Mathematical Puzzles 

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

Workings

Hint
Think about the 40%, and how many buses that represents.

Answer
72 buses.

Reasoning
The garage is currently full, with 12 more buses outside.

If we increase the garage space by 40%, there will be enough space to house those 12 buses, plus 12 more in the future – a total of 24 buses.

This means that 40% of the current garage capacity is 24 buses.

Therefore, 10% of the current garage capacity is 6 buses.

Therefore, 100% of the current garage capacity is 60 buses.

So, the current garage holds 60 buses, and there are 12 more outside.

Therefore, we have a total of 72 buses.

Double-Checking
They currently have 72 buses, 12 of which are outside, so 60 are inside.

Increasing the garage space of 60 buses by 40% will give room for 60 x 1.4 = 84 buses.

Which is the 72 they currently own, with space for 12 more buses in the future.

This confirms the answer.

Workings

Hint
How many students study chemistry and physics, but not biology?

Answer
500 students.

Reasoning
The answer is more easily seen if three intersecting circles are drawn, and the numbers inside each section are worked out.

We know that 73 students study all three disciplines, which allows us to work out the numbers.

97 students study chemistry and physics, so 97 – 73 = 24 study just chemistry and physics.
138 students study physics and biology, so 138 – 73 = 65 study just physics and biology.
152 students study chemistry and biology, so 152 – 73 = 79 study just chemistry and biology.

280 students study chemistry, so 280 – 73 – 24 – 79 = 104 study just chemistry.
254 students study physics, so 254 – 73 – 24 – 65 = 92 study just physics.
280 students study biology, 280 – 73 – 65 – 79 = 63 study just biology.

For a total of: 104 + 92 + 63 + 24 + 65 + 79 + 73 = 500 students.

Workings

Hint
The answer is 4 digits long, so what must G equal?

Answer
423 + 675 = 1098.

Reasoning
Remembering that:
   even + even = even
   odd  + odd  = even
   even + odd  = odd

To discuss individual letters, it's easiest to represent the sum as:
   A B C
   D E F +
————————
 G H I J

A + D has to be over 9, which means that G = 1.

B + E = I, is even + odd = odd, which means that we can't have a carry from C + F (otherwise it would have been even + odd + 1, which is even).

The 1 has already gone, so the smallest possible value for either C or F is 3, which means that the other can't be 7 or 9 (otherwise we'd have a carry).

Therefore, C and F are 3 and 5, but we don't know which is which. But we do now know that J = 8.

A + D = H, is even + even = even, which means that we can't have a carry from B + E. Therefore, E can't be 9, as this would force a carry. So E = 7.

I is the only remaining odd number, so I = 9.

Which means that B = 2.

Neither A nor D can be 0 (otherwise we would have two of the same digit). So, H = 0.

Therefore, A and D are 4 and 6 (but we don't yet know which is which).

Since the top row's digits have to add to 9, A can't be 6, so A = 4, making C = 3.

This makes the sum 423 + 675 = 1098.