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# 0.99999999/1.0001 - 0.99999991/1.0003 =

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EMPOWERgmat Instructor
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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03 Sep 2018, 21:43
3
Hi,

To start, since this question has no variables, we know that we're going to be doing some type of math to get to the answer. Looking at the answer choices, they're all written in the same 'format' - and can be rewritten as a decimal point followed by a bunch of 0s and then a single non-0 digit. When the GMAT gives you an 'ugly looking' fraction to work with, you can often 'rewrite' what you've been given (potentially getting rid of the fraction entirely by reducing it or reformatting it).

The first fraction is .99999999/1.0001.... we can multiply both the numerator and denominator by 10,000... which gives us....

9999.9999/10001

You might recognize a pattern here (there will almost certainly be a string of 9s here) Even if you don't spot the pattern though, with a few division steps, you'll end up with .9999

This is interesting for a couple of reasons. First, it's only 4 decimal places (notice how three of the answers fit that pattern while two of them don't). Second, you should again consider the format of the answer choices... each answer is a bunch of 0s followed by a single non-0 digit. That result won't happen if you're subtracting an 8-digit decimal from a 4-digit decimal. Thus, it's almost certain that the second fraction will ALSO be a 4-digit decimal.

Using the same approach that we used on the first fraction, we can rewrite the second fraction as...

9999.9991/10003

Before you do any math, think about how a '3' divides into a '1'.... what will the last digit in this decimal probably be? Since 7x3 = 21, that last digit will almost certainly be a '7'. With a little work, you can prove it. You'll end up with .9997

Subtracting the two decimals, you'll have .9999 - .9997 = .0002

Notice the '2' as the last decimal point? It won't take much to find that answer.

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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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20 Jul 2019, 05:44

I am unable to understand why do we not simply approximate a larger no if we are deducting
a very small no as 1 from it.

E.g. $$999,999^2$$ is same as $$10^6$$
since 1000 =$$10^3$$ and $$a^m$$ *$$a^n$$ = $$a^(m+n)$$

So effectively $$999,999^2$$ - 1 is same as $$10^6$$ (How much effect will 1 have
after deducting from such a huge no: negligible, right?)

Why do we use difference of squares instead of approximation while later is more quicker?
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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20 Jul 2019, 23:20

I am unable to understand why do we not simply approximate a larger no if we are deducting
a very small no as 1 from it.

E.g. $$999,999^2$$ is same as $$10^6$$
since 1000 =$$10^3$$ and $$a^m$$ *$$a^n$$ = $$a^(m+n)$$

So effectively $$999,999^2$$ - 1 is same as $$10^6$$ (How much effect will 1 have
after deducting from such a huge no: negligible, right?)

Why do we use difference of squares instead of approximation while later is more quicker?

Yes, certainly approximation is far more quicker. But here are the options are very very small too. Try approximating and let us know how you plan to arrive at the answer.
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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22 Aug 2019, 03:04
1
1
$$\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}=$$

(A) $$10^{(-8)}$$

(B) $$3*10^{(-8)}$$

(C) $$3*10^{(-4)}$$

(D) $$2*10^{(-4)}$$

(E) $$10^{(-4)}$$

OG 2019 #215 PS00574

$$\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}= \frac{(1 - .00000001)}{(1+.0001)} - \frac{(1 - .00000009)}{(1+.0003)} = (1-.0001) - (1-.0003) = .0002 = 2*10^-4$$

IMO D
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27 Aug 2019, 08:03
$$\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}=$$

(A) $$10^{(-8)}$$

(B) $$3*10^{(-8)}$$

(C) $$3*10^{(-4)}$$

(D) $$2*10^{(-4)}$$

(E) $$10^{(-4)}$$

OG 2019 #215 PS00574

$$\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}=$$

$$\frac{1-10^{-8}}{1+10^{-4}}-\frac{1-9*10^{-8}}{1+3*10^{-4}}=$$

$$(1-10^{-4})-(1+3*10^{-4})$$

$$2*10^{-4}$$

IMO D
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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19 Sep 2019, 11:52
Can someone tell me if I did it in the right way:

I rounded off the denominator to 1*10^-4

So now we can safely subtract the two equations:

0.00000008/10^-4

This becomes 2*10^-8+4 which gives D

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27 Nov 2019, 13:07

Hi Experts -- i see the answer is D in this question but i do have a question regarding "Estimating" this

Give estimation is a strategy for the GMAT -- i estimated :

