Technical Article => Programming => Algorithm

Dividing any number By 9, 90, 900 and so on

The
technique for Dividing any number by 9 mentally is simply to reduce a complex division
to a very simple addition. The technique can be applied from both ends i.e. from right-most digit or from left-most digit.

Dividing
by 9 into a mixed number from right-most digit uses the Divisibility Rules for
9:

1. First, add all the digits
together and divide by 9, keeping in mind the whole number and the remainder.

2. Write the remainder over 9, this
is the fraction part of the answer. (Make sure the fraction is in simplest
form.)

3. Add all the digits together
again, except for the last one. If there was a whole number from step 1, add
this too. Write the number down, carrying if need be.

4. Continue to add all the digits
together, subtracting off one digit each time, until you are left with the
first (left most) digit. Write these numbers down, carry if need be.

5. Write the last number down.
(Remember to add any carried numbers.)

Here are some examples to help
illustrate the above steps:

**1234 ÷ 9 =**__________(mixed number)

a. Add
all the digits together and divide by 9: 1+2+3+4 = 10. 10 divided by 9 is 1
with 1 remainder.

b. Write
1/9, this is the fraction of the answer.

c. Add
1+2+3+1 = 7. Write 7.

d. Add
1+2 = 3. Write 3.

e. The
last number is 1, so write 1.

f. The
answer is 137 1/9.

**8346 ÷ 9 =**__________(mixed number)

a. Add
all the digits together and divide by 9: 8+3+4+6 = 21. 21 divided by 9

is 2 with
3 remainder.

b. The
fraction part is 3/9, which reduces to 1/3.

c. Add
8+3+4+2 = 17. Write 7, carry 1.

d. Add
8+3+1 = 12. Write 2, carry 1.

e. 8+1 =
9. Write 9.

f. The
answer is 927 1/3.

**For dividing by 90, move the decimal point to 1 digit left and for dividing by 900, move the decimal point to 2 digits left after diving by 9 or following the above steps.**

However,

**the faster way to divide any number by 9 mentally**is to start it from left-most digit. Here are some examples to help illustrate the method:

**221013**

**÷**

**9 = ?**

Start putting the first (left) digit of the dividing as it is which is 2,
and you have the first answer digit.

The next one would be this answer digit plus the next digit of the
dividend. So, 2 plus 2 is 4.

The next answer digit would be this 4 plus 1 that is 5.

Now, using the same logic, the next answer digit would be 5 plus 0,
which is again 5.

Next one would be 5 plus 1 that is 6.

And then the last one would be 6 plus 3 that is 9. But do not put 9 straightaway,
because this would be the remainder, if there is a remainder. In this case,
there would be none, because 9 still goes 1 times in 9. This 1 would be added
to the last findings i.e. 6 resulting 7.

So, the answer would be 2 > 4 > 5 > 5 > (6+1) =7 that is 24557.

**32142**

**÷**

**9 = ?**

The answer would be 3,

3 plus 2 that is 5,

5 plus 1 that is 6,

6 plus 4 that is 10, now 10 is a 2 digit number so carry forward 1 (to
be added with 6 making 7) and put 0 here.

And 10 plus the last digit 2 would be 12, so do not
put 12 straightaway. Now, 9 goes how many times in 12? Nine 1 times is 9. So, this 1
would be added to 0 and 3 is the remainder.

So, the answer would be 3 > 5> (6+1) =7 > 1 that is 3571 with 3
reminder.

Source:http://mathema-tricks.blogspot.com/2012/01/dividing-by-9-90-900-and-so-on-into.html

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