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DS exponent question

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by amontobin » Mon Jul 17, 2017 10:35 am
If c=3.905 × 10^x and x is a negative integer, what is x?

(1) .00005 < c < .0005
(2) 10^−4 < c < 10^−3

What's a quick way to deal with negative exponents like this?
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Source: — Data Sufficiency |
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by GMATGuruNY » Mon Jul 17, 2017 1:04 pm
amontobin wrote:If c=3.905 × 10^x and x is a negative integer, what is x?

(1) .00005 < c < .0005
(2) 10^−4 < c < 10^−3
Test cases for x:
Case 1: x=3, with the result that c = 3.905 * 10¯³ = 0.003905.
Case 2: x=4, with the result that c = 3.905 * 10¯� = 0.0003905.
Case 3: x=5, with the result that c = 3.905 * 10¯� = 0.00003905.

Statement 1: 0.00005 < c < 0.0005
Only Case 2 satisfies this constraint, as illustrated by the digits highlighted in red below:
0.0000500 < 0.0003905 < 0.0005000.
Thus, x=4.
SUFFICIENT.

Statement 2: 0.0001 < c < 0.001
Only Case 2 satisfies this constraint, as illustrated by the digits highlighted in blue below:
0.0001000 < 0.0003905 < 0.0010000.
Thus, x=4.
SUFFICIENT.

The correct answer is D.
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by Jay@ManhattanReview » Mon Jul 17, 2017 10:10 pm
amontobin wrote:If c=3.905 × 10^x and x is a negative integer, what is x?

(1) .00005 < c < .0005
(2) 10^−4 < c < 10^−3

What's a quick way to deal with negative exponents like this?
We have c = 3.905 × 10^x

Statement 1: 0.00005 < c < 0.0005

=> 5 x 10^-5 < c < 5 x 10^-4
5 x 10^-5 < 3.905 × 10^x < 5 x 10^-4

We see that 3.9 < 5, so the decision of less/more is among 10^-5, 10^x and 10^-4.

If we deal with the inequality 5 x 10^-5 < 3.905 × 10^x, we conclude that x must be at least -4.

Similarly, if we deal with the inequality 3.905 × 10^x < 5 x 10^-4 , we conclude that x can be at the most -4.

Thus, the only value of x that satisfies the conditions: "x must be at least -4" and "x can be at the most -4," is x = -4. Sufficient.

Statement 2: 10^−4 < c < 10^−3

=> 1 x 10^−4 < 3.905 × 10^x < 10 x 10^−4

The only value x can assume is x = -4. Sufficient.

The correct answer: D

Hope this helps!

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-Jay
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by Jay@ManhattanReview » Mon Jul 17, 2017 10:37 pm
Statement 1 can be dealt in the following way too.
We have c = 3.905 X 10^x

Statement 1: 0.00005 < c < 0.0005

=> 5 x 10^-5 < c < 5 x 10^-4
5 x 10^-5 < 3.905 x 10^x < 5 x 10^-4
=> 5 x 10^-5 < 39 x 10^(x-1) < 50 x 10^-5

We did this so that the non-exponent parts can be easily compared. We see that they are arranged in an ascending order.

Thus, x-1 = -5

x = -5 + 1 = -4.

Hope this helps.

-Jay
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