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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.)

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 $9114$

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 $10752$

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 $11845$

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 $10858$

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 $1012736$

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 $983042$

sam

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.

Later, 1 was borrowed from 108 for making RHS positive.

hence LHS became 107.

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B 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) = x^{2} + 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

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)

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

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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rajkumar

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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Jayasri

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 ?

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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sam

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

So, borrowed 1 becomes 1000

No need to borrow 3 as it becomes 3000

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B 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.

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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