Tips And Tricks For Speedy Calculations Module 2 Roots



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.

1. Multiplying two numbers starting with same digit(s)

Let us take an example of one two digits and another three digit numbers;

1. 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

2. 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

2. 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.

3. 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

4. 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;

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

Let us now take some examples;

1. 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)

2. Multiply 99999 x 78

= 78-1   | 100000 – 78

= 7799922

{78 should be subtracted from (99999+1)

3. Multiply 1203579 × 9999999

= 1203579-1   | 10000000- 1203579

=12035788796421

5. 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;

1. 36 x 11 = 0+3   |   3+6   | 6+0  = 396
2. 74 x 11 =0+ 7 |  7+4 |  4+0 =  7  | ₁1 |  4 = 814   (1 of 11 is carried over and added to next digit, so 7+1 = 8 )
3. 6349 x 11 = (0+6)  |  (6+3)   |   (3+4)   |   (4+9)  |   9+0 =  69839

6. 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.

7. 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

8. 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

9. 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.

1. Multiply 45 and 46;

First part: 4 x (4+1)

Second part: (4 x 6)

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

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

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