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Parallelogram 48 • Level 5 • 31 Jul 2025Harald Bluetooth

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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.
  • Tackle each Parallelogram in one go. Don’t get distracted.
  • Your score & answer sheet will appear immediately after you hit SUBMIT.
  • Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

It’s summer and the school year is over (or maybe it’s nearly over), but that doesn’t mean that it’s the end of mathematics. If you are taking mathematics seriously and if you are having fun with it (being serious and having fun can go together), then it is important that you keep the numbers and geometry parts of your brain at least a little bit busy over the summer. I am hoping that your teacher will have set you some holiday work, but if you have missed any earlier Parallelograms then the summer is a great time to do some catching up.

1.

4 marks

1.1 In the diagram PQ=PR=RS. What is the size of angle x?

  • 54°
  • 72°
  • 90°
  • 108°
  • 144°
  • (Not answered)

Triangle PQR is isosceles and therefore PQR = PRQ = 72°.

Triangle PSR is also isosceles therefore RPS = RSP = 36° (using the external angle theorem or by calculating that PRS = 108°) and x = 180° - 36° - 36° = 108°.

2.

4 marks

2.1 The three circles in the diagram have the same centre and have radii 3cm, 4cm and 5cm.

What percentage of the area of the largest circle is shaded?

  • 20%
  • 25%
  • 28%
  • 30%
  • 3313%
  • (Not answered)

The area of the largest circle = 25π cm2.

The shaded area = (16π - 9π) cm2 = 7π cm2.

The percentage which is shaded = 725 × 100% = 28%.

3. Bluetooth

Here is Bluetooth’s origin story, courtesy of Tom Scott.

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

2 marks

3.1 Who did Harald Bluetooth unite?

  • Denmark and Germany
  • Norway and Sweden
  • Denmark and Norway
  • Denmark and Sweden
  • Norway and Germany
  • (Not answered)

4.

3 marks

4.1 Granny has been having a smashing time. Yesterday she had 12 cups and 10 matching saucers, but this morning she dropped a tray holding one third of the cups and half the saucers, breaking all of those on the tray. How many of her cups are now without saucers?

  • 1
  • 3
  • 4
  • 5
  • 6
  • (Not answered)

Granny dropped 4 cups and 5 saucers, leaving her with 8 cups and 5 saucers. Therefore 3 cups did not have matching saucers.

5.

4 marks

5.1 The ratio a:b=2:3 and the ratio a:c=3:4. What is the ratio b:c?

  • 1 : 8
  • 1 : 2
  • 8 : 9
  • 9 : 8
  • 2 : 1
  • (Not answered)

bc=ac×ba=34×32=98.

And that’s it for this academic year. I hope you’ve enjoyed doing Parallel this year and make sure you come back in September, when we will have another year of Parallelograms.

But, before that, a few important points.

Before you hit the SUBMIT button, here are some quick reminders:

  • You will receive your score immediately, and collect your reward points.
  • You might earn a new badge... if not, then maybe next week.
  • Make sure you go through the solution sheet – it is massively important.
  • A score of less than 50% is ok – it means you can learn lots from your mistakes.
  • Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon.