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# (1.0002)(0.9999) – (1.0001)(0.9998) =

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Manager
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18 Apr 2012, 08:53
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Difficulty:

75% (hard)

Question Stats:

60% (02:31) correct 40% (02:23) wrong based on 350 sessions

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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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18 Apr 2012, 08:59
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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.

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##### General Discussion
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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....

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Rich
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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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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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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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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?

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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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

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

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

$$= 1 - 1*10^{-4} + 2*10^{-4} - 2*10^{-8} - 1 + 2*10^{-4} - 1*10^{-4} + 2*10^{-8}$$

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

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

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

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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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20 Jun 2018, 04:37
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

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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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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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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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
Re: (1.0002)(0.9999) – (1.0001)(0.9998) =   [#permalink] 06 Sep 2019, 00:52
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