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(1.0002)(0.9999) – (1.0001)(0.9998) =

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(1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 18 Apr 2012, 08:53
5
1
20
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A
B
C
D
E

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Question Stats:

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(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002

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(1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 18 Apr 2012, 08:59
11
5
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002



Method #1:
Don’t actually multiply these numbers out! The key here is to substitute a variable for a tiny value that shows up in several places.

Specifically, let x = 0.0001. Now you can rewrite all the numbers as 1 plus or minus x or 2x.

(1.0002)(0.9999) – (1.0001)(0.9998)
= (1 + 2x)(1 – x) – (1 + x)(1 – 2x)

Now, distribute each product in the expression separately.

First product: (1 + 2x)(1 – x) = 1 + 2x – x – = 1 + x –

Second product: (1 + x)(1 – 2x) = 1 + x – 2x – = 1 – x –

Subtract the two products:

1 + x – – (1 – x – )
= 1 + x – – 1 + x +
= 2x

Finally, substitute back in for x. The difference is 2(0.0001) = 0.0002.

It might seem odd to solve an arithmetic problem by turning it into algebra! But in this case, doing so saves you a ton of work.

Method #2:
Another way to tackle the problem is to estimate judiciously. Of course, if you round every number to 1 in the original expression, you get 0. But consider rounding the numbers in this way:

(1.0002)(0.9999) – (1.0001)(0.9998)
≈ (1.0002)(1.0000) – (1.0001)(0.9999) — round both of the second numbers up by 0.0001, and because the first numbers in each product are approximately equal, you’ll only have a truly small rounding error
= 1.0002 – (1.0001)(0.9999). Now, if this were 1.0002 – 1.0001, you’d get 0.0001. But you’re subtracting something even smaller (since the 1.0001 is being multiplied by a number less than 1), so the difference must be larger than 0.0001.

The correct answer is E.
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(1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post Updated on: 16 Jun 2018, 14:42
Hi All,

This question actually has some nice shortcuts built into it that can help you avoid doing math the "long way":

The first part of the calculation...

(1.0002)(0.9999)

…will have 8 decimal places (4 decimal points x 4 decimal points = 8 total decimal points) and the last digit will be an 8 (since 2 x 9 = 18)

The second part of the calculation….

(1.0001)(0.9998)

….will also also have 8 decimal places (for the same reason that the first part has 8 decimal points) and the last digit will also be an 8 (1 x 8 = 8)

When subtracting the second value from the first value, the resulting number will have a '0' in the 8th decimal 'spot.' That clearly doesn't happen in the first 3 answer choices, so we can eliminate Answers A, B and C.

To get the exact correct answer, we have to do a little more work. This time, I'll start by breaking the second calculation into two pieces:

(1.0001)(0.9998) =
(1)(0.9998) + (.0001)(0.9998) =
0.9998 +
0.00009998
---------
0.99989998

With the first calculation, we have to pay a bit more attention to the number of digits involved (remember though - there's still only 8 total decimal points):

(1.0002)(0.9999) +
(1)(0.9999) + (.0002)(0.9999) =
0.9999 +
0.00019998
---------
1.00009998

1.00009998 -
0.99989998

You should notice the 5th through 8th decimal points 'cancel out':

1.0000 -
0.9998

....leaving us with a 4th decimal point that must be a 2....

Final Answer: [spoiler=]E/spoiler]

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Originally posted by EMPOWERgmatRichC on 11 Sep 2016, 18:48.
Last edited by EMPOWERgmatRichC on 16 Jun 2018, 14:42, edited 1 time in total.
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(1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 18 Feb 2018, 08:28
4
1
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002


(1.0002)(0.9999) – (1.0001)(0.9998)
= (1.0001+0.0001)(0.9999) – (1.0001)(0.9998) <Expanding 1.0002>
= (1.0001)(0.9999) + (0.0001)(0.9999) - (1.0001)(0.9998)
= (1.0001)(0.9999 - 0.9998) + (0.0001)(0.9999) <Taking out common term>
= (1.0001)(0.0001) + (0.0001)(0.9999)
= (0.0001)(1.0001 + 0.9999) <Taking out common term>
= 0.0001(2) = 0.0002(Option E)
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(1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 18 Feb 2018, 09:11
2
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002



remove the decimal before calculations and get that back in the final answer..

so we reomove 8 decimal points from .0002 and .9999 and similarly from second term..

\(10002*9999-10001*9998 = (10000+2)(10000-1)-(10000+1)(10000-2) = 10000^2-2*10000-10000*1-10000^2+2*10000-1*10000 = 40000-20000=20000\)

now get 8 decimal points back in the answer
\(\frac{20000}{10^8} = 0.0002\)

E
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New post 16 Jun 2018, 08:49
2
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


You changed the question. You have 8 decimal places both sides plus you changed the actual numbers.


