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

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Manager
Joined: 12 Jan 2013
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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, 05:46
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.
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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 Feb 2014, 20:53
10^8 is way too high from 10^2, similarly 10^7 is way too high from 10^3
No need to make any further simplification calculation, just calculate 10^8 / 10^7 = 10 = Answer = 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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08 Mar 2014, 20:49
Option B.
Factor the numerator and denominator.
{[(10^4)-10] [10^4+10]}/10^3(10^4-1)
Taking 10 common from each term in numerator.
{10^2[(10^3-1)(10^3+1)]}/10^3[(10^2-1)(10^2+1)]

After further factorization and cancellation We're left with
(111*91)/10*101=9.99 or 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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09 Sep 2015, 10:27
I think the easiest way to solve this is to use the quotient rule. Since all bases are the same (10), split the problem into two fractions, use the quotient rule on each, then subtract.

(10^8-10^2)/(10^7-10^3) =

(10^8/10^7) - (10^2/10^3) =

10 - (10^-1) =

10 - (1/10) =

10 - 0.1 =

9.9 =

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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11 Jan 2016, 15:18
Why we can't simply eliminate 10^8 with 10^7 and so on?

I did in this way and this is what I got:

1) 10^8 - 10^2 / 10^7 - 10^3;
2) I eliminate 10^8 with 10^7 and I got 10 on the up left side;
3) then I eliminate 10^2 with 10^3 and got zero on the up ritghtside and 10 on the down right side;
4) so, basically I left with 10 - 0 / - 10;
5) 10 / -10 = -1;
Because there isn't -1 on the answer choices I pick 1 that is the closest one.

What is wrong with my line of reasoning? I saw many times calculations involving exponents elimination if there is a division.

Thanks in advance for the explanation
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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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07 Jan 2017, 22:45
As the question asks for approximation, let's tackle it in the same way :

Numerator : $$10^8-10^2$$ : $$10^2$$ is 0.00001% of $$10^8$$ ,which is negligible when compared to $$10^8$$
denominator: $$10^7-10^3$$ : $$10^3$$ is 0.0001% of $$10^7$$ ,which is negligible when compared to $$10^7$$

Hence ,($$10^8$$) - (negligible) / ($$10^7$$) - (negligible)= $$10^8$$/$$10^7$$ =10

Ans : 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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19 Jul 2017, 21:02
It is not

If you multiply the denominator by 10 you get very close to the numerator. No need to do exact calculatio

999999/99990

99990X10=999900 which is only 99 less than the numerator.

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?
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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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21 Jul 2017, 08:19
10^8-10^2 similar to 10^8
10^7-10^3 similar to 10^7
Now (10^8-10^2)/(10^7-10^3) will give us 10^8/10^7 i.e. 10.
Option 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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08 Aug 2017, 22:07
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!

$$\frac{(10^8-10^2)}{(10^7-10^3)}$$

$$\frac{10^2(10^6-1)}{10^3(10^4-1)}$$

'$$1$$' is very small value compared to $$10^6$$ and $$10^4$$, hence we can approximate value to $$10^6$$ and $$10^4$$. Therefore;

$$\frac{10^2*10^6}{10^3*10^4}$$

$$\frac{10^8}{10^7} = 10^{(8-7)} = 10^1 = 10$$

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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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12 Aug 2018, 09:22
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!

OA:B
$$\frac{(10^8-10^2)}{(10^7-10^3)}=\frac{10^2(10^6-1)}{10^3(10^4-1)}=\frac{10^2(10^3-1)(10^3+1)}{10^3(10^2-1)(10^2+1)}=\frac{10^2(10-1)(10^2+10*1+1)(10^3+1)}{10^3(10-1)(10+1)(10^2+1)}$$
$$\frac{10^2(10-1)(10^2+10*1+1)(10^3+1)}{10^3(10-1)(10+1)(10^2+1)}=\frac{10^2(9)(111)(1001)}{10^3(9)(11)(101)}=\frac{(111)(1001)}{10(11)(101)}≈10$$
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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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01 Feb 2019, 18:52
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

We see that since 10^2 is MUCH SMALLER than 10^8 and 10^3 is MUCH smaller than 10^7, subtracting 10^2 from 10^8 does not appreciably change the value, and subtracting 10^3 from 10^7 does not appreciably change the value. Thus, the approximate value of (10^8-10^2)/(10^7-10^3) is:

(10^8)/(10^7) = 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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02 Feb 2019, 14:51
Hi All,

We're asked to choose the answer that is CLOSEST in value to (10^8-10^2)/(10^7-10^3). While you might be tempted to try lots of 'math steps', if you consider how the answer choices are written, you can use a bit of estimation and logic to get to the correct answer without doing too much math.

To start, when the answer choices are based around increased exponents, it's worth noting that those answers are NOT "close" to one another. Each of the answers here is a 'power of 10', and each answer is 10 TIMES greater than the one immediately above it.

With the numerator of the given fraction, instead of factoring, consider how 10^8 compares to (10^8 - 10^2)....

10^8 = 100,000,000
10^2 = 100

Difference = 99,999,900

100,000,000 is essentially the same as 99,999,900. Next, we can think about the denominator in the same terms...

10^7 = 10,000,000
10^3 = 1,000

Difference = 9,999,000

This means that 10,000,000 is essentially the same as 9,999,999...

Thus, we're essentially just asked to divide 10 million by 1 million... and that equals 10.

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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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10 Mar 2019, 07:48
1) Factor powers of 10 and reduce. 10^2*(10^6-1) / 10^2*(10^5-1)

2) Thinking about it logically, 10^6 has 7 place values from 1, so when we subtract 1 it loses a 0 place value. Same for 10^5 - 1.
Therefore, we have 6 place values / 4 place values. But we also have an additional 10 in the denominator, so it's actually 6 place values / 5 place values = 1 place value more in numerator so it should be B.
The value of (10^8-10^2)/(10^7-10^3) is closest to which of   [#permalink] 10 Mar 2019, 07:48

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