Is 5^k less than 1,000? (1) 5^(k+1) 3,000 (2) 5^(k-1) = 5^k - 500

Question

Is 5^k less than 1,000?

(1)  5^{(k+1)} > 3,000

(2)  5^{(k-1)} = 5^k - 500

Bunuel · Answer

Is 5^k less than 1,000? Is 5^k<1,000 ? (1) 5^(k+1) > 3,000 --> 5^k>600 --> if k=4 then the answer is YES: since 600<(5^4=625)<1,000 but if k=10 , for example, then the answer is NO. Not sufficient. (2) 5^(k-1) = 5^k - 500 --> we can solve for k and get the single numerical value of it, hence this statement is sufficient. Just to illustrate: 5^k-5^{k-1}=500 --> factor out 5^{k-1} : 5^{k-1}(5-1)=500 --> 5^{k-1}=125 --> k-1=3 --> k=4 . Sufficient. Answer: B. Hope it's clear.

LMKong · Answer

I would to ask why is this wrong
(1) 5^(k+1) > 3000

5^(k+1) > 5^5
 Hence, k+1= 5 , k =4

If so, 5^4 is less than 1000. The answer should b sufficient for (1).

Bunuel · Answer

First of all 5^5=3,125>3,000, thus 5^(k+1) > 3000 does NOT necessarily mean that 5^(k+1) > 5^5.

Next, even if we had 5^(k+1) > 5^5 it still does not mean that k+1=5. It means that k+1>5 --> k>4.

Hope it's clear.

cssk · Answer

Statement(1) : 5^(k+1) > 3000
The above inequality can be reduced to 5^k > 600. From this, we clearly know few possible values for k i.e., 4,5,6,..
Substituting these values in the inequality given in the question gives away both yes and no answers.
k = 4, 5^(4-1) < 1000
k = 5, 5^(5-1) < 1000
k = 6, 5^(6-1) > 1000
Hence statement(1) is not sufficient.

Statement(2): 5^(k-1) = 5^k - 500
Reducing the above inequality, 4/5 * 5^k = 500
So 5^k = 625 = 5^4. Clearly k = 4 and the original inequality is satisfied: 5^4 < 1000.
Hence statement(2) is sufficient.

Answer: B

NoHalfMeasures · Answer

The biggest take-away here should be that we don't need to solve the St-2. The moment you start solving St2, you have fallen for GMAT's classic time waster trap. See below.

Statement1: As 5^(k+1) > 3,000 --> k>4 and hence insufficient
Statement2: We dont need to solve the equation. Since this is an EQUATION (and not an inequality) with one variable, we will get the exact value of k and we will be able to answer the question one way or the other. SUFFICIENT.

EMPOWERgmatRichC · Answer

Hi All,

Right from the beginning, there are 2 things that you should note about this DS prompt:

1) At NO POINT does it state that K has to be an integer.
2) It's clearly based on exponents, so some exponent rules/patterns MUST be involved.

We're asked if 5^K is < 1,000. This is a YES/NO question.

Fact 1: 5^(K+1) > 3,000

In this Fact, notice how the exponent (K+1) differs from the exponent in the question (K). There's an exponent rule that accounts for this difference.

As an example, consider...
5^2 = 25
5^3 = 125
Notice how 5^3 is "5 times" greater than 5^2? This difference occurs because the base is 5 and we're increasing the exponent by 1. It can also be used in reverse....

5^3/5^2 = 5^(3-2) = 5^1 = 5

This is a standard rule about "dividing" exponents with the same base --> we SUBTRACT the exponents.

With Fact 1, we're dealing with 5^(K+1) and the question is dealing with 5^K. This means that DIVIDING 5^(K+1) by 5 will give us 5^K:

5^(K+1)/5^1 = 5^(K+1-1) = 5^K.

This is all meant to say that we can DIVIDE both sides of this inequality by 5, which gives us...

5^(K+1) > 3,000
5^K > 600

IF....
5^K = 601 then the answer to the question is YES
5^K = 1,001 then the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: 5^(K-1) = 5^K - 500

This is a 1 variable, 1 equation "system", so we CAN solve it (and there will only be 1 answer). Even if you did not know that, it's still easy enough to get to the solution.... Since most Test Takers are better at basic multiplication than they are at manipulating higher-level exponents, here's how you can "brute force" the solution:

5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625
5^5 = 3125

Find two consecutive powers of 5 that differ by 500 and you have the solution to the above equation.
5^4 - 5^3 = 625 - 125 = 500
Fact 2 is SUFFICIENT.

