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another manhatten problem

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by nickhar130 » Thu Nov 04, 2010 10:59 am
sorry for this post but I got stumped on a simple question as this, please help and provide step by step.

If 400,000 = 4^n+1 x 25^n then n =

a) 1.5
b) 2
c) 2.5
d) 3
e) 3.5

the "x" in the equation is times and not a variable.
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by limestone » Thu Nov 04, 2010 11:17 am
Hi,
If 400,000 = 4^(n+1) * 25^n
Then:
400,000 = 4^1 * 4^n * 25^n = 4* (4*25)^n = 4*100^n

Divided both side by 4:
100,000 = 100^n, or
100*100*10 = 100^n, or
100*100*sqrt (100) = 100^n, or
100*100* 100*(1/2) = 100^n, or
100^ (1+1+1/2) = 100^n
Thus n = 2.5

Pick C.
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by GMATGuruNY » Thu Nov 04, 2010 12:35 pm
nickhar130 wrote:sorry for this post but I got stumped on a simple question as this, please help and provide step by step.

If 400,000 = 4^n+1 x 25^n then n =

a) 1.5
b) 2
c) 2.5
d) 3
e) 3.5

the "x" in the equation is times and not a variable.
Another approach is to prime-factorize everything in order to get similar bases on each side of the equation:

400,000 = 400*1000 = 4*100*10*100 = (2*2)*(2*2*5*5)*(2*5)*(2*2*5*5) = 2� * 5�
Thus:

2� * 5� = 4^(n+1) * 25^n
2� * 5� = (2²)^(n+1) * (5²)^n
2� * 5� = 2^(2n+2) * 5^2n

Since the bases on each side of the equation are matching, the exponents also must match.
Thus, the two exponents with a base of 5 (� and 2n) must be equal:
5 = 2n
n = 2.5

The correct answer is C.
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by nickhar130 » Thu Nov 04, 2010 1:16 pm
thank you, I understand now.
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