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# Maths Week: The solutions to all of the puzzles

## Tens of thousands of you took our mathematical challenge all week. Here are the answers. "Pass Maths did not prepare me for this"
Image: Shutterstock/Africa Studio

THIS PAST WEEK was Maths Week and if you’ve clicked in here, you’re probably one of the people who has been puzzling over the fiendish maths challenges that have run on TheJournal.ie every evening over the past seven days.

So here they are: the answers you ‘ve been waiting for.

Let us know how you did in the comments.

Sunday: The One About Dividing Up The Sweets

One half is 30, one third is 20, one sixth is 10. If we take 10 from 30 and add it to 10 then everybody has 20.

Monday: The One About Organising The Building Of A Wall

The faster worker finishes his wall in 4 days that means he builds 1/4 of a wall in a day. The second worker builds 1/6 of a wall in a day and the third builds 1/12 of a wall in a day. Between them they can build 1/4 + 1/6 + 1/12 of a wall per day. That is 6/12 or 1/2 of a wall per day. This means that they can build one wall in 2 days.

Tuesday: The One About The Sugar Tax

The answer: 35g in a 330ml can is 10.6g per 100ml so the sugar tax added will be 60c (30c per litre)

You would consume 365 x 35 g = 12,775 g. That’s over 12 kg (26 pounds or 1 stone 8 pounds) of unnecessary sugar from just one can of this sugared drink a day for one year. If you read your food labels you will see that this wouldn’t be the only source of sugar intake in your diet.

Wednesday: The One About The Number Of Games In A Tennis Tournament

1) There will be 5 rounds. 32 players in 1st round, 16 in 2nd, 8 in 3rd, 4 in 4th, 2 in 5th (final).

1b)  There will be 31 matches. If you think about it – in every match there will be a unique loser. The tournament will have 31 losers in all. That means that 31 matches will have to be played.

2) There will be 10 matches – the outer vertices (corners) below represent the players, the lines each represent one match or connection between the players.

Thursday: The One About The Currency Exchange

The answer: £100 costs €100/0.85 + commission = 117.64+ 1.27 = €118.92

£100 can be sold for 1.11 €/£ = €110 and the commission has to be paid from this, so you get €110 – 1.27 = €108.73 back. So, the whole business has cost you €10.19.

Friday: The One About The Snooker Balls

The answer: For 10 rows we will need 55 balls.

The method:  You could count these in your head, but a formula can be discovered that will calculate for any number of rows.

The formula may be understood by looking at the diagram below. The balls can be rearranged into a right angled triangle. A duplicate of this can be inverted and placed alongside forming a rectangle. This has height the number of rows and as length the number of rows plus one. The number of balls in this rectangle is 6×5 which is 30 which is twice the number of balls in the 5 rows.

For one hundred rows the rectangle would be 100 x 101 = 10100. And half of this is 5050 which gives the number of balls needed for 100 rows.

Writing as a formula in mathematical form:
N = 1/2 n(n+1)
N = number of balls needed
n -= number of rows Saturday: The One About Two Friends Running a 5KM

The answer:  Ann runs 3 km. She runs one and a half times as fast as Bill, Therefore she will run one and a half times as far as Bill in the same time. That is in the ratio of 3 : 2. So Ann runs 3 km in the time that Bill runs 2 km.

Ann can lap Bill if she runs 15 km in which time Bill would run 10 km. 