Random Puzzles 

Workings

Hint
This is a very common sequence.

Answer
5.

The numbers represent the number of letters of the words ONE, TWO, THREE, etc. Therefore the answer is the number of letters in the word EIGHT.

Workings

Hint
Adding the speeds won't work.

Answer
205 mph.

Reasoning
If we take the number of miles in a lap to be L miles and use Time = Distance ÷ Speed.

The first ²/₅ are covered at 123 mph. Therefore, the time taken was [(²/₅) x L] ÷ 123 hours.

The next ²/₅ are covered at 164 mph. Therefore, the time taken was [(²/₅) x L] ÷ 164 hours.

The last ¹/₅ is covered at y mph. Therefore, the time taken was [(¹/₅) x L] ÷ y hours.

The entire trip is covered at an average of 150 mph, and we can conclude that:
 2L      2L      L     L
————— + ————— + ——— = ———
5x123   5x164    5y   150

Rearranging and simplifying gives:
 1     1    2     2
——— = —— - ——— - ———
 y    30   123   164

That means that y = 1 ÷ (¹/₃₀ − ²/₁₂₃ − ²/₁₆₄)

Using a calculator gives y = 205.

Workings

Hint
Can you see 1, 2, 3, …?

Answer
46.

Reasoning
Take the numbers in pairs, square the first, and reverse the digits.
1  1   (1 x 1 = 1,  reversed is 1)
2  4   (2 x 2 = 4,  reversed is 4)
3  9   (3 x 3 = 9,  reversed is 9)
4  61  (4 x 4 = 16, reversed is 61)
5  52  (5 x 5 = 25, reversed is 52)
6  63  (6 x 6 = 36, reversed is 63)
7  94  (7 x 7 = 49, reversed is 94)
   so we require:
8  46  (8 x 8 = 64, reversed is 46)

Workings

Hint
The first letters of the names are: G, R, E, P, M, E, A, A.

Answer
gl    = Gail
rs    = Rose or Iris
dth   = Edith
ptsy  = Patsy
mxn   = Maxine
dwn   = Edwina or Dawn
gth   = Agatha
dry   = Audrey