The document provides various shortcuts and methods for multiplication, division, and other mathematical operations involving large numbers: 1) It explains how to multiply 2-digit and 3-digit numbers using the standard multiplication algorithm but breaking it into steps. 2) It outlines methods for dividing numbers by 9, 99, or any number ending in 9 by finding the reciprocal of the divisor and converting the division into a multiplication problem. 3) Other shortcuts covered include squaring numbers ending in 5, multiplying by 12 by doubling each digit and adding to the next, and multiplying by 6 using the "half plus 5" method.

1. Multiplication Shortcuts

2. Multiplication of 2 digit with 2 digit
• 67 × 56
• Step 1: 7 × 6 = 42
• Step 2: (6×6) + (5×7) =
71 +4(carry) = 75
• Step 3: 6 × 5 = 30 +
7(carry) = 37

3. Multiplication of 3 digit with 3 digit
• 789 × 123
• Step 1: 3 × 9 = 27
• Step 2: (8×3) + (9×2) = 42
+ 2(Carry) = 44
• Step 3: (7 × 3) + (1 × 9) +
(8 × 2) = 46 + 4(Carry) =
50
• Step 4: (7 × 2) + (8 × 1) =
22 + 5(Carry) = 27
• Step 5: 7 × 1 = 7 +
2(Carry) =9

4. Divisibility by 7

5. Find % ending in 0 or 5

6. To find the % of any Number

7. Convert Fractions into %

8. Multiply the Big Numbers

9. How to divide any number by 9 ?
Before we proceed to understand this shortcut, we need to remember the reciprocals of 9.
Number Fraction Reciprocal
1 1/9 0.1111
2 2/9 0.2222
3 3/9 0.3333
4 4/9 0.4444
5 5/9 0.5555
6 6/9 0.6666
7 7/9 0.7777
8 8/9 0.8888
9 9/9 1
Dividend is a 2-digit number
Find 13 / 9
Step 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

10. 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 9
4 0

11. How to divide any number by 9 ?
• Can you solve the following :
• 89 / 9
• 97 / 9
• 91 / 9
• 96 / 9
8 9
8 17
1
089

12. 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
•

13. • Can you solve the following :
• 12389 / 9
• 933347 / 9
• 1931 / 9
• 3631 / 9
How to divide any number by 9 ?

14. 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

15. 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

16. 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.

17. 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

18. How to divide a number by 99 ?
• Can you solve the following :
• 898 / 99
• 2947 / 99
• 9451 / 99
• 9634 / 99

19. 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

20. • Find 3214675 / 99
Reminder = 115
So, the quotient is 32470+1 =32471 and reminder is 115-99 = 16.
So, 3214675 / 99 = 32471.161616
How to divide a number by 99 ?

21. • Can you solve the following :
• 4564564 / 99
• 679543 / 99
• 345346 / 99
• 45643 / 99
•
How to divide a number by 99 ?

22. Squaring a number ending in 5
• 352 =?
• The successor of 3 is 4
• Multiply 3 and 4
• Put the answer 12.
• Suffix 25.
• The answer is 1225.

23. Squaring the number beginning in 5
• 562 = ?
• Always Add 25 with the
number in units digit
25+6 =
• Suffix the square of the
unit digit =
31
36

24. Numbers ending in 6
• 662 =_________.
• a) Write down 6.
• b) 2 x (6 + 1) + 1 = 15. Write 5, carry *1.
• c) 6 x (6 + 1) = 42 + *1 = 43. Write 43.
• d) The answer is 4356.
• 862 =_________.
• a) Write down 6.
• b) 2 x (8 + 1) + 1 = 19. Write 9, carry *1.
• c) 8 x (8 + 1) = 72 + *1 = 73. Write 73.
• d) The answer is 7396.

25. 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

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

27. 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.

28. 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.

29. 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,

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

31. Multiplication by 12
• Let’s understand this method by
taking an example.
• 345 × 12
• Step 1:
• Write 345 as 0345.
• Step 2:
• Multiply each and every digit with
2 and add it to its right neighbour.
This process is started from right.
Multiply 5 with 2 and add it to its
right neighbour. 5 has no right
neighbour so add the
multiplication to 0. This gives 10
as a result. So 0 is written down
and 1 is carried forward.

32. • Step 3:
• Multiply 4 with 2 and add it to 5.
Carry 1 is added to the result. This
gives 14 as a result.
• 4 is written down and 1 is carry
forwarded.

33. • Step 4:
• Multiply 3 with 2 and add it
to 4. Carry 1 is added to the
result. This gives 11 as a
result.
• 1 is written down and 1 is
carry forwarded.

34. • Step 5:
• Multiply 0 with 2 and add it
to 3. Carry 1 is added to the
result. This gives 4 as a
result.

35. MULTIPLICATION BY SIX
Practice what is half of ? To each number add ‘half’ of the neighbour;
‘plus 5’ if it is odd
• 6
• 5
• 4
• 8
• 7
• 2
• 3
• 2
• 2
• 4
• 3
• 1
0 6 2 2 0 8 4 x 6
4052373
1
0 4 4 3 0 5 2 x 6
2138562
1