Parallelogram 48 • Level 3 • 31 Jul 2025Plane physics

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

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. Airplanes taking off

3 marks

1.1 When a plane is taking off, which of the following would the pilot prefer?

Wind blowing in the opposite direction to plane moving along the runway.
Wind blowing the same direction to the plane moving along the runway.
Wind blowing from the side.
Wind blowing from the other side.
No wind.
(Not answered)

The wings generate more lift if the air passing over them is moving faster. So the plane needs to reach a take-off speed relative to the air. If the wind is blowing in the opposite direction to the plane’s motion, then it is easier to reach the take-off speed.

2.

3 marks

2.1. Which of the following is the longest period of time?

3002 hours
125 days
1712 weeks
4 months
13 of a year
(Not answered)

One year is, at most, 366 days, so one-third of a year is less than 125 days.

No month is longer than 31 days, so 4 months is also less than 125 days, as is 17.5 weeks which equals 122.5 days.

However 3002 hours equals 125 days 2 hours, so this is the longest of the five periods of time.

3.

4 marks

3.1 Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth eats there every fifth day and the sixth eats there every sixth day.

They agree to have a party the next time they all eat together there. In how many days' time is the party?

30 days
60 days
90 days
120 days
360 days
(Not answered)

Show Hint (–1 mark)

1 mark

You need the lowest common multiple (LCM) of 1,2,3,4,5 and 6. Since every number is a multiple of 1, and every multiple of 4 is a multiple of 2, and every multiple of 6 is a multiple of 3, you really only need to consider the LCM of 4,5 and 6.

The lowest common multiple of 2, 3, 4, 5 and 6 is required. Of these numbers, 2, 3 and 5 are prime whilst 4 = 2² and 6 = 2 × 3.

So their lowest common multiple is 2² × 3 × 5, that is 60.

4.

5 marks

4.1 The diagram on the right shows a rhombus FGHI and an isosceles triangle FGJ in which GF=GJ. Angle FHI=111°.

What is the size of angle JFI?

27°
29°
31°
33°
34.5°
(Not answered)

Show Hint (–1 mark)

1 mark

Since ∠IJF and ∠FJG form a straight line, they sum to 180, so ∠FJG = 180 – 111 = 69. Since FGJ is an isosceles triangle, ∠GFJ also equals 69.

Show Hint (–1 mark)

1 mark

The remaining angle in triangle FGJ is 180 – 2 × 69 = 42. A rhombus is a quadrilateral with four equal sides, so when split along the diagonal makes two isosceles triangles.

Show Hint (–1 mark)

1 mark

In isosceles triangle FGI, ∠FGI = ∠GIF = 42. You now have two angles of the triangle FJI.

Adjacent angles on a straight line add up to 180°, so ∠GJF=180°−111°=69°.

In triangle FGJ, GJ=GF so ∠GFJ=∠GJF. Therefore ∠FGJ=180−2×69°=42°.

As the shape FGHI is a rhombus, FG=FI and therefore ∠GIF=∠FGJ=42°.

Finally, from triangle FJI, ∠JFI=180−111−42°=27°.

5. Very strange decimal

Here is an interesting mathematical oddity. Very often odd things are interesting. If you are a bit odd, then you might be a bit interesting.

3 marks

5.1 If you divide 1 by 998,001, the resulting decimal number will give you almost every three-digit number. For example, the decimal starts as follows: 0.000001002003004005006 ... and so on. However, one three-digit number gets skipped in this strange series. Which three-digit number is missing?

Clue – you can start by trying out the division on your calculator. Then, you will need to read this article to find the answer.

Correct Solution: 998

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...
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.
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.
This was our last Parallelogram of the year, but be sure to come back in September for more puzzles and problems.

Cheerio, Simon.