All from 9 and Last from 10 (To perform large subtractions)

Performing large subtractions fast could be a cumbersome for some. This method can be used for subtracting numbers from 10, 100, 1000, 10000 and so on and from 30, 500, 6000, 20000 and so on. Consider we want to subtract 376 from 1000 and 10905 from 100000. This can be done as follows:

  1. Subtract the last digit from 10 (compliment) and the rest of the digits from 9. i.e., 3 would become 6, 7 would become 2 and 6 would become 4. Thus 1000 – 376 would result in 624.
  2. Similarly 10905 would become 89095 by subtracting the last digit 5 from 10 and the rest of the digits from 9.

Note that applying this to any number ending in 0 should be done by performing the above operations ignoring the 0 and then adding the 0 in the end. Suppose we want to subtract 71840 from 100000, we get 28160 by subtracting 4 from 10 and the rest from 9 and finally adding a 0 to the end.

Also note that the formula All from nine and last from ten subtracts numbers from the next highest base number. Eg: 1000 – 864, 100 – 78 etc. Suppose we had 1000 – 67. 1000 has three 0s, but 67 is a two digit number. This can be done as 1000 – 067 and then applying the formula we get, 933.

Now consider 700 – 46. Here we have 700 instead of 100. This can be done as follows:

  1. Apply the regular formula like it is done for 100 – 46. We get 54 by subtracting 6 from 10 and 4 from 9.
  2. Now 7 is reduced to 6. Thus we get the answer as 654.
  3. Subtraction using  All from 9 and last from 10

Consider the example, 4652. Applying the Sutra, All from nine and last from ten we get 5348 as 4 6 and 5 are taken from 9 and the last digit 2 is taken from 10. Note that if 0 is present in the end, we perform the steps for rest of the number and then add the 0 to the end. Consider the example, 3450. Applying the Sutra we get, 6550. This method can be used when the numbers are close to a base (10, 100, 1000 etc) or multiples of a base (60, 200, 4000 etc). Note that we subtract numbers from the next highest unity. Thus 1000 – 462 = 538 and 10000 – 3672 = 6328.

  1. Consider the example, 1000 – 47

ð  The next highest unity for 47 is 100, but we have to subtract it from 1000. This is done by adding a 0 in front of 47 to make it a 3 digit number. Thus we write it as 1000 – 047.

ð  Now applying the Sutra, we get 953.

  1. Consider the example 4000 – 243.

ð  Consider the unity to be 1000 and solve as usual. Thus we get the answer as 757.

ð  Now since we are subtracting 243 from 4000, it will reduce to 3000 and we get the final answer as 3757.

  1. Consider other cases with subtraction not involving unity or multiples of unity. We can perform these subtractions in a single line without taking borrow or carry forward concept that we have been taught in school. Consider the example, 476 – 248.

ð  Subtracting each column from left to right  (or right to left) we get 4 – 2 = 2 ; 7 – 4 = 3 ; 6 – 8 = -2.

ð  The answer that we got by performing the above subtraction is 23(-2)

ð  When we see a negative number, we take its compliment (All from 9 and Last from 10) and reduce the previous digit by 1 since we are subtracting from it. Thus here 23(-2) becomes 228

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3 thoughts on “All from 9 and Last from 10 (To perform large subtractions)

  1. Excellent explanation. When I was studying 1’s Complement subtraction and 2’s complement subtraction when I was at EEC, and also when I studied discrete Mathematics, I wondered briefly about the possibility of doing this kind of Math. It’s brilliant that it’s been worked out so well already.

    This article has several formatting issues, most likely due to WordPress messing things up. You might want to go back and fix that. I also noticed that you have one example separate from another while both belong together! I mean the example of 1000 – 67 and 1000 – 47. You might even want to remove one.

    Keep these articles coming! 🙂

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