Random Puzzles 

Workings

Hint
Look at the second letter in each word.

Answer
Guava.

The second letter of the words are the vowels (AEIOU) in order.

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
The first nine prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23.

Answer
29.

The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

Workings

Hint
Another example, PEN is a complete word because:
   -en = ten
   p-n = pin
   pe- = pea

Answer
Two of the words are not complete: bugs, lean.

Reasoning
The remaining 8 words can be changed as follows:
hard:  -ard = card
 h-rd = herd
 ha-d = hand
 har- = harp

port:  -ort = fort
 p-rt = part
 po-t = poet
 por- = pore

duck:  -uck = muck
 d-ck = deck
 du-k = dusk
 duc- = duct

bugs:  -ugs = hugs
 b-gs = bags
 bu-s = buys
 bug- = ---- X

bail:  -ail = hail
 b-il = boil
 ba-l = bawl
 bai- = bait

mast:  -ast = cast
 m-st = most
 ma-t = malt
 mas- = mask

thin:  -hin = shin
 t-in = twin
 th-n = than
 thi- = this

lean:  -ean = mean
 l-an = loan
 le-n = ---- X
 lea- = leaf

fire:  -ire = hire
 f-re = fare
 fi-e = five
 fir- = firm

read:  -ead = bead
 r-ad = road
 re-d = reed
 rea- = real