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TIPS AND TRICKS FOR SPEEDY CALCULATIONS – MODULE III – MULTIPLICATION

In this module we deal with techniques using which we can multiply two numbers in an unorthodox but quick manner. Firstly we take up some specific cases using which we come to generalized multiplication of any two given numbers.

- Multiplying two numbers starting with same digit(s)

Let us take an example of one two digits numbers and the other three digit numbers to be multiplied with each other;

- Multiply 34 and 37

To multiply 34×37, we know they are in the base 30. Hence the reference point (base) will be 30.

Step 1.

Determine how much more is 34 from 30. The answer is 4

Determine how much more is 37 from 30. The answer is 7

Step 2.

Either add 4 to 37 = 41 or 7 to 34 = 41.

The result will be same always.

Step 3.

Multiply the resultant number from step 2 by the base, which is in this case 30

41×30 = 41x3x10 = 123×10 = 1230

Step 4.

Add to the resultant of step 3 the product of the numbers obtained from step 1. This will give you the answer.

1230+ (4×7) = 1230 + 28 = 1258

- Multiple 234 x 232

From step 1 and step 2 above, 234 +2 = 236 or 232 + 4 = 236

From step 3: 236 x 230 = 54280

From step 4: 54280 + (4×2) = 54288

- Multiplying 5 with an even number

Halve the number you are multiplying by 5 and place a zero after the number.

Example:

A) 5 × 136, half of 136 is 68, add a zero for an answer of 680.

B) 5 × 874, half of 874 is 437; add a zero for an answer of 4370.

- Multiplying 5 with an odd number

Subtract one from the number you are multiplying, then halve that number and place a 5 after the resulting number.

Example:

343 x 5 = (343-1)/2 | 5 = 1715

- Multiplying by 9, 99, 999, 9999 and so on

Let X be 9 or 99 or 999 or 9999 and so on and Y represent the other number.

This technique should follow two conditions;

- The number Y should end with 9
- The digits in number Y should not be more than that in number X.

Let us now take some examples;

- Multiply 789 × 999

You will get the answers in two parts,

The left hand side of the answer: subtract 1 from 789, which is 788

The right hand side of the answer subtract 789 from 1000 = 1000-789= 211

Thus, 999 x 789 = 789-1 | 1000-789 = 788, 211 (answer)

{for the right hand side of the answer, 789 should be subtracted from (999+1)

- Multiply 99999 x 78 = 78-1 | 100000 – 78 = 7799922

{78 should be subtracted from (99999+1)

- Multiply 1203579 × 9999999 = 1203579-1 | 10000000- 1203579 =12035788796421

- Multiplying by 11

Assume that there are two invisible 0 (zeroes), one in front and one behind the number to be multiplied with 11. Take an example of 234, assume it to be 0 2 3 4 0

Start from the right, add the two adjacent Zeros and keep on moving left 02340. Add the last zero to the digit in the ones column (4), and write the answer below the ones column. Then add 4 with digit on the left i.e. 3. Next add 3 with 2. Next 2 with 0.

0+4 = 4

4+3 = 7

3+2 = 5

2+0 = 2

So answer is 2574. Further examples;

- 36 x 11 = 0+3 | 3+6 | 6+0 = 396
- 74 x 11 =0+ 7 | 7+4 | 4+0 = 7 |
_{1}1 | 4 = 814 (1 of 11 is carried over and added to next digit, so 7+1 = 8 ) - 6349 x 11 = (0+6) | (6+3) | (3+4) | (4+9) | 9+0 = 69839

- Multiplying a two digit number by 111

To multiply a two-digit number by 111, add the two digits and if the sum is a single digit, write this digit TWO TIMES in between the original digits of the number. Some examples:

36×111= 3996

54×111= 5994

The same idea works if the sum of the two digits is not a single digit, but you should write down the last digit of the sum twice, but remember to carry if needed. So

57×111= 6327 because 5+7=12, but then you have to carry the one twice.

