Mathematical Puzzles 

Workings

Hint
What can digit 4 be?

Answer
6849.

Reasoning
Digit 4 must be a multiple of 3:
   ---3
   ---6
   ---9

Digit 3 is one more than Digit 9 ÷ 3:
   --23
   --36
   --49

Digit 2 is twice Digit 3:
   -423
   -636
   -849

But Digit 4 is one more than Digit 2, which only leaves:
   -849

And Digit 1 is three smaller than Digit 4:
   6849.

Workings

Hint
Remember that there are 12 inches in a foot.

Answer
I asked for 9 foot 2 inches (110 inches) and my friend brought me 2 foot 9 inches (33 inches).

Reasoning
I asked my friend to buy me A feet and B inches, which is a total of 12A + B inches.

He accidentally bought me B feet and A inches, which is 12B + A inches.

We know that 12B + A is actually only 30% of what I wanted, so:

     12B + A = 0.3 x (12A + B)

Multiply throughout by 10:

  120B + 10A = 3 x (12A + B)

  120B + 10A = 36A + 3B

        117B = 26A

           B = 26A ÷ 117

Don't forget that A and B can only be integers, and between 0 and 11.

Since 26 x A has to be at least 117, the lowest possible value for A is 5. We can check the remaining possibilities.
 5 x 26 = 130
 6 x 26 = 156
 7 x 26 = 182
 8 x 26 = 208
 9 x 26 = 234 (*)
10 x 26 = 260
11 x 26 = 286

The only number wholly divisible by 117 is 234, which means that A = 9, and B = 2.

So, I originally asked for 9 feet 2 inches.

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.

Workings

Hint

Answer
6423: 109191 / 17. The numbers 1234567890 have been replaced by 7182435069 respectively.