Random Puzzles 

Workings

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

Answer
Crystal Ball        (CB)
 Justin Time        (JT)
  Inga Hootz        (IH)
   Leigh Vamessage  (LV)
    May Eye         (ME)
     Mike Stup      (MS)
      Laura DeLand  (LD)
       Joe King     (JK)
        Holly Day   (HD)
         Jack Pott  (JP)

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

CB beat ME and MS LV beat ME and LD HD, MS, and JT, all beat JP MS, ME, and CB, all beat LD JK beat HD and JP LD and IH both beat JK CB and IH both beat LV JT beat IH CB beat JT JT beat HD and JP ME beat MS and HD LV and IH both beat HD
By (9): CB JT
By (8): CB JT IH
By (7): CB JT IH LV
By (2): CB JT IH LV ME
By (11): CB JT IH LV ME MS
By (4): CB JT IH LV ME MS LD
By (6): CB JT IH LV ME MS LD JK
By (5): CB JT IH LV ME MS LD JK HD
By (3): CB JT IH LV ME MS LD JK HD JP

Workings

Hint
This is quite a tricky puzzle, and knowledge of algebra would certainly help.

Answer
The children tickets were £14, and the adult tickets were £18.

Reasoning
There are a number of methods for solving this problem, including:

Using a spreadsheet. Using a computer program. Using the intersection of lines on a graph. Using an online equation solver. Solving simultaneous equations using algebra. Solving simultaneous equations using inverse matrices.
Here is my solution using simultaneous equations and algebra.

First construct two algebraic equations, where 'c' is the number of children, and 'a' is the number of adults:

[1]   4296c + 2143a = 98718
[2]   5146c + 2807a = 122570

To make the number in front of 'c' the same on both, we multiply [1] by 5146 and [2] by 4296 to give:

[3]   22107216c + 11027878a = 508002828
[4]   22107216c + 12058872a = 526560720

Now we can do [4] – [3] to give:

   1030994a = 18557892

Divide throughout by 1030994 so that:

   a = 18

Substituting a = 18 in [1] will give:

   4296c + 2143 x 18 = 98718
   4296c + 38574     = 98718
   4296c             = 60144
       c             = 14

Double-Checking
c = 14 and a = 18

4,296 x 14 + 2,143 x 18 = 98,718

and

5,146 x 14 + 2,807 x 18 = 122,570

Workings

Hint
In order, the added letters spell TELESCOPE.

Answers
The added letters spell TELESCOPE.
herb + t becomes berth bang + e becomes began keen + l becomes kneel suit + e becomes suite quad + s becomes squad whey + c becomes chewy cell + o becomes cello toil + p becomes pilot wild + e becomes wield

Workings

Hint
The word begins with the letter T.

Answer
Tattooists.

Which has 10 letters.