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Intern  Joined: 09 Feb 2011
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If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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If $$5^x – 5^{(x- 3)} = 124*5^y$$, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Originally posted by alltimeacheiver on 11 Feb 2011, 03:17.
Last edited by Bunuel on 21 Oct 2017, 09:35, edited 2 times in total.
Renamed the topic and edited the question.
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Posts: 58334
Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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16
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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1
4
$$5^x – 5^{(x- 3)} = 124*5^y$$

Re-write as

$$5^x – 5^{(x- 3)} = (125 - 1 ) * 5^y$$

$$5^x – 5^{(x- 3)} = 125* 5^y - 5^y$$

$$5^x – 5^{(x- 3)} = 5^{(y+3)} - 5^y$$

LHS = RHS, so equating the powers

y = x+3 ; Answer = C
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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1
u can also plug in x=3. makes it very simple.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Hi Bunuel,
I', trying to understand your factorization, but I find some troubles when I have to factorize some value with exponent an expression like in the question (x-3 or x+4),
Could you explain me that or do you have a deck where I could dive into this argument?
Thank you in advance
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Joined: 02 Sep 2009
Posts: 58334
Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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3
mario1987 wrote:
Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Hi Bunuel,
I', trying to understand your factorization, but I find some troubles when I have to factorize some value with exponent an expression like in the question (x-3 or x+4),
Could you explain me that or do you have a deck where I could dive into this argument?
Thank you in advance

$$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3+3}-5^{x-3}=124*5^y$$ --> $$5^{x}-5^{x-3}=124*5^y$$.

Theory: math-number-theory-88376.html

Practice:
tough-and-tricky-exponents-and-roots-questions-125956.html
tough-and-tricky-exponents-and-roots-questions-125967.html

Hope it helps.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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1
$${5^{x}}\frac{124}{125}=124(5^y)$$
$$5^{x}=5^{y+3}$$
$$x=y+3-->y=x-3$$

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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vaivish1723 wrote:
5^x – 5^(x- 3) = (124)(5^y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Oa is

Finding one variable in terms of another is very simple. Assume a value for one and find the other. Then look for the option that works.

Say, x = 3 (so that x-3 becomes 0 and we are left with simple calculations)

$$5^3 - 5^{3- 3} = (124)(5^y)$$
$$1 = 5^y$$
y = 0

Now put x = 3 in the options and find out which option gives you y = 0. Only C does so answer is (C)
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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1
alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

Given that 5^x – 5^(x- 3) = 124*5^y
So, 5^(x-3+3) - 5^(x- 3) = 124*5^y
So, 5^(x-3) [5^3 - 1] = 124*5^y
So, 5^(x-3)*124 = 124*5^y
So, 5^(x-3) = 5^y
So, x-3 = y

Hence option (C).

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Intern  Joined: 21 Feb 2015
Posts: 24
Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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4
ok this is how you solve this

you have 5^x -5^x-3 = (124) (5^y)
step 1: 5^x (1-5^-3) =(124) (5^y)
Step 2: 5^x(1 - 1/5^3) = (124) (5^y)
Step 3: 5^x(1- 1/125) = (124) (5^y)
Step 4: 5^x (125-1)/125 = (124) (5^y)
Step 5: 5^x (124/125) = (124) (5^y)
Step 6: 5^x =[(124) (5^y) (125) ]/124
Step 7: 5^x = (125) (5^y)
Step 8: 5^x = 5^3 (5^y)
Step 9: x = 3 +y --> y = x -3
for explaining purposes I broke it down in to 9 steps. It should not take you more than 5 steps to solve this in term of y.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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Rookie124 wrote:
ok this is how you solve this

you have 5^x -5^x-3 = (124) (5^y)
step 1: 5^x (1-5^-3) =(124) (5^y)
Step 2: 5^x(1 - 1/5^3) = (124) (5^y)
Step 3: 5^x(1- 1/125) = (124) (5^y)
Step 4: 5^x (125-1)/125 = (124) (5^y)
Step 5: 5^x (124/125) = (124) (5^y)
Step 6: 5^x =[(124) (5^y) (125) ]/124
Step 7: 5^x = (125) (5^y)
Step 8: 5^x = 5^3 (5^y)
Step 9: x = 3 +y --> y = x -3
for explaining purposes I broke it down in to 9 steps. It should not take you more than 5 steps to solve this in term of y.

