as a digit sum of 9 is the same as a digit sum of
zero.
Before coming
to the actual problem, shouldn’t we understand the basic meaning of above two
points? Let us take some examples to understand what meaning the above line
has.
Example: Find
the digit sum of 54653
Verification:
Digit sum of 54653= 5 + 4 + 6 + 5 +3 = 23
Since 23 is a double figure number so to get a single figure, we have to
sum it again.
Digit sum of 23 = 2 + 3 = 5
The digit sum
of 54653 can be done in other way very easily. As discussed above, we need not
take 9 into account
Digit sum of 54653 = 5 + 4 + 6 + 5 + 3
The two group
of number 5+ 4 and 6+ 3 can easily be left out while finding the digit sum of
54653, as their sum is equal to 9.
Example: Find
the digit sum of 438219
Verification:
Add all the digits
4 + 3 + 8 + 2 + 1 + 9 =
27
Digit sum of 27 = 2 + 7 = 9
Check for
Addition
Whatever we do
to the numbers, we also do to their digit sum; then the result that we have
from the digit sum of the numbers, must be equal to the digit sum of answer.
Example:
Verify 47385 + 69384 + 58769+ 38173 + 29464 = 243144
Verification: Digit
sum of number
47385 0
69384 3
58769 8
38173 4
+ 29464 3
243144
?
LH.S.
= Digit sum of 243144 = 2 + 4 + 3 + 1 + 4 + 4 = 0
R.H.S.
= Sum of the Digit sum of number = 0 + 3 + 8 + 4 + 3 = 0
Since
LHS = RHS
Result
Verified
Example:
Verify 87643 + 84397+ 38549 + 29765 =
240354
Verification
Digit
sum of number
87643 1
84397 4
38549 2
29765 2
240354 ?
Check for subtraction
The process of subtraction is same as
applied in addition. Remember that the value of the digit sum of minuend should
be greater than that of subtrahend.
Example: Verify the result, 8934 –
6758 = 2176
Verification: Digit sum of number
8934 6
- 6758 8
2176 ?
Since the digit sum of Minuend is less
than the digit sum of Subtrahend, hence we need to replace the digit sum of
8934, in such a way that the final digit sum remains the same.
Here the value of the digit sum of
8934 = 6
This digit sum 6 can also be written
in so many ways as the digit sum of 15, 24, 33, 42, 51 and 60 also gives the
same value 6. Therefore, the need of the hour is to replace the digit sum of
8934 with any of the given value 15, 24, 33,42,51,60.
Digit
sum of number
8934 15 or 24
or 33 or 42 or 51 or 60
- 6758 8
2176 ?
LHS = Digit sum of 2176 = 2 + 1 + 7+ 6
= 7
RHS = Digit sum of (15 – 8) = 7
Or, Digit sum of (24 – 8 = 16) = 1 +
6 = 7
Or, Digit sum of (33 – 8 = 25) = 2 +
5 = 7
Or, Digit sum of (42 – 8 = 34) = 3 +
4 = 7
Or, Digit sum of (51 – 8 = 43) = 4 +
3 = 7
Or, Digit sum of (60 – 8 = 52) = 5+ 2
= 7
In all the above cases
LHS = RHS
LH.S.
= Digit sum of 240354 = 2 + 4 + 0 + 3 + 5 + 4 = 0
R.H.S.
= Sum of the Digit sum of number = 1 + 4 + 2 + 2 = 0
Check
for Multiplication
Multiplication is the most error prone
fundamental operation in mathematics. Students always have doubts about the
accuracy of their result and waste time in re-checking every operation again
and again. This method will prove a panacea for all those who are not very much
sure about their result. Let us take few examples to understand the modus
operandi of this method.
We know,
Multiplicand X Multiplier = Product
Example: Verify 12876 x 43853 =
564651228
Verification:-
Digit sum of Multiplicand = 1 + 2 + 8 + 7 + 6
= 6
Digit sum of Multipliers =4 + 3 + 8 + 5 + 3 =
5
LHS
= Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS=
Digit sum of Product =5 + 6 + 4 + 6 + 5+ 1 + 2 + 2 + 8 = 3
Since
LHS = RHS
Result
Verified
Example: Verify 5972 X 4853 = 29882116
Verification:-
Digit sum of Multiplicand = 5 + 9 + 7 + 2 = 5
Digit sum of Multipliers = 4 + 8+ 5 + 3 = 2
LHS
= Digit sum of Multiplicand and Multiplier taken together (5 X 2= 10)= 1
RHS= Digit sum of Product = 2 + 9 + 8 +
8 + 2 + 1 + 1 + 6 = 1
Since
LHS = RHS
Result
Verified
Example: Verify 12589476 x 43256853 =
54058115267908
Verification:-
Digit sum of Multiplicand =1 + 2 + 5 + 8 + 9 +
4+ 7 + 6 = 6
Digit sum of Multipliers =4 + 3 +2 + 5+ 6+ 8 +
5 + 3 = 5
LHS
= Digit sum of Multiplicand and Multiplier taken together (6 x 5= 30) = 3 RHS=
Digit sum of Product = 5+ 4+ 0+ 5+ 8+1+1+ 5+ 2+ 6+ 7+ 9+ 0+ 8 = 7
Since
LHS ≠ RHS
Result
Incorrect
Check for Division
We
Know
Dividend = Divisor x Quotient +
Remainder
The casting out nines method described
in the very beginning will suffice to check the division operation effectively.
You have only one thing to do-
Find the digit sum of Dividend,
Divisor, Quotient and Remainder and put the value of digits sum in LHS and RHS.
If the same digit sum is obtained in both the side, it ultimately tells you
that you have performed the right operation. Let us take some examples to understand
how effectively this method work for Division.
Example: Verify 876543 ÷ 123, Q = 7126
and R = 45
Verification:-
Here Dividend
= 876543
Digit sum of
Dividend = 8 + 7 + 6 + 5 + 4 + 3 = 6
Divisor = 123
Digit sum of
Divisor = 1 + 2 + 3 = 6
Quotient =
7126
Digit sum of
quotient = 7 + 1 + 2 + 6= 7
Remainder = 45
Digit sum of
Remainder = 4 + 5 = 0
Putting the digit sum value in the
given formulae, we get,
L.H.S = Digit sum of Dividend = 6
R.H.S = Divisor x Quotient + Remainder
= 6 X 7 + 0 = 42
Digit sum of 42 = 6
Hence LHS = RHS
Result Verified
Example: Verify 8765 ÷ 243, Q = 36 and
R = 23
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