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Question Stats: 58% (02:11) correct 42% (02:01) wrong based on 440 sessions

### HideShow timer Statistics $$(1.00001)(0.99999) - (1.00002)(0.99998) =$$

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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Joined: 02 Sep 2009
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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

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

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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?

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

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

$$= 1 - 10^{-10} - 1 + 4*10^{-10}$$

$$=3*10^{-10}$$

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

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1
(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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8
2
I wrote it in a simple way:

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

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

$$= \frac{4}{10^{10}} - \frac{1}{10^{10}}$$

$$= \frac{3}{10^{10}}$$

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Kindly press "+1 Kudos" to appreciate 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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somalwar wrote:
I wrote it in a simple way:
(1 + 1/10^5) (1 - 1/10^5) - (1 + 2/10^5) (1 + 2/10^5) =
= 1 - 1/10^10 - (1 - 4/10^10) =
= 4/10^10 - 1/10^10
= 3/10^10

That's correct. One typo though: should be - instead of +.
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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

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

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

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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. $$3(10^{-5})$$

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

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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2
1
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?

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# Rich Cohen

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ VP  V Joined: 23 Feb 2015 Posts: 1124 (1.00001)(0.99999) - (1.00002)(0.99998) = [#permalink] ### Show Tags 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__ _________________ “The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.” Henry Wadsworth Longfellow Do you need official questions for Quant? 3700 Unique Official GMAT Quant Questions ------ SEARCH FOR ALL TAGS GMAT Club Tests EMPOWERgmat Instructor V Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 14590 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: (1.00001)(0.99999) - (1.00002)(0.99998) = [#permalink] ### Show Tags 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 _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** # Rich Cohen Co-Founder & GMAT Assassin Follow Special Offer: Save$75 + GMAT Club Tests Free
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Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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EMPOWERgmatRichC wrote:

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

$$0.001×0.004-0.002×0.001$$=?
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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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: (1.00001)(0.99999) - (1.00002)(0.99998) =  [#permalink]

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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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