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Multiplication by Nines (Speed Math)

This method can be used to multiply any number by $99, 999,$ etc. It can be divided into three cases, as explained below. (Note: speed math calculator can be used to practise problems using this method.)

Case 1: Number of Digits < Number of Nines

Example: Calculate 18 × 999

Number of digits in $18$ is less than number of digits in $999.$ Hence, write $18$ as $018$ by making number of digits equal. Subtract $1$ from $018$ to obtain left side of answer.

$018 - 1 = 017$

Subtract each digit of $017$ from $9$ to obtain right side of answer.

$9 - 0 = 9, 9 - 1 = 8, 9 - 7 = 2.$ That is, $982.$

Combine left and right sides, that is, $017$ and $982$

Answer $= 017982$

Example: Calculate 758 × 99999

Number of digits in $758$ is less than number of digits in $99999.$ Hence, write $758$ as $00758$ by making number of digits equal. Subtract $1$ from $00758$ to obtain left side of answer.

$00758 - 1 = 00757$

Subtract each digit of $00757$ from $9$ to obtain right side of answer.

$9 - 0 = 9, 9 - 0 = 9, 9 - 7 = 2, 9 - 5 = 4, 9 - 7 = 2.$ That is, $99242.$

Combine left and right sides, that is, $00757$ and $99242$

Answer $= 0075799242$

Case 2: Number of Digits = Number of Nines

Example: Calculate 24 × 99

Subtract $1$ from $24$ to obtain left side of answer.

$24 - 1 = 23$

Subtract each digit of $23$ from $9$ to obtain right side of answer.

$9 - 2 = 7, 9 - 3 = 6.$ That is, $76.$

Combine left and right sides, that is, $23$ and $76$

Answer $=2376$

Example: Calculate 2358 × 9999

Subtract $1$ from $2358$ to obtain left side of answer.

$2358 - 1 = 2357$

Subtract each digit of $2357$ from $9$ to obtain right side of answer.

$9 - 2 = 7, 9 - 3 = 6, 9 - 5 = 4, 9 - 7 = 2.$ That is, $7642.$

Combine left and right sides, that is, $2357$ and $7642$

Answer $= 23577642$

Case 3: Number of Digits > Number of Nines

Example: Calculate 897 × 99

$897 × 99 = 897(100 - 1) = 89700 - 897 = 88803$

Example: Calculate 23683 × 999

$23683 × 999 = 23683(1000 - 1) = 23683000 - 23683 = 23659317$

Comments(21)

profileDeepak
2014-10-16 13:32:35 
Great but its trick doesn't follow $(45879×999=?)$ this type simplification
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profileAniket
2015-11-19 12:29:06 
best way is $45879(1000-1)$
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profileRaj
2014-07-10 19:33:34 

How to do this $12158×999$

Bz, $12158-1=12157$
$99999-12157=87842$

Here, right Answer is $12145842,$ where center value $45$ How to get in answer?

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profiles
2015-02-11 10:12:09 
When number of 9s are less than number of digit in other number The method is slightly different

here 12158*999 .It has 3 9's and other number has 5 digit

So divide the other number in to two group from right .Where right most group contain as much digit as number of 9s

that is 12158 to 2 group 12 and 158(because 3 9's)
then add 1 to the left group 12+1=13
now subtract this number from original number : 12158 - 13 = 12145

now take right most group 158 subtract it form 999+1 : 1000 - 158 = 842

So the result is combine this group 12145842

example 2

45612*999

45 and 612
45+1=46

45612 - 46 = 45566 => 1
1000 - 612 = 388 => 2

combine 1 and 2 => 45566388

example 3

4214*999

4 and 214

4214 - 5 = 4209
1000 - 214 = 786

so answer is 4209786
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profileAnuradha
2014-07-14 20:28:46 

I don't think this method can be used for problems like $12158×999$ where number of digits > number of $9\text{s}$

One method for such cases is $12158×999=12158(1000-1)=12158000-12158=12145842$

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profileMaruthu
2014-03-13 17:58:59 
8413*9999

1) 8413-1=8412
2)9999+1=10000-8413=1587

Ans: 84121587
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profileABDUL MUKEEM
2013-10-10 14:46:20 

$758×99999$

to solve these type of question, there are $5$ nines$(9)$ so add zeros with digit as
$75800000$
i added $5$ zeros becoz of $5$ times of nine$(9)$

now subtract digit from it
$75800000-758=75799242$

this is the required ans.

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profilesunitha naidu
2016-02-17 05:21:37 
$758(100000-1)=75800000-758=75799242$
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profilehari
2013-09-18 09:12:48 
$1.~23458×999=\\2.~786×99=$
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profileSANJAY PAUL
2015-09-15 12:19:30 

Just quick trick.
$786×99$

First take last $2$ digit: $86$
$100-86=14$
Write down at right: $14$

Then add $1$ to first digit: $1+7=8$
Then $786-8=778$

Right answer is $77814$

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