Parallelogram 47 Level 4 24 Jul 2025Inflating a balloon

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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.
  • Finish by midnight on Sunday if your whole class is doing parallelograms.
  • 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.

1.

2 marks

1.1 The diagram shows a regular pentagon inside a square.

What is the value of x?

  • 48
  • 51
  • 54
  • 60
  • 72
  • (Not answered)
Show Hint (–1 mark)
1 mark

The interior angles of a regular pentagon are each 108°.

Let the vertices of the square be labelled K, L, M, N and those of the regular pentagon P, Q, R, S, T, as shown in the diagram.

We use the fact that the interior angles of a regular pentagon are each 108°. It follows that RST=STP=108°. Because it is the angle of a square, SNT=90°. By the Exterior Angle Theorem, RST=SNT+NTS. Therefore, 108°=90°+NTS. It follows that NTS=108°90°=18°.

Finally, because the angles at T on the line KN have sum 180°,

NTS+STP+PTK=180°,

that is,

18°+108°+x°=180°.

We conclude that

x=18018108=54.

2.

3 marks

2.1. How many two-digit squares differ by 1 from a multiple of 10?

  • 1
  • 2
  • 3
  • 4
  • 5
  • (Not answered)

The two-digit squares are 16, 25, 36, 49, 64 and 81. Of these, just 49 = 50 − 1 and 81 = 80 + 1 differ by 1 from a multiple of 10. So there are 2 such two-digit squares.

3. An unexpected way to inflate a balloon

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

3 marks

3.1 The video shows a graph explaining how “spherical” the balloon is as time passes. What would be a good way to describe the line in the graph?

  • Spiral
  • Saw tooth
  • Flat
  • Rollercoaster
  • (Not answered)

4.

4 marks

4.1 The football shown is made by sewing together 12 black pentagonal panels and 20 white hexagonal panels.

There is a join wherever two panels meet along an edge.

How many joins are there?

  • 20
  • 32
  • 60
  • 90
  • 180
  • (Not answered)
Show Hint (–1 mark)
1 mark

Try harder.

Each of the 12 pentagons has 5 edges, and each of the 20 hexagons has 6 edges. So, altogether, the panels have 12×5+20×6=180 edges. Each join connects 2 edges, So there are 12180=90 joins.

5.

5 marks

5.1. The diagram shows a shaded region inside a large square.

The shaded region is divided into small squares.

What fraction of the area of the large square is shaded?

  • 310
  • 13
  • 38
  • 25
  • 37
  • (Not answered)
Show Hint (–1 mark)
1 mark

We form a complete grid inside the large square by extending the sides of the small squares, as shown in the figure. In this way the large square is divided up into 61 small squares, 20 triangles along the edges whose areas are equal to half that of the small squares, and 4 triangles in the corners whose areas are equal to one quarter that of the small squares.

We form a complete grid inside the large square by extending the sides of the small squares, as shown in the figure. In this way the large square is divided up into 61 small squares, 20 triangles along the edges whose areas are equal to half that of the small squares, and 4 triangles in the corners whose areas are equal to one quarter that of the small squares.

So the area of the large square is equal to that of

61×1+20×12+4×14=72

of the small squares. The shaded area is made up of 24 small squares. Therefore the fraction of the large square that is shaded is

2472=13.

[Note that we could save some work by taking advantage of the symmetry of the figure, and just work out which fraction of, say, the top left hand corner of the square is shaded.]

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.
  • The next Parallelogram is next week, at 3pm on Thursday.
  • 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.