Is 5^k less than 1000?
1) 5^k+1 > 3000
2) 5^k-1 > 5^k - 500
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NOTE: Your transcription is somewhat misleading. I've added some brackets to avoid any ambiguity.
Statement 1: 5^(k+1) > 3000
Notice that 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125...
There are several values of k that satisfy the condition that 5^(k+1) > 3000. Here are two:
case a: k = 4 [so, 5^(k+1) = 5^5 = 3125, which is greater than 3000]. In this case 5^k = 5^4 = 625, and 625 IS less than 1000.
case b: k = 5 [so, 5^(k+1) = 5^6 = 15625, which is greater than 3000]. In this case 5^k = 5^5 = 3125, and 3125 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)[5 - 1] = 500
Simplify: 5^(k-1)[4] = 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
Target question: Is 5^k less than 1000?Is 5^k less than 1,000?
(1) 5^(k+1) > 3,000
(2) 5^(k-1) = 5^k - 500
Statement 1: 5^(k+1) > 3000
Notice that 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125...
There are several values of k that satisfy the condition that 5^(k+1) > 3000. Here are two:
case a: k = 4 [so, 5^(k+1) = 5^5 = 3125, which is greater than 3000]. In this case 5^k = 5^4 = 625, and 625 IS less than 1000.
case b: k = 5 [so, 5^(k+1) = 5^6 = 15625, which is greater than 3000]. In this case 5^k = 5^5 = 3125, and 3125 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)[5 - 1] = 500
Simplify: 5^(k-1)[4] = 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
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LIST the powers of 5:Is 5^k less than 1,000?
(1)5^(k+1) > 3,000
(2)5^(k-1) = 5^k - 500
5¹ = 5
5² = 25
5³ = 125
5� = 625
5� = 3125.
Statement 1: 5^(k+1) > 3000
It's possible that k=4, since 5^(4+1) = 5� = 3125.
In this case, 5^k = 5� = 625, which is LESS than 1000.
It's possible that k=10, since 5^(10+1) = 5¹¹ = something HUGE.
In this case, 5^k = 5¹�, which is GREATER than 1000.
INSUFFICIENT.
Statement 2: 5^(k-1) = 5^k - 500
5^k - 5^(k-1) = 500.
The terms on the left represent two powers of 5 that are 500 apart.
The required values can be seen in our list above:
5^k = 5� = 625.
5^(k-1) = 5³ = 125.
Thus, 5^k = 5� = 625, which is LESS than 1000.
SUFFICIENT.
The correct answer is B.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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Hi Rastis,
Both Brent and Mitch have provided some valuable insight into this question. Here's another set of rules to keep in mind (that involve exponent rules):
First, 5^(k+1) is 5 times the value of 5^k.
Using exponent rules, here's why:
(5^k)(5^1) = 5^(k+1)
Second, 5^k is 5 times the value of 5^(k-1)
Using a different exponent rule, here's why:
(5^k)/(5^1) = 5^(k-1)
So, in Fact 1, when we're told...
5^(k+1) > 3,000
We can divide both sides by 5^1....
[5^(k+1)]/5^1 = 5^k
3,000/5 = 600
5^k > 600
The question asks if 5^k is less than 1,000?
We know that 5^k > 600
If 5^k = 601, then the answer to the question is YES
If 5^k = 1,001, then the answer to the question is NO
Fact 1 is INSUFFICIENT
Fact 2 gives us 1 equation with 1 variable. As the other explanations show, you CAN solve it.
Fact 2 is SUFFICIENT
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
Both Brent and Mitch have provided some valuable insight into this question. Here's another set of rules to keep in mind (that involve exponent rules):
First, 5^(k+1) is 5 times the value of 5^k.
Using exponent rules, here's why:
(5^k)(5^1) = 5^(k+1)
Second, 5^k is 5 times the value of 5^(k-1)
Using a different exponent rule, here's why:
(5^k)/(5^1) = 5^(k-1)
So, in Fact 1, when we're told...
5^(k+1) > 3,000
We can divide both sides by 5^1....
[5^(k+1)]/5^1 = 5^k
3,000/5 = 600
5^k > 600
The question asks if 5^k is less than 1,000?
We know that 5^k > 600
If 5^k = 601, then the answer to the question is YES
If 5^k = 1,001, then the answer to the question is NO
Fact 1 is INSUFFICIENT
Fact 2 gives us 1 equation with 1 variable. As the other explanations show, you CAN solve it.
Fact 2 is SUFFICIENT
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