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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 16 Jun 2018, 09:24
1
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002


\((1.0002)(0.9999) – (1.0001)(0.9998) = (1 + 0.0002)(1 - 0.0001) - (1 + 0.0001)(1 - 0.0002)\)

\(= (1 + 2*10^{-4})(1 - 1*10^{-4}) - (1 + 1*10^{-4})(1 - 2*10^{-4})\)

\(= 1 - 1*10^{-4} + 2*10^{-4} - 2*10^{-8} - 1 + 2*10^{-4} - 1*10^{-4} + 2*10^{-8}\)

\(= 10^{-4} * (-1 + 2 + 2 - 1) = 0.0002\)

Answer E.


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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 16 Jun 2018, 14:44
Hi Antreev,

Thank you for catching that error. I accidentally copied over an explanation for a similar OG question (in the OG2018, it's PS question #216 on pg. 178). I've edited my original post accordingly.

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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 16 Jun 2018, 17:30
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002

One more approach:

\((1.0002)(0.9999) – (1.0001)(0.9998)\)

\(=\) \((1 + 0.0001\) \(+\) \(0.0001)\)\((1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001\) \(-\) \(0.0001)\)

\(=\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 + 0.0001)\)

\(=\) \(0.0001 - (0.0001)^2 + 0.0001 + (0.0001)^2\)

\(= 0.0002\)


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New post 17 Jun 2018, 00:44
1
GMATGuruNY wrote:
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002

One more approach:

\((1.0002)(0.9999) – (1.0001)(0.9998)\)

\(=\) \((1 + 0.0001\) \(+\) \(0.0001)\)\((1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001\) \(-\) \(0.0001)\)

\(=\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 + 0.0001)\)

\(=\) \(0.0001 - (0.0001)^2 + 0.0001 + (0.0001)^2\)

\(= 0.0002\)



OR after the last third step you could have just cancelled first and third items as they have opposite signs and solved the rest like this.
0.0001(1-0.0001+1+0.0001)
0.0001(2)
0.0002 so E


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New post 17 Jun 2018, 05:04
Antreev wrote:
GMATGuruNY wrote:
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002

One more approach:

\((1.0002)(0.9999) – (1.0001)(0.9998)\)

\(=\) \((1 + 0.0001\) \(+\) \(0.0001)\)\((1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001\) \(-\) \(0.0001)\)

\(=\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 - 0.0001)\) \(-\) \((1 + 0.0001)(1 - 0.0001)\) \(+\) \((0.0001)(1 + 0.0001)\)

\(=\) \(0.0001 - (0.0001)^2 + 0.0001 + (0.0001)^2\)

\(= 0.0002\)



OR after the last third step you could have just cancelled first and third items as they have opposite signs and solved the rest like this.
0.0001(1-0.0001+1+0.0001)
0.0001(2)
0.0002 so E


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Yes, indeed.
I considered this approach but suspected that most test-takers would intuitively distribute 0.0001 in the second and fourth terms rather than factor it out.
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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 20 Jun 2018, 04:37
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GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002



We can write decimals as..

[ \(\frac{10002}{10^4}\) * \(\frac{9999}{10^4}\) ] - [ \(\frac{10001}{10^4}\) *\(\frac{9998}{10^4}\) ]

Let say, x = 10000 = \(10^4\)

Solving for numerator, since the denominator is \(10^8\)

\((x+2)(x-1) - (x+1)(x-2)\)

\((x^2 + x - 2) - (x^2 - x - 2)\) = \(2x\)

Final solution = \(\frac{2x}{10^8}\) = \((2*10^4)/10^8\) = \(\frac{2}{10^4}\)= 0.0002
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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 20 Jun 2018, 07:12
(1.0002)(0.9999)-(1.0001)(0.9998)
=(1.0001+0.0001)(0.9998+0.0001)-(1.0001)(0.9998)
1.0001=a
0.0001=x
0.9998=b
=(a+x)(b+x)-ab
=ab+x(a+b)+x^2-ab
=x(a+b+x)
=0.0001*2
=0.0002
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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 20 Jun 2018, 08:42
siddharthabingi wrote:
GreginChicago wrote:
(1.0002)(0.9999) – (1.0001)(0.9998) =

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002



We can write decimals as..

[ \(\frac{10002}{10^4}\) * \(\frac{9999}{10^4}\) ] - [ \(\frac{10001}{10^4}\) *\(\frac{9998}{10^4}\) ]

Let say, x = 10000 = \(10^4\)




Solving for numerator, since the denominator is \(10^8\)

\((x+2)(x-1) - (x+1)(x-2)\)

\((x^2 + x - 2) - (x^2 - x - 2)\) = \(2x\)

Final solution = \(\frac{2x}{10^8}\) = \((2*10^4)/10^8\) = \(\frac{2}{10^4}\)= 0.0002




Thats a very efficient solution. Smart!!
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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =  [#permalink]

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New post 06 Sep 2019, 00:52
(1.0002)*(0.9999)-(1.0001)*(0.9998) Can also be expressed as
(1+2/10^4)*(1-1/10^4)-(1+1/10^4)*(1-2/10^4)

if 1/10^4 = a , then
(1+2a)*(1-a)-(1+a)*(1-2a)
= 1+2a-a-2a^2-1-a+2a+2a^2
=4a-2a
=2a
=2*1/10^4
=0.0002
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Re: (1.0002)(0.9999) – (1.0001)(0.9998) =   [#permalink] 06 Sep 2019, 00:52
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