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

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 \(\frac{10^6-1}{10*(10^4-1)}\approx{\frac{10^6}{10*10^4}=\frac{10^6}{10^5}=10}\).

Answer: B.

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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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.)

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.

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

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
Anand

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

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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Hi Gonaives,

While this prompt might appear 'scary', It might help to think about this question conceptually:

10^2 = 100
10^3 = 1,000

10^7 = 10,000,000
10^8 = 100,000,000

Notice how much BIGGER 10^8 and 10^7 are, relative to 10^3 and 10^2? That's the 'key' to this question. In the numerator, subtracting 100 from 100,000,000 would have almost no mathematical 'impact' on the calculation (and the same can be said about the subtraction in the denominator). As such, the question is ultimately asking...

10^8/10^7 = ?

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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It is a really easy problem... According to division rule of powers with same base. we need to subtract the powers. so the question goes like

10^8-10^2 / 10^7-10^3
subtract power 8 from power 7 We get 10^1
then, subtract power 2 with power 3 we get 10^-1
in the end we are left with 10^1-10^-1 and the answer is 10

Unfortunately this is not correct approach.

1. \(\frac{(10^{8} - 10^{2})}{(10^{7} - 10^{3})}\) does not equal to \(\frac{10^{8}}{10^{7}}-\frac{10^2}{10^3}\). For example, (6-2)/(2-1)=4 does not equal to 6/2 - 2/1 = 1. You cannot break a fraction this way.

2. 10^1 - 10^(-1) = 99/10 not 10.


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This problems is testing us on our knowledge of the relationship between bases raised to exponents as well as estimation of exponents.

Starting with the numerator, we see that 10^8 = 100,000,000 is much larger than 10^2 = 100; therefore 10^8 - 10^2 is still approximately 10^8.

In the denominator, 10^7 = 10,000,000 is much larger than 10^3 = 1,000; therefore 10^7 - 10^3 is still approximately 10^7.

Thus, (10^8 - 10^2)/(10^7 - 10^3) is approximately 10^8/10^7 = 10.

Answer: B

(Note: this problem should have asked for the approximate value of (10^8 - 10^2)/(10^7 - 10^3)).
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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

Answer: B
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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.

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Hi GMATters,

No reason to overthink this question: first thing is to note that it's estimating, so your answer will be close to simply 10^8/10^7 = 10^1.

In short, if I had 100 million dollars, I would be lighting my cigars with 100-dollar bills.

Here's a graphic explanation:

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