ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual

 

DIGIT SUM QUICK CHECK - Checking answers

Checking the accuracy of answers in problems of addition, subtraction, multiplication and division often involves doing the problem again for a second or even a third time. This was my method for many years and I always found it tedious. In his book Speed Mathematics Simplified, Edward Stoddard shows us a quick and efficient way to determine if answers are correct. It involves calculating the Digit Sum and Casting out 9s. I found that with a little practice, this check can be performed in a few seconds; even when the problem involves large strings of numbers. This Digit Sum Quick Check is especially well suited to abacus work.

Basically the Quick Check works like this.

CALCULATING THE DIGIT SUM AND CASTING OUT 9s

This is best explained by example. Take the number 139. Using addition techniques as taught by Takashi Kojima, calculate the Digit Sum by adding all three digits together, 1 + 3 + 9 = 13. Then using Kojima's subtraction techniques, cast out the 9s (subtract 9 as many times as you can). In this case 9 can only subtracted once, 13 - 9 = 4. Since no more 9s can be subtracted the Digit Sum of 139 is 4.

In another example take 7849. Calculate the Digit Sum by adding all four digits together, 7 + 8 + 4 + 9 = 28. Cast out the 9s (subtract 9 as many times as you can). In this case, subtract 9 from 28 three times leaving 1. Since no more 9s can be subtracted the Digit Sum of 7849 is 1.

MORE EXAMPLES: Add the digits in each of these numbers and cast out 9s. The result is the Digit Sum.

The Digit Sum of 549 = 0 *
The Digit Sum of 928 = 1
The Digit Sum of 7468 = 7
The Digit Sum of 42702 = 6
The Digit Sum of 332173 = 1

* The technique requires casting out 9s, and 9s reduce to zero.


QUICK CHECKING THE ANSWER

ADDITION

Rule: Add together the Digit Sums of the numbers in the problem; the resulting Digit Sum must equal the Digit Sum of the of the answer.

  Problem      Digit Sum Check

    145              1
  + 382            + 4
  + 478            + 1
   1005              6 

Digit Sum of the answer = 6

 

145 reduces to 1
382 reduces to 4
478 reduces to 1
1 + 4 + 1 = 6

Answer 1005 reduces to 6

 

  Problem      Digit Sum Check

   23,458            4
 + 35,689          + 4
 + 71,288          + 8
  130,435            7

Digit Sum of the answer = 7

 

23,458 reduces to 4
35,689
reduces to 4
71,288 reduces to 8
4 + 4 + 8 = 16 which reduces to 7

Answer
130,435 reduces to 7

 

SUBTRACTION

Rule: Subtract the Digit Sums of the numbers in the problem; the resulting Digit Sum must equal the Digit Sum of the of the answer.

  Problem      Digit Sum Check

   776               2
 - 152            
- 8
*
   624               3

Digit Sum of the answer = 3


 

 

 

776 reduces to 2
152 reduces to 8
(2 + 9*)- 8 = 3

Answer 624 reduces to 3

In this example it's clear that
8 cannot be subtracted from 2.
*Solution: Add 9 to the smaller
number above and subtract normally.

In the above example, an alternative would be to add the numbers starting at the bottom. Add 3 + 8 to equal 11, which would reduce to 2.

 

  Problem      Digit Sum Check

   2,489             5
   - 382           
4
   - 932           
5
*
   1,175             5

Digit Sum of the answer = 5
 

 

 

 

2,489 reduces to 5
382 reduces to 4
932 reduces to 5
(5 - 4)+ 9* - 5 = 5

Answer 1,175 reduces to 5

In this example, it's possible to subtract 
4 from 5 but we cannot subtract 5 from 1.
*Solution: Once again, add 9 to the smaller
number above and subtract normally.

Once again an alternative would be to add the numbers starting at the bottom. Add 5 + 5 + 4 = 14, which reduces to 5.

