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adi: Junior | Next Rank: 30 Posts; Posts: 23; Joined: Mon Jan 07, 2008 12:55 pm; Location: Frisco, Texas

by adi » Fri Mar 14, 2008 7:46 pm

If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

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Source: Beat The GMAT — Problem Solving | Fri Mar 14, 2008 7:46 pm

musicdaemon: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Wed Feb 13, 2008 10:34 am; Location: India; Thanked: 3 times

by musicdaemon » Fri Mar 14, 2008 9:10 pm

Since, (3^x) - 3^(x-1) = 162

 (3^(x-1))(3 -1) = 162
 (3^(x-1))2 = 162
 (3^(x-1)) = 162/2 = 81 = 3^4
 x-1 = 4
 x = 5
Therefore, x(x-1) = 4*5 = 20…………… your answer

Let the Game begin

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tngearhart: Senior | Next Rank: 100 Posts; Posts: 50; Joined: Tue Apr 18, 2006 5:41 pmFollowed by: 1 member; Thanked: 2 times

by tngearhart » Mon Nov 17, 2008 4:02 am

Where did the (3-1) come from?

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rohangupta83: Legendary Member; Posts: 541; Joined: Thu May 31, 2007 6:44 pmFollowed by: 3 members; Location: UK; Thanked: 21 times; GMAT Score:680

by rohangupta83 » Mon Nov 17, 2008 4:08 am

musicdaemon wrote:Since, (3^x) - 3^(x-1) = 162

the above equation can also be written as:

3x3^(x-1) - 3^(x-1) = 162
taking 3^(x-1) common
we get:

3^(x-1) [3-1] = 162

musicdaemon wrote:  (3^(x-1))(3 -1) = 162
 (3^(x-1))2 = 162
 (3^(x-1)) = 162/2 = 81 = 3^4
 x-1 = 4
 x = 5
Therefore, x(x-1) = 4*5 = 20…………… your answer

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rbansal: Senior | Next Rank: 100 Posts; Posts: 36; Joined: Tue Jul 20, 2010 7:42 pmFollowed by: 1 member; Thanked: 1 times

by rbansal » Mon May 09, 2011 5:19 pm

I am very confused

When factoring out the common factor how are you getting 3x3^x-1 - 3^x-1=162

Isnt the original question 3^x-3^x-1=162 So wouldn't the common factor be 3x can someone please explain step by step.

Thank you,

RB

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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 » Mon May 09, 2011 7:53 pm

adi wrote:If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

We can plug in the answers, which represent the value of x(x-1).

Answer choice C: x(x-1) = 20.
If x(x-1) = 20, then x = 5, since 5*(5-1) = 5*4 = 20.
Plugging x=5 into 3^x - 3^(x-1) = 162, we get:
3â�µ - 3^(5-1) = 162
3â�µ - 3â�´ = 2 * 3â�´
3â�´(3-1) = 2 * 3â�´
3â�´ * 2 = 2 * 3â�´. Success!

The correct answer is C.

Alternatively, the numbers are small enough that we could plug in the answers and expand the exponents:

Answer choice C: x(x-1) = 20.
If x(x-1) = 20, then x = 5, since 5*(5-1) = 5*4 = 20.
Plugging x=5 into 3^x - 3^(x-1) = 162, we get:
3â�µ - 3^(5-1) = 162
3â�µ - 3â�´ = 162
243 - 81 = 162. Success!

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smackmartine: Legendary Member; Posts: 516; Joined: Fri Jul 31, 2009 3:22 pmFollowed by: 13 members; Thanked: 112 times

by smackmartine » Mon May 09, 2011 11:15 pm

162 can be written as 2 * 3^4
& 3^(x-1) can be written as (3^x)/3

now given, 3^x - 3^(x-1) = 2 * 3^4
=> 3^x - (3^x)/3 = 2 * 3^4

Take 3^x common ---> 3^x[ 1-(1/3)] = 2* 3^4
3^x [2/3]== 2* 3^4

cancelling 2 on both sides (3^x)/3 = 3^4
Taking denominator 3 on other side we get 3^x = 3^5
=> x=5 (as bases are same)

now x (x-1) is 5 (5-1) = 5*4 = 20 (C)

Hope it helps!.

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rbansal: Senior | Next Rank: 100 Posts; Posts: 36; Joined: Tue Jul 20, 2010 7:42 pmFollowed by: 1 member; Thanked: 1 times

by rbansal » Tue May 10, 2011 8:59 am

Thank you, Thank you soooooooooo much, you provided me with my light bulb moment.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pmFollowed by: 614 members; Location: Toronto; Thanked: 1710 times; GMAT Score:800

by Stuart@KaplanGMAT » Tue May 10, 2011 10:12 am

adi wrote:If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

As a follow-up to Mitch's post (backsolving is a great way to solve this question), we can use common sense and a bit of arithmetic to quickly eliminate two of the five choices, making backsolving even more viable.

We know that the correct answer is the product of x and (x-1). Since this is the GMAT, it's reasonable to assume that x is going to be an interger (otherwise the problem would be well beyond the scope of the exam).

So, which answers can be written as the product of two consecutive integers?

A) 12=3*4... keep it
B) 3*4 = 12; 4*5=20... no way to write 16 as the product of 2 consecutive integers - eliminate B.
C) 20 = 4*5... keep it
D) 30 = 5*6... keep it
E) 8*9 = 72; 9*10 = 90... no way to write 81 as the product of 2 consecutive integers - eliminate E.

Since we only have 3 possible choices, we know that the worst case scenario is that we have to plug 2 of them in (if neither of those work, then the third choice is correct by default).

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Bek: Senior | Next Rank: 100 Posts; Posts: 68; Joined: Fri Dec 03, 2010 10:29 pmFollowed by: 1 member; Thanked: 5 times

by Bek » Tue May 10, 2011 1:04 pm

3^x - 3^(x-1) = 162
3^x - 3^x/3 = 162
3^x(1-1/3) = 162
3^x(2/3) = 162
3^x = 162:2/3
3^x = 162*3/2
3^x = 81*3
3^x = 243
3^x = 3^5

x(x-1) =
5(5-1) =
5(4) = 20

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tonebeeze: Master | Next Rank: 500 Posts; Posts: 158; Joined: Mon Nov 02, 2009 5:49 pmFollowed by: 3 members; Thanked: 2 times

by tonebeeze » Tue May 10, 2011 3:41 pm

1. 3^(x) - 3^(x-1) = 162
2. Factor out 3^x ----> 3^(x) {1 - 3^(-1)} = 162
3. Further simplify: 3^(x) {1 - (1/3) ] = 162
4. 3^(x) (2/3) = 162
5. 3^(x) = 243 ---> x = 5
6. x (x-1) = 5 (5 -1)----> 20

OA = C

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