- Multiplying a three digit number by 111

123×111 = 1 | 3 (=1+2) | 6 (=1+2+3) | 5 (=2+3) | 3

Similarly, 241×111 = 26751

For an example where carrying is needed

Say, 352×111=3 | 8 (=3+5) | 10 (=3+5+2)| 7 (=5+2)| 2

= 3 | 8 | 10 | 7 | 2 = 3 | 9 | 0 | 7 | 2

= 39072

- Multiplying any number by 21

The process is a bit complicated and shall be explained by means of an example;

Multiply 5392 by 21. The first digit of the answer will be equal to twice the first digit of 5392. To make the rule consistent assume there is a zero before the number.

So it looks like 05392

0 + (5 x 2) = 10

Next, add the first digit of the given number, 5, to twice the second digit, 3.

5 + (2 x 3) = 11

Since we must have a single digit at each step, the tens place of the result above will be carried over and added to the previous number.

1 | (0 +1) | 1 = 111

The first 3 digits up to this point are 111. The next digit is obtained by adding 3 to twice of 9

3 + (2 x 9) = 21

Thus the first four digits of the answer are –

1 | 1 | (1 + 2) | 1 = 1131 (carried over 2 added to the last digit of 111 )

The next digit is obtained by adding 9 to twice of 2

9 + (2 x 2) = 13

Thus the first five digits of the answer are –

1 | 1 | 3 | (1+1) | 3

The last digit of the answer will be same as the last digit of the number itself.

Hence, in this case last digit will be 2. Therefore the answer is 113232

- Multiply any two digit numbers with 10 being the sum of their unit places

Principle: You will get the answer in two parts. First part, to get left hand side of the answer: multiply the left most digit(s) by its successor. Second part, to get right hand side of the answer: multiply the right most digits of both the numbers.

- Multiply 45 and 46;

First part: 4 x (4+1)

Second part: (4 x 6)

Combined effect: (4 x 5) | (4 x 6) = 2024

- 37 x 33 = (3 x (3+1)) | (7 x 3) = (3 x 4) | (7 x 3) = 1221
- 11 x 19 = (1 x (1+1)) | (1 x 9) = (1 x 2) | (1 x 9) = 209

- Multiply any three digit numbers with 10 being the sum of their unit places

The technique as discussed above can be extended to three digit numbers also. Tis will be made clear by solving certain examples;

- Multiply 292 and 208;

Here 92 + 08 = 100, L.H.S portion is same i.e. 2

292 x 208 = (2 x 3) x 10 | 92 x 8 (Note: if 3 digit numbers are multiplied, L.H.S has to be multiplied by 10)

60 | 736 (for 100 raise the L.H.S. product by 0) = 60736.

- 848 X 852

Here 48 + 52 = 100,

L.H.S portion is 8 and its next number is 9.

848 x 852 = 8 x 9 x 10 | 48 x 52 (Note: For 48 x 52, use methods shown above)

720 | _{2}496

= 722496.

[L.H.S product is to be multiplied by 10 and 2 to be carried over because the base is 100].

C) 693 x 607 = 6 x 7 x 10 | 93 x 7 = 420 / 651 = 420651

- Multiply two numbers close to 100 but greater than 100.

Principle: You will get the answer in two parts. First part, to get left hand side of the answer we add the difference between 100 and either of the numbers to the other number. For the Second part, we multiply the difference from 100 of both the numbers. Consider the following examples;

- 103 x 104 = 10712

The answer is in two parts: 107 and 12,

107 is just 103 + 4 (or 104 + 3), and 12 is just 3 x 4.

- 107 x 106 = 11342

First part; 107 + 6 = 113 and Second part; 7 x 6 = 42

- 123 x 103 = 12669

(123 + 3) | (23 x 3) = 126 | 69 =12669

If the multiplication of the offsets is more than 100 then this method won’t work. For example 123 x 105. Here offsets are 23 and 5. Multiplication of 23 and 5 is 115 which are more than 100. So this method won’t work. But it can still work with a little modification. Consider the following examples:

- 122 x 123 = 15006

Step 1: 22 x 23 = 506 (as done earlier)

Step 2: 122 + 23 (as done earlier)

Step 3: Add the 5 (digit at 100s place) of 506 to step 2

Answer: (122 + 23 + 5) | (22 x 23) = 150 | 06 = 15006

- 123 x 105 (Different representation but same method)

123 + 5 = 128

23 x 5 = 115

128 | 115 = 12915

- Multiply two numbers close to 100 but less than 100.

Principle: You will get the answer in two parts. First part, to get left hand side of the answer: Add the difference between 100 and either of the numbers to the other number. Second part, to get right hand side of the answer: multiply the difference from 100 of both the numbers. Consider the following examples;

- Multiply 93 and 94;

First part: 93 – 100 = – 7; Add this to the other number, thus 94 + (- 7) = 87

Or you can start with the other number 94;

94 – 100 = – 6; Add this to the other number, thus 93 + (- 6) = 87

Result will be same in both the cases

Second part:

Multiply the difference from 100 of both the numbers.

Hence, (93 – 100) x (94 – 100) = -7 x -6 = 42

Combined effect: 87 | 42 = 8742

- Multiply 92 and 86;

Step 1: 92 + (86 – 100) = 78

Step 2: (92 – 100) x (86 – 100) = -8 x -14 = 112

Combined effect will look like this: 78 | _{1}12

Step 3: Add the 1 (digit at 100s place) of 112 to 78

Answer: 78 + 1 | 12 = 79 | 12 = 7912

- Multiply two numbers close to 100 one being less and the other more than 100

Principle is same as given above so we directly take up some examples;

- Multiply 96 and 103;

First part: 96 – 100 = – 4; Add this to the other number, thus 103 + (- 4) = 99

Or you can start with the other number 103;

103 – 100 = 3; Add this to the other number, thus 96 + 3 = 99

Result will be same in both the cases

Second part:

Multiply the difference from 100 of both the numbers.

Hence, (96 – 100) x (103 – 100) = -4 x 3 = – 12

Combined effect: 99 | -12 = 8742

Now to remove negative sign from the right side, we have to take one from the left hand side. 1 when shifted from left to right becomes 100. Thus we’ll have:

Combined effect: 99 – 1 | 100 – 12 = 9888

- Multiply 89 and 113;

= 89 + 13 | -11 x 13

= 102 | -143

In this case, right side number is greater than 100, so we need to subtract it from next higher 100, i.e. 200. Hence, we’ve to take 2 from left hand side, so that we get 200 on the right hand side.

= 102 – 2 | 200 – 143 = 100 | 57 = 10057

- Multiply a number with multiples of 11, that is, 22, 33, 44 and so on

As we have by now learned the short cuts to multiply any number with 11, understanding multiplication with 22, 33, 44 and so on will now be easier. These are themselves multiples of 11 and to some extent the principles of multiplications are same. This has been explained with examples given below;

- Multiplication with 22, the rule is (number +next number)*2

Let us look at it step by step –

Step 1: For sake of simplicity, assume that there are two invisible 0 (zeroes) on both ends of the given number.

Say if the number is 786, assume it to be 0 7 8 6 0

Step 2: Start from the right, add the two adjacent digits and multiply by 2. Keep on moving left.

07860

Add the last zero to the digit in the ones column (6), and multiply by 2. Write the answer below the ones column.

Then add this 6 with digit on the left i.e. 8 and multiply by 2.

Next add 8 with 7 and multiply with 2.

Next add 7 with 0 and multiply by 2.

(0+7)*2 | (7+8)*2 | (8+6)*2 | (6+0)*2

= 14 | 30 | 28 | 12

Step 3: Start from right most digit. Keep only the unit’s digit. Carryover and add the ten’s digit to the next number to the left. Doing this we get the answer as 17292.

- multiplication with 33, the rule is (number +next number)*3
- multiplication with 44, the rule is (number +next number)*4

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