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Like for example

65465-78954-12356 = -25845

Obviously the "borrowing" method that everyone learns in elementary school works, but it's slow and tedious, especially for results that come out negative.

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  • $\begingroup$ I think this is where the Common Core idea comes in of multiple steps when doing addition and subtraction. It's how we would do it in our head rather than doing rote computations on paper. You should check it out. $\endgroup$ – Cameron Williams Mar 20 '15 at 23:28
  • $\begingroup$ Is "Common Core" some kind of US thing? Never heard of it. $\endgroup$ – nathanvy Mar 21 '15 at 0:14
  • $\begingroup$ @nathanvy, it is: en.wikipedia.org/wiki/Common_Core_State_Standards_Initiative $\endgroup$ – Barry Cipra Mar 21 '15 at 0:16
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The biggest difference between what I do now and what I was taught to do in school is that when I do mental arithmetic I mostly work from the largest (leftmost) digits first. I think this is much better than the alternative. It builds the important mathematical habit of getting an order-of-magnitude answer before getting a more precise answer. For example, just eyeballing the most significant digits I know that the final answer has to be about $-20000$, so if I get something much different from this I know I did something wrong.

The idea of analyzing the biggest contribution to the final answer first is also a foundational habit for learning, for example, real analysis, where to analyze a complicated sum it's often crucial to identify the largest few terms and analyze those terms and then the sum of the rest of the terms separately.

Finally, I hate subtracting bigger numbers from smaller numbers, so instead I always do it the other way and take the negative. So first I'd compute $78954 - 65465$ as follows:

$$78954 - 60000 = 18954$$ $$18954 - 5000 = 13954$$ $$13954 - 400 = 13554$$ $$13554 - 60 = 13494$$ $$13494 - 5 = 13489.$$

Next I'd compute $13489 + 12356$ as follows:

$$13489 + 10000 = 23489$$ $$23489 + 2000 = 25489$$ $$25489 + 300 = 25789$$ $$25789 + 50 = 25839$$ $$25839 + 6 = 25845.$$

Then I'd negate to get the final answer. Of course I wouldn't be saying this all out in my head; I'd just be keeping track mentally of which digits I have and haven't used yet.

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One possible way to do this is with some underlining of portions that don't require borrowing:

$$\underline{654}6\underline{5}-\underline{789}54-1235\underline6=60-13554-12351$$ followed by some more underlining with $6$ thought of as, for example, $3+3$:

$$\underline60-135\underline54-123\underline51=-13524-12321$$

followed by simple addition, which in this case involves no carrying. Whether this is any faster or less tedious than the standard borrowing approach (not to mention how to describe it in general) is unclear.

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to subtract large numbers, think of the given numbers as money and start from left. In your case it is 65 thousand, borrowed 78 thousand +12 thousand=90 thousand , so total will be 25 thousand i.e -25000 Now 465 is with you and 356 and 954 taken.

465-356= 400-300 + 65-56=109

109-954=100-900 ,9-54=-800-45=-845 so total will be -25845.

You can read about my method in more details at this link http://mathsequation.com/how-to-subtract-numbers-mentally-fast/

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