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Worked example: Subtracting 3-digit numbers (regrouping from 0)

Sal has to borrow from a 0 to subtract 301-164. Created by Sal Khan.

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  • aqualine ultimate style avatar for user apredator504
    Is there any other methods other than regrouping?
    (9 votes)
    • primosaur seed style avatar for user Ian Pulizzotto
      There is a Vedic method that involves writing bars (or vinculums) on digits to indicate "negative" digits, as we subtract in each column. Then, for each string of bar digits, we use complements (subtract digits from 9 except the last digit from 10, within the string of bar digits) and also decrease the digit immediately to the left of the string by one, to convert the answer to normal digits.

      Example: Suppose we want to subtract 912 - 749. Remember that a bar digit occurs when the bottom digit in a column is greater than the top digit.

      Hundreds column: 9-7 = 2.

      Tens column: When we do 1-4, we end up short by 3. So we use the "negative" digit 3bar.

      Ones column: When we do 2-9, we end up short by 7. So we use the "negative" digit 7bar.

      Thus we get 2 3bar 7bar, but this has to be converted to a normal number. For the bar digits from left to right, 9-3=6 and 10-7=3. We must also decrease the hundreds digits 2 by one, to get 1. The final answer is 163.

      Note: in some subtraction problems, the differences might be 0's in some of the columns. This method is more clear if any string of 0's that occur immediately to the left of a string of bar digits are also treated as bar digits.

      Example: Suppose we want to do 93418 - 53442.

      From left to right, the differences in the columns are 4, 0, 0, 3bar, 6.
      Let's write this as 4 0bar 0bar 3bar 6.

      When we use complements, we get the normal digits 9-0=9, 9-0=9, 10-3=7 to replace the string of three bar digits. We must also decrease the digit 4 immediately to the left of this string by one, to get 3. The final answer is 39,976.
      (9 votes)
  • piceratops ultimate style avatar for user dragonloverlord
    I heard you can do subtraction using addition is this true or did I hear wrong?
    (8 votes)
  • aqualine ultimate style avatar for user GemstoneSG
    I have a problem. How do you subtract 1000-392?
    (5 votes)
    • hopper cool style avatar for user Philip
      Step 1: Subtract 1000 by 300. Since the hundred's place equals 0, something less than 3, you borrow from the thousand's place (or basically, subtract 10 by 3). This gives you 700.
      Step 2: Subtract 700 by 90. Since the ten's place equals 0, something less than 9, you will again need to borrow from next higher place value. The 7 in the hundred's place is reduced to 6, while 10 minus 9 equals 1. You put the 1 into the ten's place, and for now gives you 610.
      Step 3: Subtract 610 by 2. Since the one's place equals 0, something less 2, you once again will need to borrow from a next higher place value. Reduce the 1 in the ten's place by 1, and subtract 10 by 2 which equals 8. Put the 8 into the one's place, you will have 608 as the solution.
      (12 votes)
  • leaf yellow style avatar for user HediyehS.Kermani
    why can't you subtract 4 from 1 why do you have to borrow?
    (3 votes)
  • purple pi purple style avatar for user Mathlover9415
    My mind is blown…
    (1 vote)
    Default Khan Academy avatar avatar for user

Video transcript

Let's try to subtract 164 from 301, and I encourage you to pause this video and try it on your own first. So let's go place by place, and we can realize where we have to do some borrowing or regrouping. So in the ones place, we have an issue. 4 is larger than 1. How do we subtract a larger number from a smaller number? We also an issue in the tens place. 6 is larger than 0. How do we subtract 6 from 0? So the answer that might be jumping into your head is oh, we've got to do some borrowing or some regrouping. But then you might be facing another problem. You'd say, OK, well, let's try to borrow from the tens place here. So we have a 1. If we could borrow 10 from the tens place, it could be 11. But there's nothing here in the tens place. There's nothing to borrow, so what do we do? So the way I would tackle it is first borrow for the tens place. So we have nothing here, so let's regroup 100 from the hundreds place. So that's equivalent to borrowing a 1 from the hundreds place. So that's now a 2. And now in the tens place, instead of a 0, we are going to have a 10. Now, let's make sure that this still makes sense. This is 200 plus 10 tens. 10 tens is 100, plus 1. 200 plus 100 plus 1 is still 301. So this still makes sense. Now, the reason why this is valuable is now we have something to regroup from the tens place. If we take one of these tens, so now we're left with 9 tens, and we give it to the ones place, so you give 10 plus 1, you're going to be left with 11. And we can verify that we still haven't changed the value of the actual number. 200 plus 90 is 290, plus 11 is still 301. And what was neat about this is now up here, all of these numbers are larger than the corresponding number in the same place. So we're ready to subtract. 11 minus 4 is-- let's see, 10 minus 3 is 7, so 11 minus 4 is 7. 9 minus 6 is 3, and 2 minus 1 is 1. So we are left with 137.