 

MULTIPLICATION

First the terminology. In the problem  3 x 2 = 6:  the multiplicand is 3, the multiplier is 2 and the product is 6.

Using multiplication techniques as taught by Takashi Kojima, each of these calculations is easily done on a soroban.

Rule: Multiply the Digit Sum of the multiplier x The Digit Sum of the multiplicand; the resulting Digit Sum must equal the Digit Sum of the of the product.

  Problem      Digit Sum Check

    247              4
  x  52           
x 7
 12,844              1

Digit Sum of the product = 1

 


Multiplicand 247 reduces to 4
Multiplier 52 reduces to 7
4 x 7 = 28, reduces to 1

Product 12,844 reduces to 1
 

 

  Problem      Digit Sum Check

    3,875            5
    x 834          x 6
3,231,750            3

Digit Sum of the product = 3

 

Multiplicand 3,875 reduces to 5
Multiplier 834 reduces to 6
5 x 6 = 30, reduces to 3

Product 3,231,750 reduces to 3

 

DIVISION

First the terminology. In the problem  6 3 = 2:  the dividend is 6, the divisor is 3 and the quotient is 2.

Using division techniques as taught by Takashi Kojima, each of these calculations is easily done on a soroban.

Rule: Multiply the Digit Sum of the quotient x the Digit Sum of the divisor; the resulting Digit Sum must equal the Digit Sum of the dividend.

Problem      Digit Sum Check
        
    23             5
7/ 161          
x 7
                   8

Dividend 161 reduces to 8

 

Quotient 23 reduces to 5
Divisor 7 reduces to 7
5 x 7 = 35, reduces to 8

Dividend 161 reduces to 8

 

The following example shows us how to deal with a division problem that involves a remainder.  877 27 = 32  with a remainder of 13.

Rule: Multiply the Digit Sum of the quotient x the Digit Sum of the divisor then add the Digit Sum of the remainder; the resulting Digit Sum must equal the Digit Sum of the dividend.

 

Problem      Digit Sum Check

     32
(R 13)     5
27/ 877         
x 0
                 
  0 + R4
                        4

Dividend 877 reduces to 4

 

Quotient 32 reduces to 5
Divisor 27 reduces to 0
Remainder 13
reduces to 4
(0 x 5) = 0 + R4 = 4

Dividend 877 reduces to 4

This is a great method and it works very well for those who take the time to learn it. But it's not infallible. Mr. Stoddard does caution us to be on the lookout for two instances where the Quick Check can run into problems.

  1. Since 9 reduces to 0, this check will not catch an error in which one digit in an answer is written as 9 when it should have been a 0, or when a 0 should have been a 9.

  2. This Quick Check will not find errors where two numbers have been reversed. For example, if the answer should have been 29 but was written incorrectly as 92, the error will not be found.

In addressing this issue, Mr. Stoddard rightly points out, "...years of experience have shown that the errors not caught by the digit sum are exceedingly rare. For most needs, it is perfectly adequate..."

He goes on to say that, "In return for these shortcomings, the digit-sum check offers a substantial bonus. The digit sum will not only tell you if your answer is wrong; it will tell you by how much it is wrong. If the digit sum of your answer is 4, and you find that it should be 7, then you know that one digit of your answer is too low by exactly 3. You do not know which digit it is, but the fact that one digit is precisely 3 less than it should be is helpful in locating the error quickly."

Link to a Magic Trick that uses DigitSums

 

REFERENCES:

Stoddard, Edward.
Speed Mathematics Simplified
New York: Dover Publication Inc., 1994
Republication of the second printing (1965)
First published by The Dial Press, New York, 1962

Thanks to Edvaldo Siqueira,
Benjamin &  Jeff Kirkland

 

Abacus: Mystery of the Bead
Advanced Abacus Techniques

 November, 2005
Totton Heffelfinger  Toronto Ontario  Canada
Email
totton[at]idirect[dot]com