**Answer:** 205 mph.

we have had a lot of queries on this puzzle and many methods have been suggested but only the one below gives the correct answer.

If we take the number of miles in a lap to be x miles and using time = distance / speed.

The first 2/5 are covered at 123 mph. Therefore, the time taken was [(2/5) * x] / 123 hours.

The next 2/5 are covered at 164 mph. Therefore, the time taken was [(2/5) * x] / 164 hours.

The last 1/5 is covered at y mph. Therefore, the time taken was [(1/5) * x] / y hours.

The entire trip is covered at an average of 150 mph. we can conclude that:

2x 2x x x

----- + ----- + --- = ---

5*123 5*164 5*y 150

Working this out we get:

1 -2 -2 1

-- - --- - --- = -

30 123 164 y

The means that y = 1 / (1/30 - 2/123 - 2/164)

Therefore y = 205 QED.