direktori wisata tirta 2012 (3)

8/12/2019 Direktori Wisata Tirta 2012 (3)

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Dhaval Bathia


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Dhaval Bathia

The Base Method is a technique used for multiplication

Suppose I ask you how long will you take to multiply9999998 by 9999999 ?

Perhaps you would sink at the sheer thought of multiplying

these numbers

However, using the Base Method you can get the answer in

less than 5 seconds !

Yes, it is true !


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In the base method, we use certain bases.

Normally, we use numbers like 10, 100, 1000 etc. as

bases,.

However this is not a rule.

You can use any other number toobut generally we prefer

powers of 10


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We select a particular base depending on the numbers

given in the question

Suppose we are asked to multiply 996 by 998, in this case

we take 1000 as the base as both the numbers are closer to

1000

Suppose we are asked to multiply 105 by 112, in this case

we take 100 as the base as both the numbers are closer to

100..

In the questions that follow, we will solve the answer in two

parts

The left hand part will be called LHS and the right hand partwill be called RHS


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Let us have a look at the steps involved in the procedure..


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STEPS

Find the Base and the Difference

Number of digits in RHS = Number of zeros in Base

Multiply the differences in RHS

Put the cross answer in LHS

These are the four basic rules of the system. However,

they will only become clear when we solve some

examples..We will solve three examples simultaneously..


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Dhaval Bathia

(Q) Find the answers to the following questions.

(A) (B) (C)

9 6 9 9 8 8 9 9 9 9 9 9 9

x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9


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Dhaval Bathia

STEP AFind the Base and the Difference

The first part says find the base and the difference.

Let us have a look at example A. The numbers are 96 and

98. Since, both the numbers are closer to 100 our base in

this case will be 100.

Similarly, in example B both the numbers are closer to

10,000 so our base will be 10,000

In Example C, both the numbers are closer to 1,00,00,000

and hence we will take 10 million (1 crore) as our base


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So, our bases will be

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 9 9 8 8 9 9 9 9 9 9 9

x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9


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We are still on Step A.

Next, we have to find the differences

In Example A, the difference between 100 and 96 is 4 and the

difference between 100 and 98 is 2

In Example B, the difference of 9988 from 10,000 is 12 and thedifference of 9996 from 10,000 is 4

In Example C, the difference of both the numbers from ten million is 1

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 9 9 8 8 9 9 9 9 9 9 9

x 9 8 x 9 9 9 6 x 9 9 9 9 9 9 9


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Step A is complete. The question looks as given below :

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 9 - 1


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Dhaval Bathia

Let us make a provision for it..

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

_ _ _ _ _ _ _ _ _ _ _ _ _

We have put equivalent empty blanks in the RHS to

accommodate the answer


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Dhaval Bathia

We come to Step C which says, multiply the differences and

put it as the RHS of the answer

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

_ _ _ _ _ _ _ _ _ _ _ _ _


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

0 8

In Example A, we multiply the differences -4 by -2and get the

answer 8. But, we need a two-digit answer. So, we express thenumber 8 as 08 and put it in the RHS.


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

0 8 0 0 4 8

In Example B, we multiply the differences -12 by -4and get the

answer 48. But, we need a four-digit answer. So, we express the

number 48 as 0048 and put it in the RHS.


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

0 8 0 0 4 8 0 0 0 0 0 0 1

In Example C, we multiply the differences -1 by -1and get the

answer 1. But, we need a seven-digit answer. So, we express the

number 1 as 0000001 and put it in the RHS.


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

9 40 8 0 0 4 8 0 0 0 0 0 0 1

In Example A, the cross answer can be obtained by doing (962)or (984) in a cross manner. In either case, the answer is 94.

Thus, the complete answer of 96 x 98 is 9408


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

9 4 0 8 9 9 8 40 0 4 8 0 0 0 0 0 0 1

In Example B, the cross answer can be obtained by doing (998812) or (99964) in a cross manner. In either case, the answer is

9984.

Thus, the complete answer of 9998 x 9996 is 99840048


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Dhaval Bathia

(A) (B) (C)

100 1 0 0 0 0 1 0 0 0 0 0 0 0

9 6 - 4 9 9 8 8 - 12 9 9 9 9 9 9 9 - 1

x 9 8 - 2 x 9 9 9 6 - 4 x 9 9 9 9 9 9 91

9 4 0 8 9 9 8 40 0 4 8 9 9 9 9 9 9 80 0 0 0 0 0 1

In Example C, the cross answer can be obtained by doing(99999991) in a cross manner. In either case, the answer is

9999998.

