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# The value of (10^8-10^2)/(10^7-10^3) is closest to which of

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The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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31 May 2010, 18:33
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The value of (10^8-10^2)/(10^7-10^3) is closest to which of the following?

A. 1
B. 10
C. 10^2
D. 10^3
E. 10^4

Any thoughts on the quickest way to solve? I factored out 10^2 in the numerator and 10^3 in the denominator.

Thanks!
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Apr 2015, 02:45, edited 3 times in total.
Edited the question and added the OA
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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31 May 2010, 19:51
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B.

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

$$\frac{10^3 * (10 - 1) * (10^4 + 10^3)}{10^3 * (10 -1) * (10 + 1) * (10^2 + 1)}$$

Finally, $$\frac{1000}{101}$$ --> 10 (approx.)
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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01 Jun 2010, 03:27
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jcbruin wrote:
I can't believe how I was stumped on this problem in the GMAT Prep Practice Test 1:

The value of (10^8-10^2)/(10^7-10^3) is closest to which of the following?

a. 1
b. 10
c. 10^2
d. 10^3
e. 10^4

Any thoughts on the quickest way to solve? I factored out 10^2 in the numerator and 10^3 in the denominator.

Thanks!

Yes factoring out $$10^2$$ and $$10^3$$ and than reducing the fraction by $$10^2$$ is one way to deal with this question:

$$\frac{10^8-10^2}{10^7-10^3}=\frac{10^2(10^6-1)}{10^3(10^4-1)}=\frac{10^6-1}{10*(10^4-1)}$$.

Now, $$10^6-1$$ is very close to $$10^6$$ and $$10^4-1$$ is very close to $$10^4$$, hence $$\frac{10^6-1}{10*(10^4-1)}\approx{\frac{10^6}{10*10^4}=\frac{10^6}{10^5}=10}$$.

Or else you can notice that we need approximate value of a fraction. Now, $$10^{8}$$ is much, much, much bigger than $$10^{2}$$. So subtracting $$10^{2}$$ from $$10^{8}$$ will be very close to $$10^{8}$$, basically $$10^{2}$$ is negligible in this case. The same for for $$10^{7}$$ and $$10^{3}$$. So $$\frac{10^8-10^2}{10^7-10^3}\approx{\frac{10^8}{10^7}=10$$.

Hope it helps.
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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25 Jun 2010, 22:59
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Because the question asks for what value is "closest" the question invites approximation.

Let's look at the numerator:

10^8 - 10^2

10^8 is HUGE compared to 10^2.

So 10^8 - 10^2 is very close to 10^8 itself. (Just as 100 - 0.0001 is very close to 100 itself).

Likewise, 10^7 is HUGE compared to 10^3.

So 10^7 - 10^3 is very close to 10^7.

So we have:

10^8/10^7

or 10^(8-7) = 10.

Choose B.

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I think this is the quickest way to solve. If you see the point of the question, you can quickly solve it in this way without even having to do any scratchwork.

TAKEAWAY: don't assume algebraic approaches are always most efficient. And whenever a question asks for "approximation" or "close" be on the lookout to make deductions that will allow you to solve the question fast.
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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18 Sep 2010, 14:03
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ok. without calculation you can probably guess 10^8 and 10^7 are SIGNIFICANTLY bigger than the numbers they are subtracting. so you can just do 10^8 / 10^7 = 10. B
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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18 Sep 2010, 15:28
shaselai wrote:
ok. without calculation you can probably guess 10^8 and 10^7 are SIGNIFICANTLY bigger than the numbers they are subtracting. so you can just do 10^8 / 10^7 = 10. B

Good quick thinking on your part . I took the more traditional approach of solving it the long way
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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18 Sep 2010, 18:41
Nice approximation method. I took on canceling factors and then doing approximation. After spending more time, though i got B
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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09 Feb 2011, 08:51
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I've run across several variants of the following question:
$$\frac{10^8 - 10^2}{10^7 - 10^3}$$

Here is the approach I want to take:
$$\frac{10^2(10^6 - 1)}{10^3(10^4 - 1)}$$

But when I cancel the numerator/demoninator what I am left with is kind of ugly.
$$\frac{999999}{99990}$$

Is there something I am missing? Is there a better way? Or should I just suck it up and do the division?
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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09 Feb 2011, 09:28
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gmontalvo wrote:
I've run across several variants of the following question:
$$\frac{10^8 - 10^2}{10^7 - 10^3}$$

Here is the approach I want to take:
$$\frac{10^2(10^6 - 1)}{10^3(10^4 - 1)}$$

But when I cancel the numerator/demoninator what I am left with is kind of ugly.
$$\frac{999999}{99990}$$

Is there something I am missing? Is there a better way? Or should I just suck it up and do the division?

The way you are solving it is the right way imo. Posting the answer choices will help. You can narrow it down to one or two choices.

