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Casting out nines - an easy way to check your maths quickly
Your kids need all the help they can get...
... yet it's rare that one of the easiest ways to check your own arithmetic is taught.
Casting out nines is easy. It is a way to quickly do a little checksum operation which will tell you that you have got a sum wrong, or that it is probably right.
Step 1 - Learn how to create a checksum by example.
Give the number 123456, what is the checksum?
Method: Cross out the 6 and the 3 because they add to nine.
... leaving 1245
Now cross out the 4 and the 5 because they add to nine.
... leaving 12
Now there is not enough left to cast out another 9, so just add what's left
... the checksum is 3
You have a go:
Verify the checksums for these numbers:
checksum for 1 is 1
checksum for 3 is 3
checksum for 9 is 0
checksum for 91 is 1
checksum for 712563 is 6
checksum for 77 ? Oh dear - how to do this?
Do it in stages:
Either move 2 from one 7 and give it to the other, leaving 5
add 7 to 7 to give 14, then add 1 to 4 to give a checksum of 5 again.
Either method is fine - do whichever you like at the time.
You can add and cast out, and just cast out or a combination. All that matters is that you ultimately cast out the 9's. Here is a longer example with working:
Just find pairs that add to nine and cross them out. Below are ones in bold that are pairs. If you run out of pairs, use triples.
If you cross out the bold ones all that's left is
And that lonesome 9 gets cast out too leaving a checksum of 0 in this case.
Stage 2 - Checking the answer of addition of two numbers.
Let's say, you want to add 9 and 14. Calculate the checksums and add the checksums:
0 + 5 = 5.
The ANSWER's checksum should also be 5. Let's check it:
The checksum for 23 is 2+3 = 5 which matches the addition of the checksum of 9 and 14, so the answer is probably right.
Let's do another more complicated one.
Check this addition column:
456 + 6 +
1856 2 ---> checksum 3
Oh dear - something is wrong because 2 does not equal 3. Either the column of checksums is wrong or the sum on the left is wrong. You should try and find the error. Usually, the checksum is right because it is very easy to cast out 9s.
The checksum of the correct answer of 1866 is 3, and the checksum of our addition is wrong.
Lets say you were adding a long list of numbers and for each one you have worked out the checksum column like this:
You can cross out the 0 and the nine straight away, and any pairs that add to 9 like 7 and 2 and 6 and 3. Very quickly, you are left with just 2. It's very easy.
Stage 3: It works for multiplying too:
12 x 12 = 144
Do the same with the checksums
Checksum of 12 is 1 + 2 = 3
3 x 3 = 9 ----> checksum is therefore 0.
144 ----> all add up to 9 -----> checksum is 0.
Stage 4: It works for division if you re-arrange it back into a multiplication.
2352 / 42 = 56
56 x 42 = 2352
2 x 6 = 12 ---> checksum 3
2352 ----> checksum 3.
Now try Russian Multiplication!
Have a go at Russian Multiplication. It's quite amazing.