Random Puzzles 

Workings

Hint
Work out the total number of legs.

Answer
23 children (46 legs).

With 230 ants (1380 legs), 3 puppies (12 legs), and 1 teacher (2 legs).

If we call the number of puppies P, then there were (P + 20) children and 10 x (P + 20) ants, and a single teacher. If we now count the legs, we get:
Puppies:
 P x 4 =  4P

Children:
 (P + 20) x 2 =  2P + 40

Ants:
 10 x (P + 20) x 6 = 60P + 1200

Teacher: 2

A total of 66P + 1242

Giving:

 66P + 1242 = 1440

 66P        = 198

   P        = 3

So there were 3 puppies, hence 23 children.

Workings

Hint
Loot at each number backwards.

Answer
32.

The first nine prime numbers reversed.

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
The answer involves months of the year.

Answer
6.

The number of letters each month has, starting with January, therefore, August has 6 letters.