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Long division

How to divide any number by 9 ?Before we proceed to understand this shortcut, we need to remember the reciprocals of 9.

Number Fraction Reciprocal1 1/9 0.11112 2/9 0.22223 3/9 0.33334 4/9 0.44445 5/9 0.55556 6/9 0.66667 7/9 0.77778 8/9 0.88889 9/9 1

Dividend is a 2-digit number

Find 13 / 9Step 1 : Here 13 is the dividend and 9 is the divisor. In this example, divisor has 1 digit. So, set 1 digit from right as reminder.

Step 2 : The quotient is the sum of the digits on left. In our case, it is 1. So, the quotient is 1.

Step 3 : The reminder is the sum of all the digits. So, the reminder is 1+3 = 4.

So, the quotient is 1 and reminder is 4.

From the above table, 4/9 = 0.4444. So 13/9 = 1.4444

How to divide any number by 9 ?• Find 36 / 9• Step 1 : Here 36 is the dividend and 9 is

the divisor. In this example, divisor has 1 digit.

• So, set 1 digit from right as reminder.• Step 2 : The quotient is the sum of the

digits on left. In our case, it is 3. So, the quotient is 3.

• Step 3 : The reminder is the sum of all the digits. So, the reminder is 3+6 = 9.

• If the remainder is equal or more than 9(divisor), it is carry forward to the quotient just like in multiplication.

• So, the quotient is 3+1 =4 and reminder is 9-9 = 0.

3 6

3 94 0

How to divide any number by 9 ?

• Can you solve the following :• 89 / 9• 97 / 9• 91 / 9• 96 / 9

8 9

8 171

08 9

How to divide any number by 9 ?• Dividend is a 3-digit number or more• When the dividend is a 3-digit number or more, there is a slight variation

in the division steps.• Find 425 / 9• Step 1 : Here 425 is the dividend and 9 is the divisor. In this example,

divisor has 1 digit. • So, set 1 digit from right as reminder.• Step 2 : The quotient is the successive sum of the digits on left i.e., 4/4+2

= 46• Step 3 : The reminder is the sum of all the digits i.e., 4+2+5 = 11.• Notice, that the reminder is greater than 9. If the remainder is equal or

more than 9(divisor), it is carry forward to the quotient just like in multiplication.

• So, the quotient is 46+1 =47 and reminder is 11-9 = 2.• So, 421 /9 = 47.222


How to divide any number by 9 ?

How to divide a number by 99 ?Before we proceed to understand this shortcut, we need to remember the reciprocals of 99.

Number Fraction Reciprocal

1 1/99 0.010101

2 2/99 0.020202

… … …

9 9/99 0.090909

10 10/99 0.101010

13 13/99 0.131313

56 56/99 0.565656

78 78/99 0.787878

89 89/99 0.898989

How to divide a number by 99 ?• Dividend is a 3-digit number• Find 732 / 99.• Step 1 : Here 732 is the dividend and 99 is the divisor. • In this example, divisor(99) has 2 digit. • So, set 2 digit from right as reminder.• Group the digits on the left into groups of 2.• Step 2 : The quotient is just the digit on the left. In

our case, it is 7. So, the quotient is 7.• Step 3 : The reminder is the sum of all the digits

taken 2 digits at a time. So, the reminder is 7+32 = 79.• So, the quotient is 7 and reminder is 39.• From the above table, 39/99 = 0.393939. So 732/99 =

7.393939

How to divide a number by 99 ?• Find 498 / 99• Solution : Step 1 : Here 498 is the dividend and 99

is the divisor. In this example, divisor has 2 digit. So, set 2 digit from right as reminder.

• Step 2 : The quotient is just the digit on the left. In our case, it is 4. So, the quotient is 4.

• Step 3 : The reminder is the sum of all the digits taken 2 digits at a time. So, the reminder is 4+98 = 102. If the remainder is equal or more than 99(divisor), the difference is carry forward to the quotient just like in multiplication.

• So, the quotient is 4+1 =5 and reminder is 102-99 = 3.

• From the above table, 3/99 = 0.030303. So 498/99 = 5.030303.

How to divide a number by 99 ?• Dividend is a 4-digit number• Find 5342 / 9• Solution : Step 1 : Here 5398 is the dividend and 99 is the

divisor. • In this example, divisor has 2 digit. So, set 2 digit from right

as reminder.• Step 2 : The quotient is just the digit on the left. • In our case, it is 53. So, the quotient is 53.• Step 3 : The reminder is the sum of all the digits taken 2

digits at a time. • So, the reminder is 53+98 = 151. • Since 151 is greater than 99(divisor), the difference is carry

forward to the quotient just like in multiplication.• So, the quotient is 53+1 =54 and reminder is 151-99 = 52.

• From the above table, 52/99 = 0.525252 So 5392/99 = 54.5252

How to divide a number by 99 ?


How to divide a number by 99 ?• Dividend is a 5-DIGIT OR MORE• Example 4 : Find 777745 / 99• Solution : Step 1 : Since we are dividing the number

by 99(2 digit), set 2 digit from right as the remainder. In our case it is 45.

• Step 2 : The quotient is the successive sum taking 2-digits at a time. So, the quotient was obtained by taking the successive sum taking 1-digit on the right at a time.

• So, the quotient after the carry-forward is 7854.• Step 3 : The reminder is the sum of all the digits

taking 2-digits at a time.• i.e., 77 +77 + 45 = 199• Notice, that the reminder is greater than 99. • If the remainder is equal or more than 99(divisor),

the carry forward to the quotient is just like in multiplication.

• So, the quotient is 7854+2 =7856 and reminder is 199-99(2) = 1.

