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(1.00001)(0.99999) - (1.00002)(0.99998) =

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(1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 24 Nov 2012, 11:39
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\((1.00001)(0.99999) - (1.00002)(0.99998) =\)

A. 0

B. \(10^{-10}\)

C. \(3(10^{-10})\)

D. \(10^{-5}\)

E. \(3(10^{-5})\)

The solution suggests to turn all the expressions into scientific notation and then simplify but I was wondering if anyone here had any mind boggling tricks to do this in a more elegant manner?
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 24 Nov 2012, 12:33
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anon1 wrote:
(1.00001)(0.99999) - (1.00002)(0.99998) =

A. 0
B. 10^-10
C. 3(10^-10)
D. 10^-5
E. 3(10^-5)

The solution suggests to turn all the expressions into scientific notation and then simplify but I was wondering if anyone here had any mind boggling tricks to do this in a more elegant manner?


Apply \(a^2-b^2=(a+b)(a-b)\).

\(1.00001*0.99999-1.00002*0.99998=(1+0.00001)(1-0.00001)-(1+0.00002)(1-0.00002)=\)
\(=1^2-0.00001^2-1^2+0.00002^2=0.00002^2-0.00001^2\).

Next, \(0.00002^2-0.00001^2=(0.00002+0.00001)(0.00002-0.00001)=0.00003*0.00001=3*10^{-5}*10^{-5}=3*10^{-10}\).

Answer: C.
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 24 Nov 2012, 12:41
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anon1 wrote:
(1.00001)(0.99999) - (1.00002)(0.99998) =

A. 0
B. 10^-10
C. 3(10^-10)
D. 10^-5
E. 3(10^-5)

The solution suggests to turn all the expressions into scientific notation and then simplify but I was wondering if anyone here had any mind boggling tricks to do this in a more elegant manner?


\(= (1 + 0.00001)(1 - 0.00001) - (1 + 0.00002)(1 - 0.00002)\)

\(= (1 + 10^{-5})(1 - 10^{-5}) - (1 + 2*10^{-5})(1 - 2*10^{-5})\)

\(= 1 - 10^{-10} - 1 + 4*10^{-10}\)

\(=3*10^{-10}\)

Answer is hence C.

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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 24 Nov 2012, 12:28
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(1.00001)(0.99999) - (1.00002)(0.99998) = (100001/100000)*(99999/100000) - (100002/100000)*(99998/100000) =
Right here from the get go we see that the denominator of the expression above is 100000*100000 = 10^(-10) - Denominator, and we live it right there and focus on the numerator.
100001*99999 = (100000+1)*99999=100000*99999+99999
100002*99998 = (100000+2)*99998 = 100000*99998+2*99998
= 100000*99999+99999-100000*99998-2*99998 = 100000+99999-2*99998 = 100000+99999-2*(99999-1)=100000+99999-2*99999+2=100000-99999+2=3 - Numerator
Or, 3*(10^(-10) = 3/10^(10)
The numbers might look intimidating and cumbersome, but a little manipulation goes a long way. Or, once you got the denominator you can cross out some answer choices and guess in case you running out of time.
Please, feel free to correct me, if I went awry
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post Updated on: 27 Aug 2014, 03:12
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I wrote it in a simple way:

\((1 + \frac{1}{10^5}) (1 - \frac{1}{10^5}) - (1 + \frac{2}{10^5}) (1 - \frac{2}{10^5})\)

\(= 1 - \frac{1}{10^{10}} - (1 - \frac{4}{10^{10}})\)

\(= \frac{4}{10^{10}} - \frac{1}{10^{10}}\)

\(= \frac{3}{10^{10}}\)

Answer = C
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Originally posted by PareshGmat on 30 May 2013, 02:51.
Last edited by PareshGmat on 27 Aug 2014, 03:12, edited 3 times in total.
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 30 May 2013, 02:59
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 17 Nov 2016, 15:31
In short, this is a very messy way of dealing with factoring.

Think (a+1)(a-1), etc.

1. We need to rewrite decimals, so they are in fractional form.

{[100,001/100,000][999,999/100,000]} - {[100,002/100,000][99998/100,000]}

2. Trick here is to recognize that we can write each of the numerator as a process applied to 100,000 (i.e. x, -, +, etc)

{[(100,000+1)(100,000-1)]/(10,000,000,000)} - {(100,000+2)(100,000-2)/(10,000,000,000)}

3. Simplify by using knowledge that (a+1)(a-1) = a^2 - 1 & (a+2)(a-2) = a^2 -4

{(10,000,000,000-1) -(10,000,000,000-4)}/(10^10) --> 3x10^(-10)
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 10 Jan 2017, 04:35
anon1 wrote:
(1.00001)(0.99999) - (1.00002)(0.99998) =

A. 0
B. 10^-10
C. 3(10^-10)
D. 10^-5
E. 3(10^-5)

The solution suggests to turn all the expressions into scientific notation and then simplify but I was wondering if anyone here had any mind boggling tricks to do this in a more elegant manner?


1.00001*0.99999 .... ends in 9, and 1.00002*0.99998 ends in 6 , thus answer has to be in the form 3*10^-x , option c or E

but since multiplication of each term around the subtraction operation will yield 10^-x < 10^-5 thus x > 5 .... thus option c
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New post 03 Nov 2017, 03:47
Does anyone know of similar question types for additional practise?
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 03 Nov 2017, 04:25
Edofarmer wrote:
Does anyone know of similar question types for additional practise?


