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
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

Note that it could be E = A − D or E = D − A, but using A − D we'd end up with a negative value for E later, which isn't allowed (so we'd have to try again with D − A anyway).

As we have no information for D, it's best to find all of the other letters in terms of D. These steps are left as an exercise (e.g. use C in the equation for B), 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 things slightly clearer in the next step.

We then use these in [1] to find that:

   (12D − 51) ÷ 3 = 103
         12D − 51 = 309
              12D = 360
                D = 30

Therefore, F = (2D − 33) ÷ 3 = 9.

Workings

Hint
Look at Digit 3 first.

Answer
6394.

Reasoning
Labelling the clues:

Digit 2 is smaller than Digit 4 Digit 4 is two thirds of Digit 1 Digit 1 is two thirds of Digit 3 Digit 3 is three times Digit 2
By (3), Digit 3 is divisible by 3:

   Digit 3 = 3, 6, 9

By (3), Digit 1 is two thirds of Digit 3:

   Digit 1 = 2, 4, 6.

But, by (2), Digit 1 is divisible by 3:

   Digit 1 = 6, and Digit 3 = 9.

By (4), Digit 3 is three times Digit 2:

   Digit 2 = 3.

By (2), Digit 4 is two thirds of Digit 1:

   Digit 4 = 4.