Solution

The cook sets the the two hourglasses simultaneously. The sand grains in the small hourglass drain out completely in four minutes.

So, four minutes have elapsed. At this point of time there is three minutes of sand in the upper half of the bigger hourglass. The cook quickly inverts the smaller hourglass and it starts "ticking" again. After three minutes, the big hour glass empties completely, and the total time measured is 4 + 3 = 7 minutes. There is still one minute of sand left in the smaller hourglass.

The cook swiftly inverts the big hour glass and sets it "ticking" again. After one minute, the small hour glass empties completely and, now,the time measured is 7 + 1 = 8 minutes. The cook now needs to measure one more minute.

At this stage, one minute worth of sand has trickled down into the lower half of the bigger hourglass, and there are six minutes of sand in its upper half. Is that right? Now, all that the cook has to do is quickly invert the bigger hourglass so that in this position one minute of sand is in the upper half and six minutes of sand is in the lower half. After one minute, the bigger hour glass empties and the cook is able to measure nine minutes quite accurately. A clever cook, really!