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Parallelogram 44 • Level 4 • 3 Jul 2025Money catching

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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 What is the value of 1 − 0.2 + 0.03 − 0.004?

  • 0.826
  • 0.834
  • 0.926
  • 1.226
  • 1.234
  • (Not answered)

We have 1 − 0.2 = 0.8 and so 1 − 0.2 + 0.03 = 0.8 + 0.03 = 0.83, and therefore 1 − 0.2 + 0.03 − 0.004 = 0.83 − 0.004 = 0.826.

Note that in the context of the IMC where we just need to decide which of the options is correct, it isn’t really necessary to do this exact arithmetic. Looking at the terms making up the sum it is evident that the answer is just greater than 0.8, and the presence of the term −0.004 shows that the digit in the thousandths column is 6. So 0.826 is the correct option.

2.

3 marks

2.1. Which of the following numbers is not a square?

  • 16
  • 25
  • 34
  • 43
  • 52
  • (Not answered)

We have 25=32 which is not a square.

In the context of the IMC we can assume that only one option is correct and that it is sufficient just to find it. However, for a complete solution we need to check that none of the other options is a square. Note that we can show this without evaluating any of them, as follows. We have 16=132, 34=322, 43=223=26=232, and so these are all squares, and it is immediate that 52 is a square.

3. Money catching

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

2 marks

3.1 Tadashi explained that an object falls a distance equal to (12×g×t2), which is (12×10×t2), which is 5×t2.

How many meters will an object fall in 1 second?

Correct Solution: 5 metres

3 marks

3.2 How many seconds will it take for an object to fall 80m?

Correct Solution: 4 seconds

If distance = 5 × (time)2, then we know that 80 = 5 × (time)2.

That means that 16 = time2.

So, the time required is 4 seconds.

4.

4 marks

4.1. A flag is in the shape of a right-angled triangle, as shown, with the horizontal and vertical sides being of length 72 cm and 24 cm respectively.

The flag is divided into 6 vertical stripes of equal width.

What, in cm2, is the difference between the areas of any two adjacent stripes?

  • 96
  • 72
  • 48
  • 32
  • 24
  • (Not answered)
Show Hint (–1 mark)
–1 mark

I did not follow the approach of the solution you will see after you have completed the Parallelogram, but here is something to get you started. If the difference in area is the same for any two stripes, then just consider the two smallest stripes. They make up a sub-triangle that is only a third the size of the whole triangle. And the shaded mini-triangle is half the length, so a quarter of the area of the outlined sub-triangle.

The lines we have added in the figure on the right divide the flag into 15 congruent rectangles and 6 congruent triangles. Let the area of each of the 15 rectangles be x cm2. The area of each of the 6 congruent triangles is half that of the rectangles, that is, 12x cm2.

The difference between the area of the adjacent vertical stripes of the flag is the area of one of the rectangles.

From its dimensions, we see that the area of the flag is 1272×24 cm2 =36×24 cm2. It is made up of 15 rectangles, each with area x cm2, and 6 triangles, each with area 12x cm2.

Therefore

15x+612x=36×24,

that is,

18x=36×24,

and it follows that

x=36×2418=2×24=48.

Hence the difference between the area of adjacent stripes of the flag is 48 cm2.

5.

4 marks

5.1 A voucher code is made up of four characters. The first is a letter: V, X or P. The second and third are different digits. The fourth is the units digit of the sum of the second and third digits.

How many different voucher codes like this are there?

  • 180
  • 243
  • 270
  • 300
  • 2700
  • (Not answered)

There are 3 choices for the first character, namely V, X or P. There are 10 choices for the second character, namely one of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. Because the second and third characters are different digits, the third character can be any one of the 10 digits, other than the second character. Therefore there are 9 choices for the third character. There is just 1 choice for the fourth character as this is the units digit of the sum of the second and third digits.

It follows that the total number of voucher codes is 3×10×9×1=270.

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.