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

Question

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

Bunuel · Accepted Answer

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}  to get  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 .

Answer: C.

PareshGmat · Answer

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

KarishmaB · Answer

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)

BrentGMATPrepNow · Answer

One option is to rewrite the left side of the equation by factoring out 5^(x-3) 
So, we get: 5^(x-3)  = (124)(5^y)
Evaluate to get: 5^(x-3)  = (124)(5^y)
Simplify to get: 5^(x-3)  = (124)(5^y)
Divide both sides by 124 to get: 5^(x-3) = 5^y
So, x-3 = y

Answer: C

-----------------------------------------------------
ASIDE: A lot of students struggle to see how we can factor 5^x - 5^(x-3) to get 5^(x-3) 
Sure, they may be okay with straightforward factoring like these examples:
k^5 - k^3 = k^3(k^2 - 1) 
m^19 - m^15 = m^15(m^4 - 1) 
But they have problems when the exponents are variables.

IMPORTANT: Notice that, each time, the greatest common factor of both terms is the term with the  smaller exponent .

So, in the expression 5^x - 5^(x-3), the term with the  smaller exponent  is 5^(x-3, so we can factor out 5^(x-3)

Likewise, w^x + x^(x+5) = w^x(1 + w^5) 
And 2^x - 2^(x-2) = 2^(x-2) 
------------------------------------------------------------

Cheers,
Brent

BrentGMATPrepNow · Answer

Another option is to PLUG IN a value for x and see what kind of relationship we get between x and y. 
There are two "nice" x-values to plug in. They are x = 0 and x = 3, since we can easily use these to evaluate 5^x and 5^(x-3). Of these two values, x = 3 is the easier one to plug in. 
So, let's plug in  x = 3 
We get: 5^ 3  - 5^( 3 -3) = (124)(5^y)
Simplify to get: 125 - 1 = (124)(5^y)
Simplify to get: 124 = (124)(5^y)
Divide both sides by 124 to get: 1 = 5^y
Solve for y to get:  y = 0

So, when  x = 3 ,  y = 0 .

Now we'll check the answer choices to see which one satisfies this relationship. 
 A) y = x ... So, we get  0  =  3  (NOPE)
 B) y = x - 6 ... So, we get  0  =  3  - 6 (NOPE)
 C) y = x - 3 ... So, we get  0  =  3  - 3  IT WORKS! 
 D) y = 2x + 3 ... So, we get  0  = 2( 3 ) + 3 (NOPE)
 E) y = 2x + 6 ... So, we get  0  = 2( 3 ) + 6 (NOPE)

Answer: C

Cheers,
Brent

144144 · Answer

u can also plug in x=3. makes it very simple.

marioslash · Answer

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

Bunuel · Answer

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.

mbaiseasy · Answer

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

Answer: C

OptimusPrepJanielle · Answer

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) = 124*5^y So, 5^(x-3)*124 = 124*5^y So, 5^(x-3) = 5^y So, x-3 = y Hence option (C). -- Optimus Prep's GMAT On Demand course for only $299 covers all verbal and quant. concepts in detail. Visit the following link to get your 7 days free trial account: https://www.optimus-prep.com/gmat-on-demand-course

Rookie124 · Answer

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
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.

GMATinsight · Answer

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

Answer: Option C

ScottTargetTestPrep · Answer

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(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

Answer: C

u1983 · Answer

5^x  – 5^(x–3) = (124) (5^y) 
or  5^x  –  \frac{5^x}{5^3}  =  (125-1)  (5^y) 
or  \frac{(5^x*5^3-5^x)}{5^3}  =  (5^3-1)  (5^y) 
or  5^x*(5^3-1)  =  (5^3-1)  (5^y)(5^3) 
or  5^x  = 5^(y+3)
==>  x=y+3 
==> y=x-3  ..................Thus the answer is  option C .

JeffTargetTestPrep · Answer

5^x - 5^(x - 3) = (124)(5^y)

We see that 5^(x - 3) is common to both terms on the left side of the equation, so we factor it out:

5^(x - 3)(5^3 - 1) = (124)(5^y)

5^(x - 3)(124) = (124)(5^y)

5^(x - 3) = 5^y

Recall that when the bases are equal, we can equate the exponents:

x - 3 = y

Answer: C

travelwithbrandon · Answer

Could you help me understand the specific steps you took to get from the first bold line to the second? I'm not sure how you managed to get rid of the fraction here. Thanks!

ScottTargetTestPrep · Answer

Response: 
 
Notice that in the first bold line, the fraction 124/125 is not in the exponent; that fraction is simply being multiplied by 5^x (as a matter of fact, nothing besides x is in the exponent on and after the third line). Thus, to get rid of the fraction 124/125 on the left hand side, I simply multiplied each side of the equation by 125/124.

GMATinsight · Answer

travelwithbrandon It's mentioned in post that it is a sub-600 level question however I find it better than sub 600 level.

yashikaaggarwal · Answer

Its mentioned in the left hand corner just before the question starts. sub 600

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

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GMAT Problem Solving (PS) Questions

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

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