Final Answer:  B

GMAT assassins aren't born, they're made,
Rich

BrentGMATPrepNow · Answer

Target question:  Is 5^k less than 1000?

Statement 1 : 5^(k+1) > 3000 
First notice that 5^(k+1) = (5^k)(5^1)
So, we can take 5^(k+1) > 3000 and divide both sides by 5 to get: 5^k > 600
There are several possible cases to consider. Here are two:
case a: 5^k = 601, in which case  5^k is less than 1000 .
case b: 5^k = 1001, in which case  5^k is  not  less than 1000 .
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2 : 5^(k-1) = 5^k - 500 
 IMPORTANT : Notice that we're given an EQUATION, which means we can solve the equation to find the definitive value of k. If we can find the value of k, then we can instantly tell whether or not  5^k is less than 1000.  So, it SEEMS that we can conclude that statement 2 is sufficient WITHOUT performing any calculations. HOWEVER, if it's the case that the equation yields 2 possible values of k, then it may be the case that one value of k is such that  5^k is less than 1000,  and the other value of k is such that  5^k is greater than 1000 . So, at this point, we need only determine whether or not the equation will yield 1 or 2 values of k.

Rearrange to get the k's on one side: (5^k) - 5^(k-1) = 500
Factor the left side: 5^(k-1)  = 500
Simplify: 5^(k-1)  = 500
 STOP!! 
At this point, we can see that this equation will yield only one value of k. So, IF WE WERE to solve the equation for k, we would definitely be able to determine whether or not  5^k is less than 1000 .
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.

Answer = B

Cheers,
Brent

hovhannesmkrtchyan · Answer

Correct me if my reasoning is flawed. first of all the question asks is 5^k<1000, in other words is k<=4, since 5^4= 625 and 5^5=3125. Then we have (1) 5^(k+1)>3000, since 5^5=625, we have k>=4, NOT k>4, because when k=4, the inequality holds true. So, here we have one scenario that k=4, because in the original question we know that k<=4 and here we have k>=4, therefore only k=4 holds true. Therefore, D should be the answer.

EMPOWERgmatRichC · Answer

Hi hovhannesmkrtchyan,

When dealing with Quant questions, you have to be careful about distinguishing between what you KNOW and what DON'T KNOW. This question does NOT tells us that K is an integer, so you have to be open to the idea that it might not be a whole number. By extension, if K is a NON-INTEGER, then 5^K will not actually be a multiple of 5. If you read the other explanations in this thread, you'll see why Fact 1 is insufficient.

GMAT assassins aren't born, they're made,
Rich

TheUltimateWinner · Answer

EMPOWERgmatRichC 
Thanks for the explanation with kudos.
But, is it necessary to testify (as you said that k could be non-integer too) the value of k (k=4 and k=10) in statement 1?

Can you see my above post, please?

Question says:
Is  5^k<1000 ?
Statement 1 says:  5^k>600  (e.g.,  5^k=700 -->YES; or  5^k=10000000 -->NO). 
Why do we need to verify the value of k (like k=any value)?
Thanks__

ccooley · Answer

You don't need to, but it's a valid way to think about the problem

Some folks may see this problem and immediately realize that 5^k can have any positive value, from very close to 0 all the way up to 1,000,000,000+. You may notice this without even thinking about the specific value of k. Or, you might not have that thought immediately, in which case, plugging in different numbers for the unknown variable can help you get a handle on it. Try plugging in some different values for k, and notice that 5^k is small when k = 4, but large when k = 10. The first approach is going to be faster, but it means you have to think quickly and with confidence about exponent rules in a general sense! Sometimes it's safer to try specific numbers just to double check that your reasoning makes sense.

TheUltimateWinner · Answer

ccooley 
Thanks for the explanation with  kudos .
One more query in statement 2.
Here is the question prompt again..

