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Workings

Hint
Can the fractions be simplified?

Answer
667.

Reasoning
 3        8        1
———  of  ———  of  ———  of 2001
 4        9        2

which simplifies to:
24
——  of 2001
72

which simplifies to:
 1
———  of 2001
 3

Giving 667.

Workings

Hint
Find anagrams of the words in this order: PLEASE, TRANCE, LISTEN, WARNED, VEINED, BRUISE.

Answer
ANSWER.

Reasoning
The initial letters are RWASEN, which is an anagram of ANSWER.
bruise becomes rubies warned becomes wander (or warden) please becomes asleep listen becomes silent veined becomes envied (or endive) trance becomes nectar
The word DEBATE can also be made, but this requires an obscure anagram.

Workings

Hint
There are lots of vowels (AEIOU).

Answer
A.

We have the vowels (AEIOU) intermingled with the reverse (UOIEA).

Workings

Hint
The answer is not 15.

Answer
19 bottles.

Reasoning
From the initial 77 bottles, 75 bottles can be recycled for 15 new full bottles.

Once those are drunk, Alex will have 17 empty bottles (15 plus the 2 extra empty bottles).

From the 17 empty bottles, 15 bottles can be recycled for 3 new full bottles.

Once those are drunk, Alex will have 5 empty bottles (3 plus the 2 extra empty bottles), which can be recycled for one final full bottle.

Giving a total of 15 + 3 + 1 = 19.