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by Hmna » Thu Apr 27, 2017 3:00 am
If p and q are nonzero integers and 81p=27q what is the value of p\q
A)3/4
B)4/3
C)27/7
D)7/27
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Help plz!!

by [email protected] » Thu Apr 27, 2017 4:36 pm
Hi Hmna,

In the future, you should make sure to post your questions in the appropriate sub-Forum. The Problem Solving Forum can be found here:

https://www.beatthegmat.com/problem-solving-f6.html

This question is all about exponent rules. When comparing exponents, it helps if the numbers have the same BASE. Here, the bases are 81 and 27, so we can rewrite those bases as (3^4) and (3^3). Now the given equation can be written as...

(3^4)^P = (3^3)^Q

We're raising 'a power to a power', so we can rewrite the equation one more step...

3^(4P) = 3^(3Q)

Now we know that 4P = 3Q

The question asks for the value of P/Q....

4P = 3Q
P = 3Q/4
P/Q = 3/4

Final Answer: A

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by Matt@VeritasPrep » Thu Apr 27, 2017 11:16 pm
One way:

(3�)ᴾ = (3³)ᴼ

3�ᴾ = 3³ᴼ

4p = 3q

p/q = 3/4
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:19 pm
Another way:

81ᴾ = 27ᴼ

81ᴾ/27ᴼ = 1

(27ᴾ * 3ᴾ)/27ᴼ = 1

27ᴾ�ᴼ * 3ᴾ = 1

(3³)ᴾ�ᴼ * 3ᴾ = 1

3³ᴾ�³ᴼ * 3ᴾ = 3�

3p - 3q + p = 0

4p = 3q

p/q = 3/4
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:21 pm
Yet another way:

81ᴾ = 27ᴼ

Try the answers. Using A we have

81³ = 27�

81 * 81 * 81 = 27 * 27 * 27 * 27

divide by 27 three times:

(81/27) * (81/27) * (81/27) = 27

3 * 3 * 3 = 27

This is true, so A is our answer!
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:21 pm
One way:

(3�)ᴾ = (3³)ᴼ

3�ᴾ = 3³ᴼ

4p = 3q

p/q = 3/4
The formatting on this post is screwed up, the superscript O should be a superscript Q. (The forum gives an error glyph for the Q.)
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:22 pm
81ᴾ = 27ᴼ

81ᴾ/27ᴼ = 1

(27ᴾ * 3ᴾ)/27ᴼ = 1

27ᴾ�ᴼ * 3ᴾ = 1

(3³)ᴾ�ᴼ * 3ᴾ = 1

3³ᴾ�³ᴼ * 3ᴾ = 3�

3p - 3q + p = 0

4p = 3q

p/q = 3/4
Same issue here, those super Os should super Qs. Hope the idea is still clear, though!
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:25 pm
Another way of seeing the one above, if it's easier:

81ᴾ = 27ᴼ

(27 * 3)ᴾ = 27ᴼ

27ᴾ * 3ᴾ = 27ᴼ

27ᴾ/27ᴼ = 1/3ᴾ

27ᴾ�ᴼ = 3�ᴾ

3(p - q) = -p

p/q = 3/4
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by Matt@VeritasPrep » Thu Apr 27, 2017 11:28 pm
... and of course those should be superscript Qs again. Yeesh, I give up! (Why is super Q so hated by forum text? Must be a Scrabble thing :D)
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