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M24 Q 7 explanation : Retired Discussions [Locked]

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If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

Re: M24 Q 7 explanation [#permalink]
12 Mar 2009, 18:10

6

This post received KUDOS

icandy wrote:

Which of the following is closest to \frac{4}{2.001} ?

(C) 2008 GMAT Club - m24#7

* 1.997 * 1.998 * 1.999 * 2.000 * 2.001

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

4/(2+a) = {4*(2-a)}/ {(2+a)(2-a)} = 4(2-a)/{4-a^2} ~ 4(2-a)/4 = 2-a

since a is too small.. a^2 .. is negligible.. ~zero. _________________

Your attitude determines your altitude Smiling wins more friends than frowning

Re: M24 Q 7 explanation [#permalink]
12 Jan 2012, 05:12

2

This post received KUDOS

Alternative method, D and E are not possible since denominator >2. So, pick the value in the center, 1.998 and multiply by 2.001 = 3.997998. Since number is less than 4. Answer is C.

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

Which of the following is closest to \frac{4}{2.001}?

Now, since 0.001^2 is very small number then 4-0.001^2 is very close to 4 itself, so 0.001^2 is basically negligible in this case and we can write: \frac{4(2-0.001)}{4-0.001^2}\approx{\frac{4(2-0.001)}{4}}=2-0.001=1.999.

Re: M24 Q 7 explanation [#permalink]
12 Mar 2009, 18:46

x2suresh wrote:

icandy wrote:

Which of the following is closest to \frac{4}{2.001} ?

(C) 2008 GMAT Club - m24#7

* 1.997 * 1.998 * 1.999 * 2.000 * 2.001

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

4/(2+a) = {4*(2-a)}/ {(2+a)(2-a)} = 4(2-a)/{4-a^2} ~ 4(2-a)/4 = 2-a

since a is too small.. a^2 .. is neglible.. ~zero.

K Thanks. Essentially, the solution is equating 4 and (4-a^2).

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

My approach was (I had to think a while - I did not solve this in 2 mins)

4/2.001 = (2.001+1.999)/2.001 which then = 1 + (1.999/2.001)

Now I definitely know that the 2nd part is less than 1 and is equal to 1999/2001.

Courtesy of Manhattan GMAT FDP => If the fraction is originally smaller than 1, the fraction increases in value as it approaches 1. ie., 10/11 < 11/12 < 1011/1012 Taking it in reverse 1011/1012 > 11/12 > 10/11

In our case 1999/2001 > 999/1001 (subtracting 1000) Instead subtract 1001 to get 998/1000. Now 1999/2001 > 998/1000 (0.998)

The 2nd part now is less than 1 and greater then 0.998 which gives our answer C.

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

My approach was (I had to think a while - I did not solve this in 2 mins)

4/2.001 = (2.001+1.999)/2.001 which then = 1 + (1.999/2.001)

Now I definitely know that the 2nd part is less than 1 and is equal to 1999/2001.

Courtesy of Manhattan GMAT FDP => If the fraction is originally smaller than 1, the fraction increases in value as it approaches 1. ie., 10/11 < 11/12 < 1011/1012 Taking it in reverse 1011/1012 > 11/12 > 10/11

In our case 1999/2001 > 999/1001 (subtracting 1000) Instead subtract 1001 to get 998/1000. Now 1999/2001 > 998/1000 (0.998)

The 2nd part now is less than 1 and greater then 0.998 which gives our answer C.

I like your thinking and I think it's all correct, except how would you know whether 1+1999/2001 is closer to 2.000 or to 1.999?

Re: M24 Q 7 explanation [#permalink]
30 Aug 2012, 19:02

Another simple method guys:

4/(2.001) = 4/(2+0.001)

Multiply top and bottom by (2-0.001) The formula here is (a-b)(a+b) = a^2-b^2. We have all learnt this from OG.

^ denotes: raised to the power of

You get 4(2-0.001)/(2^2 - (10^-3)^2)

The denominator can be rounded off to 4 because 10^-6 is negligeble; ateast compared 10^-3 in the numerator.

So the equation shortens to : 4 (2-0.001)/(4) = 2-0.001 = 1.999.

This method is beautiful because it doesn't need any of the hardcore maths and uses only the techniques described in the OG. I hope this answers the question whether the answer is closer to 1.999 or 2.000. It is obviously closer to 1.999 because the 10^-6 is negligeble compared to 10^-3.

If a is small, \frac{4}{2 + a} is close to 2 - a . This is because (2 - a)(2 + a) = 4 - a^2 , which is close to 4 if a is small. So, \frac{4}{2.001} = \frac{4}{2 + 0.001} is approximately 2 - 0.001 = 1.999 . The correct answer is C.

I am posting the OE because I did not understand the OE. I arrived at the solution though. I need to understand the above logic of

4/2+a is close to 2-a

Just go by the gut. 4/2 would be 2 so D and E are gone. And since the denominator is slightly bigger, the result would be minutely less than 2. Thus, 1.999. _________________

1.999*2.001 has the structure of (2-a)(2+a) = 4-a2 (a=0.001) In A and B, the products of 2.001 with 1.998 and 1.997 are smaller than that in C-> A and B are out

Eliminate E for the same reason when compared with D. D = 2*2.001 = 4+0.002. 0.002 > (0.001)^2 -> choose C