Thus, the complete answer of 99999980000001


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Let us do one complete example.

(Q) Multiply 99980 by 99980


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9 9 9 8 0

x 9 9 9 8 0

Both the numbers are closer to 1,00,000so the base is 1, 00,000The difference between the base and 99,980 is 20 each

The base has 5 zeros and so RHS will be a 5 digit answer

Multiplying the differences will yield 20 x 20 = 400. We write the

number as 00400

The cross answer is 99,98020 = 99,960.

The final answer is 9996000400

1,00,000

- 20

- 20

_ _ _ _ _0 0 4 0 09 9 9 6 0


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9 9 9

x 8 5 0

Both the numbers are closer to 1,000so the base is 1, 000The difference between the base and the numbers is (-1) and (-

150) respectively


Multiplying the differences will yield 1 x 150 = 150. We write thenumber as 0150

The cross answer is 8501 = 849

1,000

- 1

- 150

_ _ _ _0 1 5 08 4 9


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Dhaval Bathia

Let us take an example where both the numbers are above

the base



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1 0 0 2

x 1 0 0 3

Both the numbers are closer to 1,000so the base is 1, 000The difference between the base and the numbers is (+2) and (+3)

respectively


Multiplying the differences will yield 2 x 3 = 6. We write the numberas 006

The cross answer is 1002 + 3 = 1005. The complete answer is

1005006.

1,000

+ 2

_ _ _0 0 61 0 0 5

+ 3


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Let us take an example where both the numbers are above

the base



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1 0 0 1 0

x 1 0 0 1 0

Both the numbers are closer to 10,000so the base is 10,000The difference between the base and the numbers is (+10) and

(+10) respectively


Multiplying the differences will yield 10 x 10 = 100. We write thenumber as 0100

The cross answer is 10010 + 10 = 10020. The complete answer is

100200100.

10,000

+ 10

_ _ _ _0 1 0 01 0 0 2 0

+ 10


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Sometimes you might have a number where there is a

carry-over involved. Let us see how to deal with such

cases.

Suppose you have to multiply 950 by 950


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9 5 0

x 9 5 0

Both the numbers are closer to 1,000so the base is 1,000

The difference is (-50) and (-50) respectively


Multiplying the differences will yield (-50) x (-50) = 2500. The number

2500 is a four-digit number but we can fit only 3 digits. So, we write downthe last three digits (500) and carry over 2.

The cross answer is 95050 = 900 (+2 carry over) is 902.

The final answer is 902500

1,000

- 5 0

_ _ _9 0 0

- 5 0

5 0 02

9 0 2 5 0 0


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1 2 0 0

x 1 0 2 0

Both the numbers are closer to 1,000so the base is 1,000

The difference between the base and the numbers is (+200) and(+20) respectively


Multiplying the differences will yield (+200) x (+20) = 4000. The

number 4000 is a four-digit number but we can fit only 3 digits. So,

we write down the last three digits (000) and carry over 4.

The cross answer is 1200 + 20 (+4 carry over) is 1224.

1,000

+ 2 0 0

_ _ _1 2 2 4

+ 2 0

0 0 0


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Dhaval Bathia

We come to the last part of this topic..

How to multiply one number above the base and one

number below the baseSuppose you have to multiply 95 by 115.

In this case, 95 is below the base (100) and 115 is above

the base (100)

In such examples, where one number is above the base

and one number is below the base. we use the same

technique with a different last step.

Let us have a look at an example :


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9 5

x 1 1 5

The base is 100 and the differences are (-5) and (+15)

respectively. RHS will be a 2 digit answer

Multiplying the differences will yield (-5) x (+15) = (-75). The LHS

will be 110. Our final answer is 110(-75). However, minus sign isnot permitted in the final answer.

In such cases, we multiply LHS with Base and subtract RHS to get

the final answer.The final answer will be 110 x 10075 = 10925.

100

- 5

_ _1 1 0

+ 1 5

(- 7 5 )

1 0 9 2 5


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1 0 4 2

x 9 9 8

The base is 1000 and the differences are (+42) and (-2)

respectively. RHS will be a 3 digit answer

Multiplying the differences will yield (+42) x (-2) = (-084). The LHS

will be 1040. Our final answer is 1040(-084).We multiply LHS with Base and subtract RHS to get the final

answer.The final answer will be 1040 x 1000(084) = 1039916.

1000

+ 4 2

_ _ _1 0 4 0

- 2

(- 0 8 4)

1 0 3 9 9 1 6


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Thus, we have seen many variations with the Base Method.

You can use this method for a variety of multiplication problems


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