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Source: — Data Sufficiency |

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by [email protected] » Sun Jul 07, 2013 11:30 am
Hi Cazubuine,

This DS question is great because it has a "question behind the question." Since you're told that x is an integer and 4^x < 100, that really means that x is less than or equal to 3. The prompt asks for the value of x? You do NOT want to try to solve this algebraically.

1) Fact 1 gives you a real ugly inequality, so instead of trying to do math, use the basic math that you already know and TEST IT.

Plugging in values for x, you'll find...
If x = 2 ---> total = 60 which is NOT greater than 100, so x CANNOT = 2
If x = 3 ---> total = 240, which IS greater than 100

There's no point in testing smaller integers, since the total won't be > 100 and you're NOT ALLOWED to go bigger than 3 (because of the information in the prompt.

Therefore X must = 3. Fact 1 is SUFFICIENT

2) Fact 2 provides you a similar situation, with just a slight change in the math.
Plug in values here and you'll find that X MUST = 3.

Fact 2 is SUFFICIENT.

Final Answer: D (Both are SUFFICIENT).

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by GMATGuruNY » Sun Jul 07, 2013 1:39 pm
cazubuine wrote:Can this be solved algebraically:

If x is an integer and 4^x < 100, what is x?

(1) 4^(x + 1) - 4^(x - 1) > 100
(2) 4^(x + 1) + 4^x > 100 [/b]
Plugging in values seems easier, but here's an algebraic approach.

Statement 1: 4^(x + 1) - 4^(x - 1) > 100
Factor out 4^x:
4^x * (4¹ - 4¯¹) > 100

Since (4¹ - 4¯¹) = 4 - 1/4 ≈ 4, we can ballpark:
4^x * 4 > 100
4^x > 25.

Thus, 25 < 4^x < 100, implying that x=3.
SUFFICIENT.

Statement 2: 4^(x + 1) + 4^x > 100
Factor out 4^x:
4^x * (4¹ + 1) > 100
4^x * 5 > 100
4^x > 20.

Thus, 20 < 4^x < 100, implying that x=3.
SUFFICIENT.

The correct answer is D.
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by cazubuine » Sun Jul 07, 2013 3:46 pm
Thank you very much Rich and Mitch. This definitely makes sense now!
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