Random Puzzles 

Workings

Hint
Remember that Time = Distance ÷ Speed, and try setting the track length to be a random length.

Answer
119mph.

Reasoning
It doesn't matter how long a lap is, so let's set the track length to be a random number, say 5 miles.

Using:
   Time = Distance ÷ Speed

The first two laps took:
   ¹⁰/₁₀₀ hours

The second two laps took:
   ¹⁰/₁₀₂ hours

The next two laps took:
   ¹⁰/₁₄₀ hours

The final two laps took:
   ¹⁰/₁₅₀ hours

Therefore, the total time for the 8 laps was:
   ¹⁰/₁₀₀ + ¹⁰/₁₀₂ + ¹⁰/₁₄₀ + ¹⁰/₁₅₀ hours
   = ¹/₁₀ + ⁵/₅₁ + ¹/₁₄ + ¹/₁₅ hours

Looking at the denominators:
   10 = 2 x 5
   51 = 3 x 17
   14 = 2 x 7
   15 = 3 x 5

The lowest common multiple (LCM) of these numbers is 2 x 3 x 5 x 7 x 17 = 3570.

   = ³⁵⁷/₃₅₇₀ + ³⁵⁰/₃₅₇₀ + ²⁵⁵/₃₅₇₀ + ²³⁸/₃₅₇₀ hours
   = ¹²⁰⁰/₃₅₇₀ hours
   = ¹²⁰/₃₅₇ hours
   = ⁴⁰/₁₁₉ hours

The 8 laps were 40 miles, and using Speed = Distance ÷ Time:
   average speed = 40 ÷ (⁴⁰/₁₁₉)
   average speed = 40 x (¹¹⁹/₄₀)
                 = 119 mph

Workings

Hint
This is quite a common alphabet.

Answer
Z.

From Zeta, first letters of the Greek alphabet, alpha, beta, gamma, delta, epsilon, zeta, eta.

Workings

Hint
Going down and across the floor isn't the only route.

Answer
40 feet.

Explanation Diagram
If you imagine the room to be a cardboard box, you can 'unfold' the room in various ways, and each route gives a different answer.



We can use Pythagoras' theorem (a² + b² = c²) to calculate the distances:
   distance² = horizontal² + vertical²
   distance = √(horizontal² + vertical²)

Route #1
   distance = 1 + 30 + 11 = 42 feet.

Route #2
   horizontal = 6 + 30 + 6 = 42 feet.
   vertical = 10 feet.
   distance = √(42² + 10²) ≈ 43.174 feet.

Route #3
   horizontal = 1 + 30 + 6 = 37 feet.
   vertical = 6 + 11 = 17 feet.
   distance = √(37² + 17²) ≈ 43.178 feet.

Route #4
   horizontal = 1 + 30 + 1 = 32 feet.
   vertical = 6 + 12 + 6 = 24 feet.
   distance = √(32² + 24²) = 40 feet.

Workings

Hint
One of the words is Railway.

Answer
Railway, rainbow, recycle, replace, respect, royalty, referee, rejoice.