Confused about direction of current in bjt question
For reference and to protect against edits to the question, here is the schematic being discussed:



they said that current through 3k and 9k resistors are same

That is a legitimate first approximation.  It is basically saying that the gain of Q3 is infinite.  Since the base current is the collector current divided by the gain, the base current is 0 in this approximation.

Assuming that the transistor gain is infinite is often useful just to get a basic idea what the circuit is doing.  Once you solve for the voltages and currents with this simplifying assumption, you get some idea what parts are sensitive to transistor gain.  Then you can go back and figure things more accurately in that area.

Another point is that good BJT circuits work with the transistor gains anywhere from the guaranteed minimum to infinite.  Therefore analyzing at infinite gain is verifying one of these criteria.  Often there is little difference between a gain of, say 100, and infinite.  Whether the collector current is 99% of the emitter current or 100% of the emitter current shouldn't make much difference in a good design.

Let's see what we get with infinite gain, then compare that to the fixed gain of 100.  With infinite gain, the base current is 0, so there will be 9 V across the 9K resistor, and 3 V across 3K.  This leaves about 8.3 V across 2.77K, for a collector and emitter current of 3 mA.

With a gain of 100, the resistance connected to the emitter is reflected to the base times (gain + 1).  The base therefore looks like a 280 k&Omega; load to -11.3 V.  The base voltage then comes out to -3.07 V.  The voltage across the emitter resistor is 8.23 V, and the emitter current 2.97 mA.  Again using a gain of 100, the collector current is therefore 2.94 mA.

This is a good example showing how the infinite gain approximation can lead to useful first pass results.  With Q3 having infinite gain, I0 is 3.00 mA.  With it having the finite gain of 100, I0 is 2.94 mA.  If this transistor has a minimum guaranteed gain of 100 at this operating point, then the rest of the circuit has to be able to tolerate the 2.94 to 3.00 mA variation.  That's life with real BJTs.  Fortunately, that level of uncertainty should not be a hardship.