Mathematical Puzzles 

Workings

Hint
How can the answer end in 9?

Answer
4,217 x 5,237 = 22,084,429.

Reasoning
The only way the answer can end in 9 is if the two numbers both end in 7 (7 x 7 = 49).

The only way we can reach 22 million is if the two numbers begin with 4 and 5 respectively.

This leaves each number with just 2 options, and the answer is quickly found.

Workings

Hint
A beetle has six legs, a spider has eight legs, and a mouse has four legs.

Answer
102 animals, 34 of each.

Reasoning
As there are the same number of each animal, for every beetle, spider, and mouse counted, there are a total of 18 legs.

612 ÷ 18 = 34 of each animal.

Which gives a total of 34 x 3 = 102 animals.

Workings

Hint
How much fuel am I using every hour?

Answer
150 miles.

Reasoning
I was travelling at 30mph at a rate of 20 mpg, so I was using 1.5 gallons every hour for the driving itself.

I was also losing 0.5 gallons every hour. In total, I was using 2 gallons every hour.

Therefore, to use all of the 10 gallons I started with, I travelled for 5 hours.

5 hours at 30mph is 150 miles.

Workings

Hint
You might find Pythagoras' theorem very useful [a² + b² = c²].

Answer
148 inches.



We can use Pythagoras' theorem if we draw an imaginary line across to create a right-angled triangle.

The hypotenuse is equal to the rope's length R. The bottom of the triangle is 48 inches. The vertical side is R - 8 (the difference between 20 inches and 12 inches).

Pythagoras' theorem tells us that:

   a² + b² = c²

Where c is the hypotenuse.

This gives us:

          48² + (R - 8)² = R²

   2304 + (R - 8)(R - 8) = R²

    2304 + R² - 16R + 64 = R²

         2304 - 16R + 64 = 0

                     16R = 2368

                       R = 148

As required.