Mathematical Puzzles 

Workings

Hint
How many tyre miles were travelled?

Answer
1,005 miles.

Reasoning
A total of 2,345 miles were travelled and at any one time, three tyres were on the tricycle.

Therefore, 3 x 2,345 = 7,035 tyre miles were travelled, which was shared equally by the 7 tyres.

And 7,035 ÷ 7 = 1,005.

Workings

Hint
The number is less than 20.

Answer
8.

Reasoning
We need a number (N) such that:

2 x N = 14 + N
             —
             4

Multiply throughout by 4 to give:

8 x N = 56 + N

So:

7 x N = 56

So N = 8.

Double-Checking
Its double is 16.

Its quarter is 2.

So its double is fourteen more than its quarter.

Workings

Hint
Think of the pages from 1-9, and then 10-99.

Answers
There are 147 pages. The number 3 appears 35 times.
Answer #1
For pages 1-9, there are 9 pages, which is 9 digits.

For pages 10-99, there are 90 pages, which is 180 digits.

This is a total so far of 189, therefore we require another 333 – 189 = 144 digits, which is another 144 ÷ 3 = 48 pages.

Taking us to 9 + 90 + 48 = 147 pages in total.

Answer #2
From page 1 to 147 we have 15 pages that end in 3:

   3, 13, 23, …, 143

We also have the ten pages that start with 30:

   30, 31, 32, …, 39

Plus the ten pages that start with 130:

   130, 131, 132, …, 139

For a total of 15 + 10 + 10 = 35 number 3's.

Workings

Hint
The fourth player is the key to this tricky question.

Answer
9 points.

Respectively the scores were 7, 14, 20, 30, 23, 9.

Reasoning
If we label the six players A, B, C, D, E, and F, we know that:

[1]  A + B + C + D + E + F = 103

and from the clues:

   A = B ÷ 2
   B = C – 6
   C = D x 2 ÷ 3
   E = D – A
   F = E – 14

If instead we choose E = A – D, we'd later see that we end up with a negative value for E, which isn't allowed.

Since D is the letter we're missing information for, it's best to find all of the other letters in terms of D.

These steps are left as an exercise (use C in the equation for B, etc), but the result is:

   A = ( D –  9)  ÷ 3
   B = (2D – 18)  ÷ 3
   C = (2D     )  ÷ 3
   D = (3D     )  ÷ 3
   E = (2D +  9)  ÷ 3
   F = (2D – 33)  ÷ 3

Writing it as D = 3D ÷ 3 makes it slightly clearer to see, when adding in the next step.

We can then use these in [1] to find that 12D = 360, so D = 30.

Therefore, F = 9.