Mathematical Puzzles 

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
You don't need to use a calculator.

Answer
Cube root of 2.

Reasoning
We could use a calculator to see that the tenth root of 10 = 1.2589 and the cube root of 2 = 1.2599.

But, if you've not got your calculator handy…

We have:

   10^1/₁₀ or 2^1/₃

Raise each to the power of 30 (since 10 x 3 = 30) to give:

   (10^1/₁₀)³⁰ or (2^1/₃)³⁰

   10^{1/₁₀ x 30} or 2^{1/₃ x 30}

   10³ or 2¹⁰

   1000 or 1024

Therefore, the cube root of 2 is larger.

Workings

Hint
A fathom is 6 feet.

Answer
Hours in a month.

Reasoning
A fathom is 6 feet, which is 1.8288 metres.

So there are 1000 ÷ 1.8288 ≈ 546.8 fathoms in a kilometre.

The shortest month has 28 x 24 hours = 672.

So even the shortest month has more hours than there are fathoms in a kilometre.

Workings

Hint
Try finding the cube root of 190,747.

Answer
53, 59, and 61.

Reasoning
Start by taking the cube root of 190,747, which is approximately 57.56.

The next highest prime number is 59:

   190,747 ÷ 59 = 3,233

Let's try the next prime number, which is 61:

     3,233 ÷ 61 = 53

53, 59, and 61, are all prime, and are consecutive, so this is the answer.

Double-Checking
   53 x 59 x 61 = 190,747