Random Puzzles 

Workings

Hint
Look at the start of each word.

Answer
augment.

The first three letters are the same as the first three letters of the months 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
Be careful with the middle of the last word.

Answer
Business.
Committee.
Sincerely.
Archaeologist.

Workings

Hint
Remember that the bottle count doubles each time.

Answer
10pm.

Reasoning
The skip started with 1 lonely bottle.

At Noon: 1 person came along and added a bottle, making the total 2 bottles.

At 1pm: 2 people came along and each added a bottle, making the total 2 + 2 = 4 bottles.

At 2pm: 4 people came along and each added a bottle, making the total 4 + 4 = 8 bottles.

At 3pm: 8 people came along and each added a bottle, making the total 8 + 8 = 16 bottles.

Therefore, the number of bottles in the skip is doubling every hour, as it was full at 11pm, it must have been half full at 10pm.