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Multiplication Using a Base - Part 1 (Speed Math)

In this chapter, we can learn to multiply two numbers using base method when both numbers are close to a power of ten. (Note: speed math calculator can be used to practise problems using this method.)

Example : Calculate 98 × 93

Select $100$ as base. Subtract base from both numbers. That is, $98 - 100 = -2$ and $93 - 100 = -7$

$98$$-2$
$93$$-7$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$93 - 2 = 91$

Multiply differences to obtain right side of answer.

$-2 × -7 = 14$

Combine left and right sides, that is, $91$ and $14$

Answer is $9114$

Example : Calculate 96 × 112

Select $100$ as base. Subtract base from both numbers. That is, $96 - 100 = -4$ and $112 - 100 = 12$

$96$$-4$
$112$$12$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$112 - 4 = 108$

Multiply differences to obtain right side of answer.

$-4 × 12 = -48$

Make right side positive by borrowing $1$ from left side. This borrowed $1$ becomes $100$ when coming to right side (because base is $100$ and $1 × 100 = 100$) and therefore right side becomes $100 - 48 = 52.$ Since we borrowed $1,$ left side becomes $108 - 1 = 107$

Combine left and right sides, that is, $107$ and $52$

Answer is $10752$

Example : Calculate 103 × 115

Select $100$ as base. Subtract base from both numbers. That is, $103 - 100 = 3$ and $115 - 100 = 15$

$103$$3$
$115$$15$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$115 + 3 = 118$

Multiply differences to obtain right side of answer.

$3 × 15 = 45$

Combine left and right sides, that is, $118$ and $45$

Answer is $11845$

Example : Calculate 122 × 89

Select $100$ as base. Subtract base from both numbers. That is, $122 - 100 = 22$ and $89 - 100 = -11$

$122$$22$
$89$$-11$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$122 - 11 = 111$

Multiply differences to obtain right side of answer.

$22 × -11 = -242$

Make right side positive by borrowing $3$ from left side. This borrowed $3$ becomes $300$ when coming to right side (because base is $100$ and $3 × 100 = 300$) and therefore right side becomes $300 - 242 = 58.$ Since we borrowed $3,$ left side becomes $111 - 3 = 108$

Combine left and right sides, that is, $108$ and $58$

Answer is $10858$

Example : Calculate 1024 × 989

Select $1000$ as base. Subtract base from both numbers. That is, $1024 - 1000 = 24$ and $989 - 1000 = -11$

$1024$$24$
$989$$-11$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$1024 - 11 = 1013$

Multiply differences to obtain right side of answer.

$24 × -11 = -264$

Make right side positive by borrowing $1$ from left side. This borrowed $1$ becomes $1000$ when coming to right side (because base is $1000$ and $1 × 1000 = 1000$) and therefore right side becomes $1000 - 264 = 736.$ Since we borrowed $1,$ left side becomes $1013 - 1 = 1012$

Combine left and right sides, that is, $1012$ and $736$

Answer is $1012736$

Example : Calculate 997 × 986

Select $1000$ as base. Subtract base from both numbers. That is, $997 - 1000 = -3$ and $986 - 1000 = -14$

$997$$-3$
$986$$-14$

Find any diagonal sum to obtain left side of answer. Note that both diagonal sums will be same.

$986 - 3 = 983$

Multiply differences to obtain right side of answer.

$-3 × -14 = 42$

Since base $1000$ has three zeros, write $424$ as $042$

Combine left and right sides, that is, $983$ and $042$

Answer is $983042$

Comments(47)

profilemani
2016-02-01 09:04:18 
i can't understand the example 2
107 how comes.
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profilesam
2016-02-02 20:38:49 
Initially we got 108 as LHS.

Later, 1 was borrowed from 108 for making RHS positive.
hence LHS became 107.
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profilevijay
2015-06-25 16:07:16 
405 x 397
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profileB RAJEN TOPNO
2015-08-01 13:43:43 
405 x 397

This can be done using base 100 or 1000. But it will hardly help in solving speedily.

Using base 100

405 :305
397 :297

Left side answer = 702 (405+297 or 397+305)
Right side answer = 90585

Final answer
Right side answer = 85 ( 2 left side of the number 90585)
Left side answer = 1607 (702+ 905)

1607 85

A better method is to apply the formula (x + a)(x -b) = x2 + x(a-b) - ab

405= (400+5)
397= (400-3)

405 × 397 = (400+5)×(400-3)=1600+400(5-3)-15=160000+800-15=160785
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profileswethashri
2014-09-27 19:20:59 
how to know whether to borrow 3 or 1
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profilerajkumar
2014-12-29 15:21:57 
if you multiply  24*(-11) the answer will be -264. to make it as a positive,we need to borrow 3,since it makes 300. then 300-264=36,the answer will be positive.if we borrow 1,it makes 100,then 100-264 will be again negative number
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profileJayasri
2016-01-09 07:42:12 
Consider Example 5.

In this sum,why we have borrowed 1 rather than 3?

If we have borrowed 3 then it may become 300-264=36.
36 is also +ve number know.Then why we have borrowed 1 ?
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profilesam
2016-01-11 21:13:27 
In example 5, base is 1000

So, borrowed 1 becomes 1000
No need to borrow 3 as it becomes 3000
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profileketan
2014-07-02 08:28:23 
how we caculate the figure of 557*613??
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profileB RAJEN TOPNO
2015-08-01 13:07:16 
Taking base as 1000 ( Power of 10)

557 : -443 (1000-443)
613 : -387(1000-387)

Left side of the answer will be the diagonal sum including their signs : 170(557-387 or 613-443)

Right side of the answer will be the product of the differences : -443 × -387 = 171441 (First three digits will add up with the left side of the answer)

So the final answer is 341(170+171) 441 (Left side answer Right side answer) ie 341 441

Although this procedure is applicable but it is rather lengthy for this example.
I feel it is more appropriate for nos. which are nearer to 10,100,1000, etc.
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