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 48 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 39 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 http://mathforum.org/library/drmath/view/55926.html I checked, and I do explain the pitfalls of the method. Whew! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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