Division by 9 with regrouping

In an article in our previous issue of the MATH-Inic Newsletter about dividing by 9 (see: https://www.math-inic.com/blog/division-by-9-2/), we showed that the quotient is the running total of the digits of the dividend. However, when the digits of the dividend are large, the running total may have more than one digit and so, regrouping is required.

Take the case of 4728 ÷ 9.

We first separate the last digit by a remainder bar:

4   7   2   |   8

Bring down the first digit of the dividend as the first digit of the quotient:

4   7   2   |   8

4

The succeeding digits of the quotient is determined by the running total of the digits of the dividend: 4 + 7 = 11; 11 + 2 = 13; and remainder, 13 + 8 = 21.

4   7   2   |   8

4 11  13 r. 21

We can see that we have double digits in the 2nd and 3rd columns of the quotient and in the remainder. To get the answer in proper form, we need to regroup.

First, we have a remainder of 21, from which we can get two 9s and a remainder of 3.

We add the 2 (from 2 9s) to 13 to get 15.

Then we add the 1 of 15 to the next figures to the left, 11 to get 12.

Finally, we add the 1 of 12 to the leading 4 to make 5.

Thus, we have:

4   7   2   |   8

5   2   5  r. 3

We can avoid getting double digit sums by anticipating the needed regroupings.  After placing the remainder bar, our first step is to bring down the first digit of the dividend, 4, as the first digit of the quotient. But before writing it down, mentally add the next digit to it, 4 + 7 = 11. Since the result is more than one digit, we must add 1 to 4 in anticipation of the “carry” operation. Thus, we have,

4   7   2   |   8

The next step is to add 7 to 5 to get 12, but since we have done the “carry” operation earlier, only 2 is written as the second digit of the quotient. Note that adding the next digit of the dividend 2 to the current figure in the quotient will result only to 4 and no adjustment is needed.

4   7   2   |   8

5   2

The next step is to add 2 to 2 as described in the previous paragraph, getting 4. But before writing down 4 we add the last digit of the dividend 8 to get another double digit answer 12. Thus, we have to add 1 to 4.

4   7   2   |   8

5   2   5

The final step is to get the remainder: add 8 to 5 to get 13 and since we have done the carry operation in the previous step, the remainder is only 3.

Try these following division:

1. 56 ÷ 9 =
2. 86 ÷ 9 =
3. 467 ÷ 9 =
4. 582 ÷ 9 =
5. 4732 ÷ 9 =
6. 7514 ÷ 9 =
7. 38643 ÷ 9 =
8. 55455 ÷ 9 =
9. 662772 ÷ 9 =
10. 283746 ÷ 9 =

Answers to exercises in the “Division by 9” article (https://www.math-inic.com/blog/division-by-9-2/)

1.  52 ÷ 9 = 5 r. 7
2. 71 ÷ 9 = 7 r. 8
3. 123 ÷ 9 = 13 r. 6
4. 142 ÷ 9 = 15 r. 7
5. 2121 ÷ 9 = 235 r. 6
6. 3113 ÷ 9 = 345 r. 8
7. 11221 ÷ 9 = 1246 r. 7
8. 22111 ÷ 9 = 2456 r. 7
9. 300211 ÷ 9 = 33356 r. 7
10. 321001 ÷ 9 = 35666 r. 7