Medium Puzzles 

Workings

Hint
Rewrite the clues so that those who finished ahead are written first.

Answer
Pat Piper                 (P)
 Alex Ardle               (A)
  Zeb Zebra               (Z)
   Frankie Flint          (F)
    Max Merrymood         (M)
     Jesse James          (J)
      Glenn Goodfellow    (G)
       Toby Trent         (T)
        Billie Brick      (B)
         Kelly Kingfisher (K)

Reasoning
If we number the clues, write each name using a single letter, and rewrite the clues so that those who finished ahead are written first:

M beat T and J P beat J, T, and A P beat Z G beat T Z beat F F and P beat G T beat B A beat Z, K, and G G and M beat K B beat K A and Z beat M F beat T, M, and B J and A beat T J beat G and B
We can start with the first clue that contains just two runners.

By (3): P Z
By (5): P Z F
By (6): P Z F G
By (4): P Z F G T
By (7): P Z F G T B
By (10): P Z F G T B K

And...

By (2) and (8): P A Z.

We can now insert these, to give: P A Z F G T B K

And:

By (1): M J
By (12): F M J
By (14): F M J G

We can now insert these, to give: P A Z F M J G T B K, and we have found all of the finishing positions.

Workings

Hint
Each of the letters surrounding the central P becomes the second letter of a word.

Answer
Pancake, physics, pinball, plaster, popular, premium, psychic, pyramid.

Workings

Hint
The answer is a whole number less than 10.

Answer
3.

Reasoning
We can multiply throughout by 12,
    X     X     X     X
    -  -  -  +  -  =  -
    X     2     4     12

to give:

   12 - 6X + 3X = X

        12 - 3X = X

             12 = 4X

              3 = X

As required.

Double-Checking
    3     3     3     3
    -  -  -  +  -  =  -
    3     2     4     12

    1 - 1.5 + 0.75 = 0.25

              0.25 = 0.25, which confirms that 3 is correct.

Workings

Hint
The first letters of one half of the words are: A, B, D, C, F, D, H, A, I.

Answer
ABUNDANT   + SCARCE
BRAVERY    + COWARDICE
DANGER     + SAFETY
CONTRACT   + EXPAND
FAMOUS     + UNKNOWN
DISTRIBUTE + GATHER
HORIZONTAL + VERTICAL
ACCIDENTAL + INTENTIONAL
IGNORANCE  + KNOWLEDGE