Here, the question asks about inequality. But, the statement 2 is all about equation ("=" sign). So, If I make some calculation in statement 2, it'll definitely give specific value of k. The value of k could be less than 1000 or greater than 1000. The value of k will answer either YES or NO (but not both simultaneously!) of the original question prompt. So, if we see less than or greater than in the question prompt and equal sign ("=" sign) in the statement(s), can we say it is definitely SUFFICIENT?
Thanks__

EMPOWERgmatRichC · Answer

Hi Asad,

DS questions are interesting because they're built to 'test' you on a variety of skills (far more than just your 'math' skills), including organization, accuracy, attention-to-detail, thoroughness and the ability to prove that your answer is correct. DS questions also have no 'safety net' - meaning that if you make a little mistake, then you will convince yourself that one of the wrong answers is correct. Thankfully, the 'math' behind most DS questions isn't that complicated, but you have to be thorough with your work and take advantage of any 'shortcuts' that are built into the prompt.

When dealing with inequalities in a DS question, you have to consider both what you are told and the specific question that is asked. In this prompt, the information in Fact 1 can be rewritten as:

1)  5^k>600  
 
So we know that the SMALLEST INTEGER value of K is 4 (although K could technically be smaller than 4 and still fit this 'restriction'). EVERY number greater than 4 is also a possible value (because of the inequality).

Once you have the proof that 5^4 = 625 and the answer to the question (Is  5^k<1000 ?) is YES..... then you can reasonable deduce that a much larger value of K will lead to a result that is LARGER than 1000, so the answer to the question would be NO. I would still recommend that you document your deductions though (by writing them down on your pad), because the moment you choose to not take notes, you greatly increase your chances of making a little mistake (and risk getting the question wrong - and losing points - as a result). 
 
GMAT assassins aren't born, they're made,
Rich

TheUltimateWinner · Answer

EMPOWERgmatRichC 
Thanks for the explanation with  kudos. 
Suppose, 
The question says:
Is  5^k×99^z×1000000^{abc}<1000 ?
Statement 1 says:  5^k×99^z×1000000^{abc}>600 
So, should we care about the  value  of variable  k,z,a,b,c ? Actually, I don't see any point to be bother for the  value  of the variable  k,z,a,b,c . Can't we directly say (without knowing the value of  k,z,a,b,c ) that statement 1 gives the response sometimes YES and sometimes NO?
Thanks__

EMPOWERgmatRichC · Answer

EMPOWERgmatRichC 
Thanks for the explanation with  kudos. 
Suppose, 
The question says:
Is  5^k×99^z×1000000^{abc}<1000 ?
Statement 1 says:  5^k×99^z×1000000^{abc}>600 
So, should we care about the  value  of variable  k,z,a,b,c ? Actually, I don't see any point to be bother for the  value  of the variable  k,z,a,b,c . Can't we directly say (without knowing the value of  k,z,a,b,c ) that statement 1 gives the response sometimes YES and sometimes NO?
Thanks__

Hi Asad,

You've come up with a rather extreme example - and while it's essentially the same issue that we've already discussed, I assume that you're looking to justify why you would not take notes. If you're arguing "I can SEE that this information is Insufficient to answer the given question, so do I really have to go through those extra 'steps'?", then I would then have to ask "how many points are you willing to potentially lose because you're choosing not to take notes?"

NOTHING about a GMAT question is ever "random" - each prompt is specifically written to test you on a concept/pattern (and sometimes more than one). So while you might think "I don't have to care about these variables", I guarantee you that whoever wrote the question did so with those specific variables in mind. If you're going to be dismissive of information that's presented to you, then you might end up overlooking some subtle pattern or issue - and getting questions wrong as a result.

If you can score a Q51 on every exam regardless of the circumstances, then feel free to approach the Quant section however you like. However, if you're NOT earning that type of result every time - and you're looking to maximize your performance in the Quant section (and across the entire GMAT) - then you should get in the habit of proper note-taking on every question that you face. The good news is that work is remarkably easy AND it will benefit you in a number of ways over the course of a 3.5 hour Exam. In addition, beyond the GMAT, you're going to be taking LOTS of notes in Business School and beyond - so this whole process isn't something that you should be trying to avoid; you should embrace it (as it will help you to become more successful in a variety of different areas).

GMAT assassins aren't born, they're made,
Rich

totaltestprepNick · Answer

B https://www.youtube.com/watch?v=5iIXMXVlFtM Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

Is 5^k less than 1,000? (1) 5^(k+1) > 3,000 (2) 5^(k-1) = 5^k - 500

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