Thanks
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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11 Feb 2011, 06:25
My attempt :

10^2 (10^6 - 1)
----------------
10^3(10^4 - 1)

= 1/10 * (10^3 - 1)(10^3 + 1)/((10^2 - 1)(10^2 + 1))

= 1/10 * 999/99 * 1001/101 (1001 ~ 1000 and 101 ~ 100)

= 1/10 * 111/11 * 1000/100 (now 111 ~ 110 )

= 1/10
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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16 May 2012, 10:08
(10^8-10^2)/(10^7-10^3) =
10^2 *(10^6-1)/10^3(10^4-1)=10^6/10^4*10 =10^2/10=10
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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16 May 2012, 10:16
gmontalvo wrote:
I've run across several variants of the following question:
$$\frac{10^8 - 10^2}{10^7 - 10^3}$$

Here is the approach I want to take:
$$\frac{10^2(10^6 - 1)}{10^3(10^4 - 1)}$$

But when I cancel the numerator/demoninator what I am left with is kind of ugly.
$$\frac{999999}{99990}$$

Is there something I am missing? Is there a better way? Or should I just suck it up and do the division?

This is how I would do it:

Factorise the numerator and denominator

$$\frac{10^2(10^6 - 1)}{10^2(10^5 - 10)}$$

Cancel, and you get

$$\frac{(10^6 - 1)}{(10^5 - 10)}$$

Now, some approximation:

$$\frac{(10^6 - 1)}{(10^5 - 10)} \approx \frac{10^6}{10^5}$$

This gives 10. The great part is that you are dealing with such large numbers, that 1 and 10 are immaterial.

The point here is not accuracy, it is to get a sense of what is right or wrong. If you find you have to do long division, then you are definitely missing a trick. Even if a question seems fiendish, there is always a shortcut to the solution!

Hope this helps.

EDIT: Scrolled through Bunuel's post, who has hit the nail on the head
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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16 May 2012, 18:59
Factor out 10^1 from denom cancel divide top and bottom parenthesis, left with 10^2 on top and 10 on bottom. Divide once more and left with 10
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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19 May 2013, 03:05
srimila wrote:
10^8 – 10^2 / 10^7 – 10^3 is closest to which of the following
a. 1
b. 10
c. 10^2
d. 10^3
e. 10^4

Merging similar topics. Please refer to the solutions above.

Similar questions to practice:
which-of-the-following-is-closest-to-10180-1030-a-110224.html
the-value-of-10-8-10-2-10-7-10-3-is-closest-to-which-of-95082.html
tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029229
if-x-10-10-x-2-2x-7-3x-2-10x-2-is-closest-to-143897.html
m24-q-7-explanation-76513.html

Hope it helps.

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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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02 Jul 2013, 00:17
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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02 Jul 2013, 02:01
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jcbruin wrote:
The value of (10^8-10^2)/(10^7-10^3) is closest to which of the following?

A. 1
B. 10
C. 10^2
D. 10^3
E. 10^4

The given expression can be written as : $$\frac {10^8(1-\frac{10^2}{10^8})} {10^7(1-\frac{10^3}{10^7})}$$ = $$\frac {10^8(1-\frac{1}{10^6})} {10^7(1-\frac{1}{10^4})}$$

We know that $$\frac {1} {10^6}\ll1$$ and similarly,$$\frac {1} {10^4}\ll1$$, thus, we can safely approximate the given expression as$$\frac {10^8}{10^7}$$ =10.
B.
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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14 Nov 2013, 05:16
jcbruin wrote:
The value of (10^8-10^2)/(10^7-10^3) is closest to which of the following?

A. 1
B. 10
C. 10^2
D. 10^3
E. 10^4

Any thoughts on the quickest way to solve? I factored out 10^2 in the numerator and 10^3 in the denominator.

Thanks!

Here is how I did it

(10^4-10)(10^4+10)/(10^7-10^3)

Factorize denominator:

10(10^6-10^2)
(10)(10^3-10)(10^3+10)
(10)(990)(1010)

Factorize numerator
(10^4-10)(10^4+10)
(9900)(10010)

So you end up with

(9900)(10010)/(10)(990)(1010)

Simplify and it should give you 10.

Hope it helps
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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22 Dec 2013, 18:04
nsp007 wrote:
B.

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

$$\frac{10^3 * (10 - 1) * (10^4 + 10^3)}{10^3 * (10 -1) * (10 + 1) * (10^2 + 1)}$$

Finally, $$\frac{1000}{101}$$ --> 10 (approx.)

Shouldn't 10^3 in the numerator be 10^1?
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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22 Dec 2013, 23:02
TooLong150 wrote:
nsp007 wrote:
B.

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

$$\frac{10^3 * (10 - 1) * (10^4 + 10^3)}{10^3 * (10 -1) * (10 + 1) * (10^2 + 1)}$$

Finally, $$\frac{1000}{101}$$ --> 10 (approx.)

Shouldn't 10^3 in the numerator be 10^1?

Yes. Correct algebraic approach is given here: the-value-of-10-8-10-2-10-7-10-3-is-closest-to-which-of-95082.html#p732067
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Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of [#permalink]

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28 Dec 2013, 04:46
jcbruin wrote:
The value of (10^8-10^2)/(10^7-10^3) is closest to which of the following?

A. 1
B. 10
C. 10^2
D. 10^3
E. 10^4

Any thoughts on the quickest way to solve? I factored out 10^2 in the numerator and 10^3 in the denominator.

Thanks!

Factor both numerator and denominator:

10^2(10^6 - 1) / 10^3(10^4 - 1), because the subtractions in the parenthesis are extremely small, we ignore the subtraction, so we have: 10^8 / 10^7

Now, multiply the expression by 10^-7/10^-7 and thus we have 10^1/1 = 10. Answer B.
Re: The value of (10^8-10^2)/(10^7-10^3) is closest to which of   [#permalink] 28 Dec 2013, 04:46

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