Your answer may be right. But it is not good practice to do long steps in GMAT. Time is critical.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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6
I have a very hard time believing this is a sub 600 level question
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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secterp wrote:
I have a very hard time believing this is a sub 600 level question

The trick with GMAT quant is not the level of questions but the time constraint. For PS, you either adopt the traditional algebraic method that can sometimes very time consuming or you can use the options to your advantage as mentioned below.

Let x=3 as you have a power of 3 and I wanted non negative powers of 5. Let see if we get values of 3 or -3 or 0 or 9 or 12 (based on the 5 options given) for y when we do RHS = LHS from the given equation of 5^x – 5^(x- 3) = 124*5^y.

When x =0, the LHS becomes

5^x-5^(x-3)=5^3-5^0=125-1=124.

RHS = 124*5^y and for LHS=RHS , the only way this is possible is when y=0 , giving you y=x-3 as the correct answer. Thus, C is the correct answer.

Hope this helps.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Hi Bunuel

Would factoring it like this be correct?

$$5^x-5^{x-3}=124*5^y$$
$$5^{x}(1-5^{-3})=124*5^y$$ --> x-3 = y
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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Here is how I did it:
1) 124=5^3-1; 5^x-5^x-3=(5^3-1)*5^y; 5^x-5^x-3=5^3y-5^y
2) x=3y; x-3=y

The correct answer is C.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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Bunuel wrote:
alltimeacheiver wrote:
If 5x – 5x - 3 = (124)(5y), what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?
A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

$$5^x-5^{x-3}=124*5^y$$ --> factor out $$5^{x-3}$$: $$5^{x-3}(5^3-1)=124*5^y$$ --> $$5^{x-3}=5^y$$ --> bases are equal so we can equate the powers: $$y=x-3$$.

alltimeacheiver format the questions correctly.

Factorizing this way also works but do you have any advice for performing factorizations in the simplest way possible. Why or how did you know to factor out 5^(x-3) and not 5^x? Should you take factor out the most complicated term?

5^x – 5^(x- 3) = 124*5^y
5^x(1 - 5^-3) = 124*5^y
5^x(1 - 1/125) = 124*5^y
5^x(124/125) = 124*5^y
Diving through by 124: (5^x)*(1/125) = 5^y
(5^x)*(5^-3) = 5^y
5^(x-3) = 5^y
x-3=y
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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peterpark wrote:
If 5^x - 5^x-3 = (124)(5^y), what is y in terms of x?

A) x

B) x - 6

C) x - 3

D) 2x + 3

E) 2x + 6

$$5^x - 5^{x-3} = (124)(5^y)$$

i.e. $$5^{x-3}(5^3 - 1) = (124)(5^y)$$

i.e. $$5^{x-3}(124) = (124)(5^y)$$

i.e. $$5^{x-3} = (5^y)$$

i.e. y = x-3

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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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5^(x-3) can be written as 5^x / 5^3. So 5^x - 5^x/5^3 = 5^x (1 - 1/5^3)

or 5^x (1 - 1/125) or 5^x (124/125)

or 124 (5^x/5^3) or 124 [5^(x-3)]

Compare with right hand side of the given equation, y must be = (x-3)

Hence C
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GMAT 1: 730 Q49 V41 Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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5^x - 5^x-3
= 5^x[1 - 5^(-3)]
= 5^x[1 - 1/(5^3)]
= 5^x[1 - 5^(-3)]
= 5^x[1 - 1/125]
= 5^x*124/125
= 5^x*124/5^(-3)
= 5^(x-3)*124

Again,
(124)(5^y) = 5^(x-3)*124

Hence, y = x-3.
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Re: If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?  [#permalink]

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alltimeacheiver wrote:
If 5^x – 5^(x- 3) = 124*5^y, what is y in terms of x?

A. x
B. x - 6
C. x - 3
D. 2x + 3
E. 2x + 6

We can simplify the given equation:

5^x – 5^(x- 3) = 124*5^y

5^x – (5^x)(5^- 3) = 124*5^y

5^x(1 – (5^- 3)) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(1 – 1/125) = 124*5^y

5^x(124/125) = 124*5^y

5^x = 124*5^y*(125/124)

5^x = 5^y(125)

5^x = 5^y(5^3)

5^x = 5^y+3

x = y + 3

x - 3 = y

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