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How to Use "Casting Out Nines" Faster

By
Jason Marshall, PhD,
November 29, 2011

Want to get even faster at using the "casting out nines" technique for checking your arithmetic that we've been talking about in the Math Dude podcast? Of course you do. But before we learn how, brush up on using casting out nines to check addition and multiplication problems.

How can you speed up casting out nines? Just learn to calculate check digits (more technically known as digit sums) faster. After all, calculating check digits is the part of casting out nines that takes a lot of time. So learning to do that faster will speed up the whole process! Here's how it works...

It’s easiest to see how to do this with an example. Specifically, let’s say you’re trying to find the check digit of the number 525,829. One way to do this is to add up all the digits from left to right, like this: 5+2+5+8+2+9. So that’s

  • 5+2=7, then …
  • 7+5=12, then …
  • 12+8=20, then …
  • 20+2=22, then …
  • 22+9=31, and finally …
  • turn this into a check digit by finding that 3+1=4.

While this works just fine, the numbers we're keeping track of can start to get kind of big. How can we avoid that? The trick is to know that you don’t have to add up all the numbers from left to right before finding a check digit. In fact, you can find check digits all along the way. And doing so will help keep the numbers you're keeping track of small. For example, instead of adding the digits of 525,829 from left to right, let's try this:

  • 5+2=7, then …
  • 7+5=12 … which has a check digit of 3, then we’re back to …
  • 3+8=11 … which has a check digit of 2, then …
  • 2+2=4, and finally …
  • 4+9=13 … which has a check digit of 4.

Of course, the answer had better come out the same either way. But the beauty of this second method is that we're always dealing with small numbers (even when dealing with huge numbers!), so it’s almost always going to be a lot faster.

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