Similar questions:
https://gmatclub.com/forum/1-130872.html
https://gmatclub.com/forum/999-999-2-1- ... 05499.html
https://gmatclub.com/forum/topic-143744.html
https://gmatclub.com/forum/3-109271.html
https://gmatclub.com/forum/topic-144735.html
https://gmatclub.com/forum/the-value-of ... 30682.html
https://gmatclub.com/forum/1-0-0001-0-0 ... 59398.html

For more check the questions under the tag of Arithmetic
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New post 03 Nov 2017, 22:11
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 06 Nov 2017, 17:30
anon1 wrote:
\((1.00001)(0.99999) - (1.00002)(0.99998) =\)

A. 0

B. \(10^{-10}\)

C. \(3(10^{-10})\)

D. \(10^{-5}\)

E. \(3(10^{-5})\)


Let’s rewrite the expressions using scientific notation:

(1 + 10^-5)(1 - 10^-5) - (1 + 2 x 10^-5)(1 - 2 x 10^-5)

1^2 - (10^-5)^2 - [1^2 - 2^2 x (10^-5)^2]

1 - 10^-10 - 1 + 4 x 10^-10

3 x 10^-10

Answer: C
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 28 Jan 2018, 15:00
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Hi All,

This question actually has a really big shortcut built into it that will allow you to avoid most of the "long math":

The first part of the calculation...

(1.00001)(0.99999)

…will have 10 decimal places (5 decimal points x 5 decimal points = 10 total decimal points) and the last digit will be a 9 (1 x 9 = 9

The second part of the calculation….

(1.00002)(0.99998)

….will also have 10 decimal places (for the same reason that the first part has 10 decimal points) and the last digit will be a 6 (2 x 8 = 16)

From the answers, we know that we'll be dealing with 10 to some "negative power"; subtracting the second number from the first would give us…

._ _ _ _ _ _ _ _ _ 9
._ _ _ _ _ _ _ _ _ 6
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._ _ _ _ _ _ _ _ _ 3

So, which answer has a "3" in it and implies 10 decimal points?

Final Answer:

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(1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 02 Jul 2019, 03:14
EMPOWERgmatRichC wrote:
Hi All,

This question actually has a really big shortcut built into it that will allow you to avoid most of the "long math":

The first part of the calculation...

(1.00001)(0.99999)

…will have 10 decimal places (5 decimal points x 5 decimal points = 10 total decimal points) and the last digit will be a 9 (1 x 9 = 9

The second part of the calculation….

(1.00002)(0.99998)

….will also have 10 decimal places (for the same reason that the first part has 10 decimal points) and the last digit will be a 6 (2 x 8 = 16)

From the answers, we know that we'll be dealing with 10 to some "negative power"; subtracting the second number from the first would give us…

._ _ _ _ _ _ _ _ _ 9
._ _ _ _ _ _ _ _ _ 6
__________________
._ _ _ _ _ _ _ _ _ 3

So, which answer has a "3" in it and implies 10 decimal points?

Final Answer:

GMAT assassins aren't born, they're made,
Rich

EMPOWERgmatRichC
I wanted to dobthis math like you, but can't figure out the decimal part! Why not E?
Does the highlighted part always introduce the first part like option C (3(10^-10) and E (3(10^-5) ?
Thanks__
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 02 Jul 2019, 14:51
Hi Asad,

To start, it's important to remember a couple of things about GMAT questions:

1) NOTHING about a GMAT question is ever "random"; each prompt is carefully designed to 'test' you on specific concepts - and in many cases, provide you with potential shortcuts that you can use to avoid a long-winded, math-heavy approach (and that's because the Quant section is NOT a 'math test' - it's a 'critical thinking test' that requires lots of little calculations as you work through it).

2) Even the answer choices are specifically designed - often with patterns in mind - that can help you to think about what the question is asking for. The exact format that the answers take can vary - and sometimes it's actually beneficial to 'rewrite' the answer choices in a different format (for example, changing fractions to decimals; in this case, it's thinking of "negative powers of 10" as decimals).

Answer E won't be correct because it only has 5 decimal places - and the correct answer will have TEN decimal places

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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 02 Jul 2019, 15:04
EMPOWERgmatRichC wrote:
Hi Asad,

To start, it's important to remember a couple of things about GMAT questions:

1) NOTHING about a GMAT question is ever "random"; each prompt is carefully designed to 'test' you on specific concepts - and in many cases, provide you with potential shortcuts that you can use to avoid a long-winded, math-heavy approach (and that's because the Quant section is NOT a 'math test' - it's a 'critical thinking test' that requires lots of little calculations as you work through it).

2) Even the answer choices are specifically designed - often with patterns in mind - that can help you to think about what the question is asking for. The exact format that the answers take can vary - and sometimes it's actually beneficial to 'rewrite' the answer choices in a different format (for example, changing fractions to decimals; in this case, it's thinking of "negative powers of 10" as decimals).

Answer E won't be correct because it only has 5 decimal places - and the correct answer will have TEN decimal places

GMAT assassins aren't born, they're made,
Rich

So, if the question asks:
\(0.001×0.004-0.002×0.001\)=?
Can I easily write the answer like 2×10^(-6) ?
Calculation:
The last digit of first part (after multiplication) is 4
and
The last digit of 2nd part (after multiplication) is 2
So, 4-2=2
and the decimal point is total 6.
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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New post 02 Jul 2019, 19:09
Hi Asad,

You picked a remarkably simple example, but that example DOES fit. Keep in mind that if you change ANY of those individual 0's to a non-0 digit, then the calculation will NOT fit that pattern.

One of the reasons why I was able to use the approach I used was because the answer choices (with the exception of Answer A) were all written as a decimal, followed by a bunch of zeroes, and ending in a non-0 digit. Thus, I knew that almost all of the digits in the overall calculation would 'cancel out' and I'd be left with a decimal, a string of 0s and a single digit.

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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =   [#permalink] 02 Jul 2019, 19:09
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