• So, 777745 / 99 = 7856.0101

• Find 3214675 / 99

Reminder = 115So, the quotient is 32470+1 =32471 and reminder is 115-99 = 16.So, 3214675 / 99 = 32471.161616


• Can you solve the following :• 4564564 / 99• 679543 / 99• 345346 / 99• 45643 / 99•


How to divide the number when the divisor ends in digit 9?

• 87÷129 =?• The first step is to round the denominator to the next digit.

Long Division Method Shortcut Technique

• Divide the following :• 47/89• 12/69• 132/179• 123/189

Divide 95÷159.• Reduce 95÷159 as follows :

Now divide 9.5 by 16 like normal division, instead of putting a zero in each step, you replace it with the last quotient to continue the division.

Divide 436÷159.Reduce 436÷159 as follows :

Now, there is a small difference compared to the previous 2 examples. In the previous 2 examples, the numerator was less than the denominator. However, in this case, we have the numerator which is greater than the denominator.In this division, we need to add the quotient with the rest of the digit (6) and put this digit next to the reminder.

Divide 3462÷179.Notice that numerator is greater than the denominator. There is a slight variation compared to the previous example

Notice the carry-forward in step 2 of the division. So, therefore,

• Divide the following :• 4347/149• 1222/239• 1349/179• 2323/249

Divisors are different from 9’s but Close to TEN• 108 7• Leave one digit on the right since

divisor has a single digit.• Complement of 7 is 3.• Put the dividend 1 below, multiply it

by 3 and place the product below 0.• Add 0 and 3 and place it below.• Multiply this by 3 and place the

product below 8 and add to obtain 17.• Now R = 17 > 7 (divisor). 17 contains

2 times 7. Place 2 below 3 and the product below 17 (with – sign).

• Add the results to get Q = 15 and R = 3

7 1 0 8

3 3 9

1 3 17

2 –14

1 5 3

• 863 7 • Put the complement of 7 below 7.• Bring down 8 of the dividend • multiply it by 3 to get 24• Add 6 to 24 to get 30• Put 0 on the Q–line and 3 on the

carry over line.• Multiply 30 by 3 and write 90 on

the RHS • add it to 3 to get 93.• As 93 = 7x13 +2 put 13 as shown

and the product 91 also. • Q = 110 + 13 = 123• R = 93 – 91 = 2

Divisors are different from 9’s but Close to TEN

7 8 6 3

3 24 90

8 0 93

3

11 0 93

+1 3 –91

12 3 2

Divisors are different from 99 but Close to 100

87 2 59

13 26

2 85

259 87 = ?40854 878 = ?

878 4 0 854

122 4 88

4 4 222

2

4

244

6 466

488

LONG DIVISION

7 1 6 7 6 9

1 3 2 7 3 5

2 2 4 4 3 205

4

7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17. Then 17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14 and then 14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 447 - 4 x 2 = 39 and then 39 ÷ 5 = 7 remainder 4.Put down the 7 and carry over the 446 - 4 x 7 = 18 and then 18 ÷ 5 = 3 remainder 3.

Put down the 3 and carry over the 339 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point 27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 220 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the divisionHence the answer is 13273.5

45026 ÷ 47 = 958.04 9 7 5

47 4 5 0 2 60 9 5 8 0

4 ÷ 4 = 0 remainder 4. Put the quotient 0, the first digit of the solution, in the first box of the bottom row and carry over the remainder 4The product of the flagged number (7) and the previous quotient (0) must be subtracted from the next number (45) before the division can proceed. 45 - 7 x 0 = 45 and then 45 ÷ 4 = 9 remainder 9.Put down the quotient 9 and carry over the remainder 9.

Remarks: In the first step we have written that 4 ÷ 4 = 0 remainder 4 instead of 4 ÷ 4 = 1 remainder 0. Otherwise, in the following step, we would have to subtract 7*1=7 from 05 which is not possible.

Again subtract the product of the flagged number (7) and the previous quotient (9), 90 - 7 x 9 = 27 and then 27 ÷ 4 = 5 remainder 7. Put down the quotient 5 and carry over the remainder 7.72 - 7 x 5 = 37 and then 37 ÷ 4 = 8 remainder 5.Put down the quotient 8 and carry over the remainder 5.56 - 7 x 8 = 0. There is nothing left to divide, so this cleanly completes the division.Hence the answer is 958.

To divide 716769 by 156: Split divisor as 15 and 6

• ------ 11 17 12 12 15 9• 156 71 6 7 6 9 0 0• ----- 4 5 9 4 6 7 3• Answer: 4594.673

7 1 6 7 6 9

1 3 2 7 3 5

2 2 4 4 3 205

4

7 ÷ 5 = 1 remainder 2. Put the quotient 1, the first digit of the solution, in the first box of the bottom row and carry over the remainder 2The product of the flagged number (4) and the previous quotient (1) must be subtracted from the next number (21) before the division can proceed. 21 - 4 x 1 = 17. Then 17 ÷ 5 = 3 remainder 2. Put down the 3 and carry over the 2Again subtract the product of the flagged number (4) and the previous quotient (3), 26 - 4 x 3 = 14 and then 14 ÷ 5 = 2 remainder 4. Put down the 2 and carry over the 447 - 4 x 2 = 39 and then 39 ÷ 5 = 7 remainder 4.Put down the 7 and carry over the 446 - 4 x 7 = 18 and then 18 ÷ 5 = 3 remainder 3.

Put down the 3 and carry over the 339 - 4 x 3 = 27. Since the decimal point is reached here, 27 is the raw remainder. If decimal places are required, the division can proceed as before, filling the original number with zeros after the decimal point 27 ÷ 5 = 5 remainder 2. Put down the 5 (after the decimal point) and carry over the 220 - 4 x 5 = 0. There is nothing left to divide, so this cleanly completes the divisionHence the answer is 13273.5

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