Random Puzzles 

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
This is just a sequence of letters.

Answer
V.

Move 3 and then 4 letters alternately.

Workings

Hint
This is a simple substitution cipher.

Answer
50.

This puzzle is a simple test to see how easy a ROT13 part encrypted puzzle can be. Every other letter has been converted to ROT13 and the other letters are left as they were. But, you may ask, what is the puzzle? Well, how many words are in this puzzle!

ROT13 replaces each character with the one 13 places away.
A = N, B = O, C = P, D = Q, E = R, F = S, G = T, H = U, I = V, J = W, K = X, L = Y, M = Z, N = A, O = B, P = C, Q = D, R = E, S = F, T = G, U = H, V = I, W = J, X = K, Y = L, Z = M.

Workings

Hint
Try a random number of over 50s to see what happens.

Answer
20,064 apples.

Reasoning
It doesn't matter how old the people are.

As only ³/₄ of the over 50s claimed 16 apples, this averages out to 12 apples each.

Similarly, only ¹/₈ of the under 50s claimed 96 apples, this again averages out to be 12 apples each.

Therefore, the total number of apples was 1,672 x 12 = 20,064.