# Base Multiplication or Close-together Method

Multiplying Numbers Close to the base 10, 100, 1000 and so on

A)   When both the numbers are below the base

To multiply numbers that are close to the bases of powers of 10 i.e. 10, 100, 1000 and so on easily at extremely fast speed, just follow the steps as below:

Step-1:  Find deficits of both the numbers from base (10, 100 or 1000).

Step-2:  Subtract the difference between one multiplicand with the deviation of the other (from base). This is the first part of the answer. It can be arrived at in any one of the four ways.
i) Cross-subtract deviation on the second row from the first multiplicand.
ii) Cross-subtract deviation on the first row from the second multiplicand.
iii) Subtract the base from the sum of the given numbers.
iv) Subtract the sum of the two deviations from the base.

Step-3:  The last part of the answer is the product of the deviations of the numbers from base. It contains the number of digits equal to number of zeroes in the base i.e. for numbers near base 10; 1 place to go, since 10 has 1 zero, for numbers near base 100; 2 places to go since 100 has 2 zeros, accordingly since 1000 has 3 zeros, 3 places to go for numbers near base 1000 and so on. So, carry forward or put extra zero(s) if necessary to place the digits in accurate number.

The general form of the multiplication:

Let N1 and N2 be two numbers near to a given base in powers of 10, and D1 and D2 are their respective deviations from the base. Then N1 X N2 can be represented as

N1        D1
N2        D2
------------------------------------
(N1+D2) OR (N2+D1) / (D1xD2)

Suppose you need 8 x 7.

Here base is 10. See how far the numbers are below 10, subtract one number's deficiency from the other number, and multiply the deficiencies together.

8 is 2 below 10 and 7 is 3 below 10.

The diagram below shows how you get it.

You subtract crosswise 8-3 or 7-2 to get 5, the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6, the last figure of the answer.

So, 7 x 6 =?

Here there is a carry: the 1 in the 12 goes over to make 3 into 4.

Suppose you want to multiply 96 by 92

Both the numbers are near 100, so the base here is 100.

96 is 4 below the base and 92 is 8 below.

We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.

So, 88 × 98 =?

Here base is 100. 88 is 12 below 100 and 98 is 2 below 100.

You can imagine the sum set out like this:

86 comes from subtracting crosswise: 88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way, you will always get the same answer).

And, multiplying vertically both the differences (12 and 2) from 100 results in 24.
Find 75 × 95.

Deviation of 75 from 100 is -25
Deviation of 95 from 100 is -05

75           -25       [BASE 100]

95           -05
-------------------
(75-05) or (95-25) / (25X5)
--------------------
70 / 125

Since the base is 100, we write down 25 and carry 1 over to the left giving us 70 / 125 = (70+1) / 25

So, the final answer is 7,125.

786 × 998 =?

Here base is 1000

Complement of 786 is 214.

Complement of 998 is 002 (7 from 9 is 2 and 8 from 9 is 1 and 6 from 10 is 4).

786         -214     [BASE 1000]

998         -002
-------------------
(786-002) or (998-214) / (214X2)
-------------------
784 / 428

Find 994 × 988.

994         -006     [BASE 1000]

988         -012
-------------------
(786-002) or (998-214) / (214X2)
-------------------
982 / 072

Find 750 × 995.

750        -250                 [BASE 1000]

995        -005
-------------------
(750-005) or (995-250) / (250X005)
-------------------
745 / 1250

Since the base is 1000, we write down 250 and carry 1 over to the left giving us 745 / 1250 = (745+1) / 250

So, the final answer is 746,250.

568 × 998 =?

Complement of 568 is 432
Complement of 998 is 002.

568        -432     [BASE 1000]

998        -002
______________
568 - 2 / 864

=> 566 / 864

B)   When both the numbers are above the base

The technique works equally well for numbers above the base. Here we add the differences in step-2.

Find 13 × 12.

Here, base is 10 and both the numbers are above 10. So instead of subtracting we will add the difference.

13        +3        [BASE 10]

12        +2
-------------------
(13 + 2) or (12 + 3) / (3 X 2)
-------------------
15 / 6

18 × 14 =?

18        +8        [BASE 10]
14        +4
-------------------
(18 + 4) or (14 + 8) / (8 X 4)
-------------------
22 / 32

Since the base is 10, we write down 2 and carry 3 over to the left giving us 22 / 32 = (22+3) / 2

104 × 102 =?

104      +04      [BASE 100]
102      +02
-------------------
(104 + 02) or (102 + 04) / (04 X 02)
-------------------
106 / 08

103 x 104 =?

The answer is 107 and 12,

107 is just 103 + 4 (or 104 + 3), and 12 is just 3 × 4.
So, 103 x 104 = 10,712

Similarly 107 × 106 = 11,342
107 + 6 = 113 and 7 × 6 = 42

And, 105 x 111 = 11,655

Find 1275 × 1004.

1275    +275                [BASE 1000]
1004    +004
-------------------
(1275 + 004) or (1004 + 275) / (275 X 004)
-------------------
1279 / 1100

Since the base is 1000, we write down 100 and carry 1 over to the left giving us 1279 / 1100 = (1279+1) / 100

C)   When one number is above and the other is below  the base

For multiplying such numbers:
Ø  Subtract (from the number which is above the base) or add (with the number which is below the base) crosswise. Then append as many ‘0’s to the right as there are ‘0’ in base and subtract the product of both the deviation from this result.

You can also do this calculation in a very simple way: calculate ‘Vertically and Cross-wise’ as before. Subtract 1 from the first part to get the first part of the answer and find the complement of the product (last part) with base to get the last part of the answer.

Example: Find 13 × 7.

13        +3        [BASE 10]
7           -3
----------------
10    /    -9

One deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted.

Since the base 10 has one ‘0’ so put one ‘0’ after the first part of the answer 10 to get 100. Now subtract 9 from 100 to get the final answer (100-9=) 91.

Suppose you want to multiply 102 by 97

102 is 2 more than 100 and 97 is 3 less than 100.
So, subtract (102-3) or add (97+2) crosswise to get 99.
Append 2 zeros to 99 to get 9900.

Now, subtract the product of both the differences that is 6 (2x3=6) from 9900. You get 9900-6 = 9,894

So, 102 x 97 = 9894.
Another example, 107 x 98 =?

107 is 7 more than 100 and 98 is 2 below 100.
So, subtract (107-2) or add (98+7) crosswise to get 105.
Append 2 zeros to 105 to get 10500.
Now, subtract the product of both the differences (7x2=) 14 from 10500.
You get 10500-14 = 10,486

Find 108 × 94.

108         +08      [BASE 100]

94         -06
----------------
102   /   -48

Complement of 48 is 52 and 102 is decreased by 1
(102-1) / Complement of 48 = 10152

So, 998 × 1025 =?

998      -002    [BASE 1000]

1025      +025
-------------------
1023 / -050

Complement of 50 is 950 and 1023 is decreased by 1
(1023-1) / Complement of 50 = 1022950

So, the final answer is 1,022,950.

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