Mathematical Puzzles 

Workings

Hint
What do the 3 numbers add and multiply to give so far?

Answer
$0.50.

Reasoning
The three prices add to 1.50 + 3.00 + 4.00 = 8.50.

The three prices multiply to 1.50 x 3.00 x 4.00 = 18.00.

You might be able to see that adding another 0.50 will take the total to 9.00, and multiplying by another 0.50 will take the product to 9.00.

The answer can also be found using algebra. If we call the missing price P, we have:
1.50 + 3.00 + 4.00 + P = 1.50 x 3.00 x 4.00 x P

    8.50 + P = 18.00 x P

        8.50 = 18.00 x P – P

        8.50 = 17.00 x P

8.50 ÷ 17.00 = P

           P = 0.50.

Workings

Hint

Answer
The bottle cost 35 pence and the straw cost 25 pence.

Workings

Hint
How many leaves will I have collected on day 5?

Answer
On the 23^rd.

Reasoning
We could simply keep adding until we get the required number:

   1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23

   = 276 leaves.

But a more mathematical method might help to answer the Bonus Question – as this might take a while if we keep adding!

So, let's create a method by imagining that we are adding the numbers from 1 to 30.

   1 + 2 + 3 + … + 28 + 29 + 30

If we now take the numbers in pairs, taking one from each end, we have:

   (1 + 30) + (2 + 29) + (3 + 28) + … + (15 + 16)

Each pair adds to 31, and we have 15 pairs. So the total sum is 31 x 15 = 465.

The total sum from 1 to any number (N) can be found using this technique, and we will have:

   Each pair adds to (1 + N), and there are N ÷ 2 pairs. So the total is:

   (1 + N) x N
             —
             2

In this puzzle, we know that this equals 276.

So:

   (1 + N) x N = 276
             —
             2

We can expand the brackets, and multiply both sides by 2, to give:

   N + N² = 552

Rearranging we get:

   N² + N – 552 = 0

And 552 = 2 x 2 x 2 x 3 x 23, so this can be factorised as:

   (N + 24) x (N – 23) = 0

Because we need to find a positive number of days, the only possible answer is:

   (N – 23) = 0

So N = 23 days.

Bonus Question
To answer the bonus question, we have:

   (1 + N) x N = 56616
             —
             2

Rearranging we get:

   N² + N – 113232 = 0

And 113232 = 2⁴ x 3 x 7 x 337, so this can be factorised as:

   (N – 336) x (N + 337) = 0

Because we need to find a positive number of days, the only possible answer is:

   (N – 336) = 0

So N = 336 days (I did say that I liked collecting leaves!).

Workings

Hint
How many apples were left after Charlie had taken some?

Answer
162 apples.

Alex hid 54, leaving 108.

Billie then hid 36, leaving 72.

Charlie then hid 24, leaving 48 available to share in the morning.

Reasoning
To work these numbers out, we start at the end and work backwards.

At the end there were 16 apples each, therefore there were 48 apples in total.

Charlie removed one third leaving 48, so there must have been 72 apples before.

Billie removed one third leaving 72, so there must have been 108 apples before.

Alex removed one third leaving 108, so there must have been 162 apples before.