Parallelogram 5 Year 11 8 Oct 2020Winners

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Intermediate Maths Challenge Problem (UKMT)

3 marks

1.1 The average (mean) weight of five giant dates was 50g. Kate ate one and the average (mean) weight of the four remaining dates was 40g. What was the weight of the date that Kate ate?

  • 10g
  • 50g
  • 60g
  • 90g
  • more information is needed

The total weight of the original five dates was 250g and the total weight of the four remaining dates was 160g.

2. Learning from successful people

Watch this video from the BBC about why you might not always want to emulate successful people.

(If you have problems watching the video, right click to open it in a new window)

1 mark

2.1 To protect aeroplanes from being shot down, which of these should you do?

  • Reinforce the areas where the returning planes were hit the most.
  • Reinforce the areas where the returning planes were hit the least.
3 marks

2.2 160 people finished a marathon. 120 of them drank an energy drink along the way. The manufacturers of the drink claim that this shows that their drink had a positive effect. However, 40 runners didn’t finish the marathon. In fact, with the data of all the runners, it can be shown that you were just as likely to finish the race having drunk an energy drink, as you were to finish the race had you not drunk one. How many people who started the race drank an energy drink during the marathon?

Correct Solution: 150

Show Hint (–1 mark)
–1 mark
Finished the marathon Didn’t finish TOTAL
Drank an energy drink 120 x 120 + x
Didn’t drink one 40 40 - x 80 - x
TOTAL 160 40 200
Show Hint (–1 mark)
–1 mark

The probability that you finish, given that you have an energy drink is equal to the probability that you finish, given that you don’t have an energy drink, ie 120120+x=4080x.

Finished the marathon Didn’t finish TOTAL
Drank an energy drink 120 x 120 + x
Didn’t drink one 40 40 - x 80 - x
TOTAL 160 40 200

The probability that you finish, given that you have an energy drink is equal to the probability that you finish, given that you don’t have an energy drink, ie 120120+x=4080x.

Solving this gives x=30, which means that total number of people who drank an energy drink is 150.

3. Intermediate Maths Challenge Problem (UKMT)

5 marks

3.1 The product of Mary's age in years on her last birthday and her age now in complete months is 1,800. How old was Mary on her last birthday?

  • 9
  • 10
  • 12
  • 15
  • 18
Show Hint (–1 mark)
–1 mark

We need to express 1,800 as the product of two factors, one of which (her age in complete months) is between 12 and 23 times the other (her age in complete years).

Show Hint (–2 mark)
–2 mark

These are 150 and 12 respectively.

We need to express 1800 as the product of two factors, one of which (her age in months) is between 12 and 23 times the other (her age in complete years). These are 150 and 12 respectively. Mary is 150 months old, ie she was twelve on her last birthday and she is now 12 years and 6 months old.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

4.1 The populations of five cities A, B, C, D, E in 1988 and 1998 are shown on these scales:

Which of the five cities showed the largest percentage increase in population from 1988 to 1998?

  • A
  • B
  • C
  • D
  • E

The percentage increases are A: 25%; B: 40%; C: 42 67%; D: 30%; E: 33 13%.

5. A game you can always win

Here’s a nice party trick you can learn with some hidden mathematics. Watch this video from YouTube channel Vsauce2.

(If you have problems watching the video, right click to open it in a new window)

In the game, each player picks numbers from 1 to 10 to add to a shared total. The winner is the player who makes the total up to 100. You may want to practice this game with a friend to be sure you can always win.

2 marks

5.1 Let’s suppose you’re playing with a friend and it's your go. The total is currently 47. What number should you pick?

Correct Solution: 9

1 mark

5.2 What is the 3-letter name of the game from ancient times which these games are based on?

Correct Solution: Nim

I hope you enjoyed this Parallelogram. There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.