Parallelogram 33 Year 10 24 Jun 2021Juice and Water

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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.
  • 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. Juice and water

At break, every student in the class chooses to drink either water or juice.

Use this information to work out the values of A, B, C, and D in the table.

  • There are 20 students at break
  • There are 13 girls at break
  • 8 students are drinking Juice
  • 2 boys are drinking water
1 mark

1.1 A = Correct Solution: 3

1 mark

1.2 B = Correct Solution: 10

1 mark

1.3 C = Correct Solution: 5

1 mark

1.4 D = Correct Solution: 2

It probably helps to extend the table, so that we have space to talk about the total numbers of boys, girls, juices and waters.

And we know that the total number of drinks and the total number of students both add up to 20.

So, we know from the question that:

  • total girls = 13, so total boys = 7
  • total juice = 8, so total water = 12
  • 2 boys drink water = 2, so D = 2.

If total water is 12, and 2 boys drink water, then 10 girls drink water, so B = 10.

It is now obvious that A = 3 and C = 5.

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

2.1 A certain company offers "750 hours of free Internet use for new subscribers". On closer inspection it becomes clear that this time must be used during the new subscriber's first month of membership!

What is the maximum number of hours in any one month of the year?

  • 168
  • 692
  • 720
  • 744
  • 750

The maximum number of days in any month is 31, and 31 × 24 = 744.

3. Intermediate Maths Challenge Problem (UKMT)

4 marks

3.1 The diagram shows a semicircle containing a circle which touches the circumference of the semicircle and goes through the midpoint of its diameter.

What fraction of the semicircle is shaded?

  • 23
  • 12
  • 1π
  • 2π
  • 3π

Let the radius of the circle be r. This implies that the radius of the semicircle is 2r. The area of the semicircle is therefore 12×π×2r2=2πr2 which is twice the area of the circle.

4. Squares on a chess board

4 marks

4.1 Of course, we know there are 64 small squares on an 8x8 chess board, but how many squares are there in total if we count squares of all sizes?

For example, the board above shows a 1x1 square (top left), a 2x2 square (bottom left) and a 4x4 square, but how many squares are there in total?

Correct Solution: 204

Show Hint (–1 mark)
–1 mark

How many squares of size 1x1 can you find? How many squares of size 2x2 can you find? And so on. Is there a pattern?

Show Hint (–1 mark)
–1 mark

There are 64 squares of size 1x1. There are 49 squares of size 2x2. And so on.

You need to break down the problem by looking at different sizes of squares and counting the number of each size of square.

There are 64 squares of size 1x1. There are 49 squares of size 2x2. And so on.

So you need to add up all the square numbers from 1 to 64. And the answer is 204.

5. Intermediate Maths Challenge Problem (UKMT)

5 marks

5.1 In the star shown here the sum of the four numbers in any "line" is the same for each of the five "lines".

The five missing numbers are 9, 10, 11, 12, and 13.

Which number is represented by K?

  • 9
  • 10
  • 11
  • 12
  • 13
Show Hint (–1 mark)
–1 mark

The ten numbers in the star total 75. Each number appears in two "lines" and therefore the five "lines" total 150, which implies the sum of the numbers in each "line" is 30.

The ten numbers in the star total 75. Each number appears in two "lines" and therefore the five "lines" total 150, which implies the sum of the numbers in each "line" is 30.

This means that K + C = 24 and therefore K = 11 and C = 13, or vice versa.

If K = 11, then U = 12; I = 10 and M = 11, which is impossible since K = 11.

If K = 13, then U = 10; I = 12; M = 9 and C = 11, which is correct.

6. Siphons

Here is an interesting video about siphons. If you find siphons boring, then you will just have to suck it up.

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

And here is a video that tells you more about the Pythagoras cup.

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

2 marks

6.1 What liquid does the professor suggest that a greedy student should first put in the cup in order to then fill it with wine?

  • water
  • chilled wine
  • hot wine
  • mercury
  • helium

There will be more next week, 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.