Gmat_mission wrote:$$Is\ \ \ 10^m<5,000?$$ $$(1)\ \ \ 10^{m+1}>9,000$$ $$(2)\ \ \ \ 10^{m-1}=10^m-900$$ [spoiler]OA=B[/spoiler].

Could someone explain how to solve this DS question to me? Thanks for your help.

Hello Gmat_mission.

Let's take a look at your question.

Note that we don't know if m is an integer (positive or negative) or not. But, we will assume that m is an integer.

$$(1)\ \ \ 10^{m+1}>9,000$$

We can rewrite this expression and find the possible values for m. $$10^{m+1}>9000\ \ \ \Rightarrow\ \ \ 10^m\cdot10>9000\ \ \ \Rightarrow\ \ 10^m>900\ \ \ \Rightarrow\ \ m\ge3.$$

If m=3 then 10^3=1,000<5,000. YES.
If m=4 then 10^4=10,000>5,000. NO.

Since we've got two different answers, then this statement is NOT SUFFICIENT.

$$(2)\ \ \ \ 10^{m-1}=10^m-900$$

Again, we can solve the inequality for m as follows: $$10^{m-1}=10^m-900\ \ \Rightarrow\ \ 10^m\cdot10^{-1}-10^m=-900\ \ \Rightarrow\ \ 10^m\left(\frac{1}{10}-1\right)=-900\ \ \ \Rightarrow\ 10^m\left(-\frac{9}{10}\right)=-900$$ $$10^m=1000\ <5000.\ \ \ YES.$$ Therefore, this statement is SUFFICIENT.

The correct answer is the option B.

I hope it helps.