Mathematical Puzzles 

Workings

Hint
Look at the first and third columns first.

Answer
The missing sum is 40.
Circle   = 12
Star     = 3
Square   = 7
Triangle = 9

Reasoning
Looking at Column 1 and Column 3 we have:

[1]   Square + Star + Circle + Square = 29
[2]   Square + Square + Triangle + Star = 26

If we subtract [1] – [2] we get:

[3]   Circle – Triangle = 3

And Row 3 is:

[4]   Circle + Circle + Triangle + Circle = 45

If we add [3] + [4] we get:

Circle + Circle + Circle + Circle = 48

So, Circle = 12.

Using [3] we find that Triangle = 9.

Using Column 2 we find that Star = 3.

Using Row 1 we find that Square = 7.

Double-Checking
The rows add to: 35 + 26 + 45 + 25 = 131.

The columns add to: 29 + 36 + 26 + 40 = 131.

The rows and columns must add to the same number, so this could have been used a shortcut to the missing sum.

In total, there are:
   5 Squares   = 5 x 7  = 35
   5 Circles   = 5 x 12 = 60
   3 Stars     = 3 x 3  =  9
   3 Triangles = 3 x 9  = 27

Which is also a total of 131.

So all of the totals are the same, as expected.

Workings

Hint
How many miles were left at the start of yesterday?

Answer
336 miles.

Reasoning
To work these numbers out, it is perhaps easier to start at the end and work backwards.

Yesterday I travelled ¹/₃ of the distance to leave 21 miles, therefore there were 31.5 miles (=21 ÷ ²/₃) at the start of yesterday.

On day 3 I travelled ³/₄ of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 ÷ ¹/₄) at the start of day 3.

On day 2 I travelled ¹/₂ of the distance to leave 126 miles, therefore there were 252 miles (=126 ÷ ¹/₂) at the start of day 2.

On the first day, I travelled ¹/₄ of the distance to leave 252 miles, therefore there were 336 miles (=252 ÷ ³/₄) at the start of the first day.

Double-Checking
On the first day, I travelled one quarter of the distance (84, leaving 252).

On day two, I travelled one half of the remaining distance (126, leaving 126).

On day three, I travelled three quarters of the remaining distance (94.5, leaving 31.5).

Yesterday I travelled one third of the remaining distance (10.5).

I now have 21 miles left to travel.

Workings

Hint
Remember that Time = Distance ÷ Speed, and Speed = Distance ÷ Time.

Answer
48 miles per hour.

Reasoning
Using:
   Time  = Distance ÷ Speed
   Speed = Distance ÷ Time

The outward journey took 174 miles ÷ 40 mph = 4.35 hours.

The return journey took 174 miles ÷ 60 mph = 2.9 hours.

So I travelled a total of 2 x 174 = 348 miles. I did this in a total of 4.35 + 2.9 = 7.25 hours.

My overall average speed was therefore 348 miles ÷ 7.25 hours = 48 miles per hour.

Workings

Hint
Start with the possible answers where the first digit is a prime number, and then look at the third digit.

Answer
23,475.

Reasoning
By (1), the first digit is prime:
   2----
   3----
   5----
   7----

By (3), the third digit is twice the first digit, so we can eliminate 5---- and 7----:
   2-4--
   3-6--

By (4), the fourth digit is the third digit plus three:
   2-47-
   3-69-

By (5) the fifth digit is the difference between the first digit and the fourth digit:
   2-475
   3-696

We know from the introduction that no digit is repeated, so we can eliminate 3-696. And, by (2) the second digit is the fifth digit minus the first digit:
   23475