How does regrouping actually work?

I understand how to use the process of regrouping to solve a subtraction problem, but I do not understand how this process actually works. How does borrowing 10 from a 7 in the hundreds place render that 7, which was actually worth 700, a 6 -- or, as it is actually worth, a 600 (shouldn't it become 690 if we're borrowing 10 from it?), and why do we always borrow 10 anyway? Anybody who cares to explain the answers to these questions would be appreciated.

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  • 10 years ago
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    It is very much like this problem, which is about hours minutes and seconds, but might be easier to understand.

    Suppose a task takes 3 hours 15 minutes and 30 seconds to complete. The first person works on the task for 1 hour 20 minutes and 50 seconds, then without missing a second of time another person takes over and finishes the task. How long did that second person work?

    So you have to figure out this subtraction problem:

    (3 hours 15 minutes and 30) - (1 hour 20 minutes and 50 seconds)

    You cannot subtract 50 seconds from 30 seconds (the answer would be negative and we don't want that), so you interpret 1 of the 15 minutes as 60 seconds, so you have "borrowed 1 minute" but change it to 60 seconds and put that together with the 30 seconds giving 90 seconds (and you're square, i.e. the original amount of time looks different, but is the same amount of time). Lets now write it vertically:

    3 hours 14 minutes. 90 seconds

    1 hour . 20 minutes. 50 seconds

    --------- ------------- ---------------- subtract

    ?? .......... ?? ........ 40 seconds

    Now you cannot subtract 20 minutes for 14 minutes so now you "borrow" 1 hour from the 3 hours and since that equals 60 minutes you put it together with the 14 minutes getting 74 minutes, so now:

    2 hours 74 minutes 90 seconds

    1 hour . 20 minutes 50 seconds

    --------- ------------- --------------- subtract

    1 hour . 54 minutes 40 seconds

    And that is your answer.

    Now, if it is numbers instead of time, it is very similar for example if you need to do 673 - 485

    673 = 600 + 70 + 3

    485 = 400 + 80 + 5

    ----- . ----- ----- ----- subtract

    You cannot subtract 5 from 3 so borrow 10 from 70 and put it with the 3 making 13, so now you have

    673 = 600 + 60 + 13

    485 = 400 + 80 + . 5

    ----- -------------------- subtract

    .?. .... ?? ....?? ... 8

    Now 80 cannot be subtracted from 60, so "borrow" 100 from the 600 and put it with the 60 giving:

    673 = 500 + 160 + 13

    485 = 400 + . 80 +. 5

    ----- . --------------------- subtract

    ........ 100 + . 80 + . 8

    Which equals 188, so that is the answer.

    That is how it works. I hope this helped.

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