ErikaPrepScholar wrote:A key thing to recognize here is that P, Q, R, and S are on the number line in that order. So we know going into the problem that the distance between P and S is 10, and that Q and R fall somewhere between them, with Q closer to P and R closer to S.

Statement 1:

The distance between P and R is 3. This also means that the distance between R and S is 7. But we don't know anything about Q: it's between P and R, but it could be 1 away from P and 2 away from R, or 2 away from P and 1 away from R, or 2.9 away from P and 0.1 away from R, etc. Insufficient.

Statement 2:

The distance between Q and S is 9. This also means that the distance between P and Q is 1. But this has the opposite problem of Statement 1: now we don't know where R is. It could be 1 away from Q and 8 away from S, or 4 away from Q and 5 away from S, or 7.05 away from Q and 1.95 away from S, etc. Insufficient.

Both:

Now we know where both Q and R are. If the distance between P and R is 3, and the distance between P and Q is 1, the distance between Q and R is 2. Sufficient.

I highly recommend drawing it out for number line problems! These can be hard to visualize in your head.