"Two years ago, jeans were priced at 72 dollars and 121,000 units were sold. Last year, the price was lowered to 68 and sales increased to 132,000....

In a sense we use both. I think what may be tripping you up is that normally when we estimate elasticities, we don't use the starting or ending point as the baseline; we use the midpoint as the baseline.So instead of ((q2-q1)/q1)/((p2-p1)/p1)as you did, we'd use the averages of q and p:((q2-q1)/((q1+q2)/2))/((p2-p1)/((p1+p2)/2))The 2s cancel out, so we have:((q2-q1)/(q1+q2))/((p2-p1)/(p1+p2))So  instead of what you had, we would do this:e =...

In a sense we use both. I think what may be tripping you up is that normally when we estimate elasticities, we don't use the starting or ending point as the baseline; we use the midpoint as the baseline.

So instead of
((q2-q1)/q1)/((p2-p1)/p1)
as you did, we'd use the averages of q and p:
((q2-q1)/((q1+q2)/2))/((p2-p1)/((p1+p2)/2))
The 2s cancel out, so we have:
((q2-q1)/(q1+q2))/((p2-p1)/(p1+p2))

So  instead of what you had, we would do this:
e = ((132000-121000)/(132000 +121000))/((68-72)/(68+72)) = (11000/253000)/(-4/140) = (0.043478)/(-0.0285714) = -1.522

Now it asks for what would happen if we drop the price an additional dollar, so we should start with where we currently are, which is (132000, $68).

Generally we'd assume that our elasticity is constant, so we simply need to know what the percentage change in p is, which we then multiply by the elasticity to get the percentage change in q.

Dropping the price $1 from $68 to $67 is a change of -1.47%.
Thus, the quantity will change by (-0.0147)(-1.522) = 0.0224
The quantity sold will therefore increase by 2.24%.