Data Sufficiency

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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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