Manhattan Gmat problem

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Wed Dec 14, 2011 4:44 pm

âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?

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rijul007: Legendary Member; Posts: 588; Joined: Sun Oct 16, 2011 9:42 amFollowed by: 9 members; Location: New Delhi, India; Thanked: 130 times; GMAT Score:720

by rijul007 » Wed Dec 14, 2011 5:03 pm

winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?

âˆš{3âˆš{4*4*5} + 3(9-4âˆš5)/(9-4âˆš5)(9+4âˆš5)}
âˆš{12âˆš5 + 3(9-4âˆš5)/81-80?
âˆš{12âˆš5 + 27 - 12âˆš5}
âˆš27
3âˆš3

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GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members; Location: Austin, TX; Thanked: 236 times; GMAT Score:770

by GmatMathPro » Wed Dec 14, 2011 5:03 pm

winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?

First, focus on the expression 3/(9+4âˆš5). To simplify this, multiply the top and bottom by the conjugate of the bottom. The conjugate of an expression like this is just the same expression with the opposite sign in the middle. So in this case, it would be 9-4âˆš5.

[3/(9+4âˆš5)][(9-4âˆš5)/(9-4âˆš5)] = (27-12âˆš5)/(81-36âˆš5+36âˆš5-80) = (27-12âˆš5)/1 = 27-12âˆš5. Plug this in to the original expression to get:

âˆš{3âˆš80+27-12âˆš5}. Notice that 3âˆš80 = 3âˆš(16*5) = 12âˆš5, so the entire expression simplifies to âˆš27, which simplifies to [spoiler]3âˆš3[/spoiler]

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by Anurag@Gurome » Wed Dec 14, 2011 8:47 pm

winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?

âˆš{3âˆš80 + 3/(9+4âˆš5)} = âˆš{3 * 4âˆš5 + 3/(9 + 4âˆš5)} = âˆš{12âˆš5 + 3/(9 + 4âˆš5)}
Now, 3/(9 + 4âˆš5) = 3(9 - 4âˆš5)/(9 + 4âˆš5)(9 - 4âˆš5) = 3(9 - 4âˆš5)/(81 - 80) = 3(9 - 4âˆš5)= 27 - 12âˆš5
Therefore, âˆš{3âˆš80 + 3/(9+4âˆš5)} = âˆš{12âˆš5 + 27 - 12âˆš5} = âˆš27 = [spoiler]3âˆš3[/spoiler]

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Dec 15, 2011 3:57 am

winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?

If the answer choices are sufficiently different, we can ballpark the answer quickly:

âˆš(3âˆš80 + 3/(9+4âˆš5))

â‰ˆ âˆš( 3*âˆš81 + smaller number/bigger number )

= âˆš(3*9 + fraction)

= âˆš(27 + fraction).

The fraction will have little effect on the result.
Thus, we can ignore it in our calculations:

âˆš27 â‰ˆ 3âˆš3.

Last edited by GMATGuruNY on Thu Dec 15, 2011 6:53 pm, edited 2 times in total.

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Thu Dec 15, 2011 6:18 pm

is there no way other than conjugating that this problem can be solved?

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Thu Dec 15, 2011 6:22 pm

GMATGuruNY wrote:
winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?
If the answer choices are sufficiently different, we can ballpark the answer quickly:

âˆš(3âˆš80 + 3/(9+4âˆš5))

â‰ˆ âˆš(3*9 + fraction)

â‰ˆ âˆš27

â‰ˆ 3âˆš7.

Hi Mitch, I didn't understand how did you get to the step in bold red above.
Just to correct âˆš27 = 3âˆš3

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rijul007: Legendary Member; Posts: 588; Joined: Sun Oct 16, 2011 9:42 amFollowed by: 9 members; Location: New Delhi, India; Thanked: 130 times; GMAT Score:720

by rijul007 » Thu Dec 15, 2011 6:37 pm

winniethepooh wrote:
GMATGuruNY wrote:
winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?
If the answer choices are sufficiently different, we can ballpark the answer quickly:

âˆš(3âˆš80 + 3/(9+4âˆš5)

â‰ˆ âˆš(3*9 + fraction)

â‰ˆ âˆš27

â‰ˆ 3âˆš7.
Hi Mitch, I didn't understand how did you get to the step in bold red above.
Just to correct âˆš27 = 3âˆš3

âˆš(3âˆš80 + 3/(9+4âˆš5))

to make the calculation much easier we replace âˆš80 with âˆš81 as they are approximately the same

âˆš(3âˆš81 + 3/(9+4âˆš5))
âˆš(3*9 + 3/(9+4âˆš5))

4âˆš5 = 2.23 * 4 = 9 approx.

3/(9+4âˆš5) â‰ˆ 3/(9+9) â‰ˆ 1/6 â‰ˆ 0.17
this is almost zero, so he neglected it

so the expression would be âˆš(3*9 + 0.17) â‰ˆ âˆš3*9 â‰ˆ 3âˆš3

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Dec 15, 2011 6:56 pm

winniethepooh wrote:
GMATGuruNY wrote:
winniethepooh wrote:âˆš{3âˆš80 + 3/(9+4âˆš5)} = ?
If the answer choices are sufficiently different, we can ballpark the answer quickly:

âˆš(3âˆš80 + 3/(9+4âˆš5))

â‰ˆ âˆš(3*9 + fraction)

â‰ˆ âˆš27

â‰ˆ 3âˆš7.
Hi Mitch, I didn't understand how did you get to the step in bold red above.
Just to correct âˆš27 = 3âˆš3

Please revisit my post above, in which I fleshed out the explanation.

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winniethepooh: Master | Next Rank: 500 Posts; Posts: 370; Joined: Sat Jun 11, 2011 8:50 pmFollowed by: 2 members; Location: Arlington, MA.; Thanked: 27 times

by winniethepooh » Fri Dec 16, 2011 4:29 am

Wow Mitch, as always your explanations make such a lot of sense!!

That's really helpful in saving time!!

Thanks a lot.

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