1.0001 == estimated down to 1.0000
1.0003 == estimated down to 1.0000

Hence i was left with

$$\frac{0.99999999}{1}$$ - $$\frac{0.99999991}{1}$$ =

0.99999999 −0.99999991 =0.00000008 or 8 * $$10^{-8}$$

Question : why isn't the estimation method getting me close /somewhat close to option D ...option D seems 1000 times larger than my estimation (option B and option C sees closer)

Any idea where the logic is wrong in this case for such a HUGE difference ?
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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27 Nov 2019, 20:09
jabhatta@umail.iu.edu wrote:

Hi Experts -- i see the answer is D in this question but i do have a question regarding "Estimating" this

Give estimation is a strategy for the GMAT -- i estimated :

1.0001 == estimated down to 1.0000
1.0003 == estimated down to 1.0000

Hence i was left with

$$\frac{0.99999999}{1}$$ - $$\frac{0.99999991}{1}$$ =

0.99999999 −0.99999991 =0.00000008 or 8 * $$10^{-8}$$

Question : why isn't the estimation method getting me close /somewhat close to option D ...option D seems 1000 times larger than my estimation (option B and option C sees closer)

Any idea where the logic is wrong in this case for such a HUGE difference ?

Note that all the numbers are very close to 1. So you cannot estimate some of them to 1 and not others. All options are very very close to each other too. So you cannot estimate here.
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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27 Nov 2019, 20:31
jabhatta@umail.iu.edu wrote:

Hi Experts -- i see the answer is D in this question but i do have a question regarding "Estimating" this

Give estimation is a strategy for the GMAT -- i estimated :

1.0001 == estimated down to 1.0000
1.0003 == estimated down to 1.0000

Hence i was left with

$$\frac{0.99999999}{1}$$ - $$\frac{0.99999991}{1}$$ =

0.99999999 −0.99999991 =0.00000008 or 8 * $$10^{-8}$$

Question : why isn't the estimation method getting me close /somewhat close to option D ...option D seems 1000 times larger than my estimation (option B and option C sees closer)

Any idea where the logic is wrong in this case for such a HUGE difference ?

Note that all the numbers are very close to 1. So you cannot estimate some of them to 1 and not others. All options are very very close to each other too. So you cannot estimate here.

Hi karishma

Thank you so much for replying

Understand every answer is close to 1

But you would agree though the estimation method is OFF by times 1000 compared to option D

Any idea why the estimation is so off by any chance

I think every one would make the estimation that 1.0001 can be written up as 1.0000 and the same with 1.0003 as 1.0000

Posted from my mobile device
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Posts: 10113
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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27 Nov 2019, 21:02
jabhatta@umail.iu.edu wrote:
jabhatta@umail.iu.edu wrote:

Hi Experts -- i see the answer is D in this question but i do have a question regarding "Estimating" this

Give estimation is a strategy for the GMAT -- i estimated :

1.0001 == estimated down to 1.0000
1.0003 == estimated down to 1.0000

Hence i was left with

$$\frac{0.99999999}{1}$$ - $$\frac{0.99999991}{1}$$ =

0.99999999 −0.99999991 =0.00000008 or 8 * $$10^{-8}$$

Question : why isn't the estimation method getting me close /somewhat close to option D ...option D seems 1000 times larger than my estimation (option B and option C sees closer)

Any idea where the logic is wrong in this case for such a HUGE difference ?

Note that all the numbers are very close to 1. So you cannot estimate some of them to 1 and not others. All options are very very close to each other too. So you cannot estimate here.

Hi karishma

Thank you so much for replying

Understand every answer is close to 1

But you would agree though the estimation method is OFF by times 1000 compared to option D

Any idea why the estimation is so off by any chance

I think every one would make the estimation that 1.0001 can be written up as 1.0000 and the same with 1.0003 as 1.0000

Posted from my mobile device

Note that .99999999 is also almost 1. In fact it is closer to 1 than 1.0001. So if using approximation, the first fraction becomes 1/1 = 1. Same thing for second fraction. So if using approximation, you will get 1 -1 = 0. Note that every option is very close to 0.

The point is this:
You cannot say that 99/102 = 99/100 = .99
when your options have .99, .98. .97, .96, .95

Actually, the answer is .97 here. The margin of error is so small between the numbers and options that approximation will not give you the correct answer.
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =  [#permalink]

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05 Feb 2020, 00:32
Does anyone have similar problems at hand - thanks!
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Re: 0.99999999/1.0001 - 0.99999991/1.0003 =   [#permalink] 05 Feb 2020, 00:32

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