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When Casting Out Nines Fails

Date: 11/29/2001 at 21:05:19
From: Teresa Miller
Subject: Casting out 9's, division

My son is doing Abeka 6th grade math. I was intrigued with checking 
math by casting out 9's. I introduced this technique to a friend. 
Just to prove that casting out 9's worked, I intentionally used an 
incorrect answer (quotient). I was shocked when the problem checked 
out correctly. Here is the problem. Divisor 6, Dividend 5223, and 
Quotient 875. Have I done something wrong? I know the correct answer 
is 870 R. 3.

Really stumped,
Teresa Miller

Date: 11/29/2001 at 23:30:30
From: Doctor Peterson
Subject: Re: Casting out 9's, division

Hi, Theresa.

It's not at all surprising that casting out nines would fail; it puts 
all numbers into one of 9 categories, so it will catch an error that 
puts it into one of the 8 wrong categories, but not one that happens 
to land it back in the "right" category. This means there is a 1 in 9 
chance that a random error will look okay. When the method is taught, 
it should always be pointed out that it can tell you if an answer is 
wrong, but can't be trusted to tell you that it is right!

In your example, your check should look something like this:

    Claim: 875 * 6 + 0 = 5223
    Check:  8  * 6 + 0 =? 3
                     3 =? 3

(There are different ways to arrange the work, of course. I calculated 
the digit sums, then calculated the left side as 48 and found its 
digit sum to be 3.)

The correct answer, checked, would be

    Claim: 870 * 6 + 3 = 5223
    Check:  6  * 6 + 3 =? 3
                     3 =? 3

You have added 5 to the quotient, which adds 5*6 (or 3) to the digit 
sum; and you dropped the remainder of 3, which subtracted 3 from the 
digit sum, so you came out even.

In case you are interested, here is my explanation of the method:

   Casting Out Nines to Check Arithmetic   

I checked, and I do explain the pitfalls of the method. Whew!

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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