Parallelogram 35 Year 8 8 Jul 2021Summer Sums

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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 on top of that here is a longer than usual Parallelogram.

Most of the questions are from past Junior Maths Challenge papers that you have seen before, but I have also added a sprinkling of other things to make the Parallelogram (even) more interesting.

It will take you an hour or two to complete this Parallelogram, so maybe tackle it across three or four sessions... but don’t forget to complete it before term starts.

And... if you have missed any earlier Parallelograms then the summer is a great time to do some catching up.

1.

1 mark

1.1 Which of these calculations produces a multiple of 5?

  • 1×2+3+4
  • 1+2×3+4
  • 1×2+3×4
  • 1+2×3×4
  • 1×2×3×4
  • (Not answered)

The results of the five calculations are 9, 11, 14, 25, 24 respectively.

2.

1 mark

2.1 Which of these diagrams could be drawn without taking the pen off the page and without drawing along a line already drawn?

  • A
  • B
  • C
  • D
  • E
  • (Not answered)

For it to be possible to draw a figure without taking the pen off the paper and without drawing along an existing line, there must be at most two points in the figure at which an odd number of lines meet.

Only E satisfies this condition.

3.

1 mark

3.1 All of the Forty Thieves were light-fingered, but only two of them were caught red-handed.

What percentage is that?

  • 2
  • 5
  • 10
  • 20
  • 50
  • (Not answered)

240=120=5100=5%

4.

1 mark

4.1 In this diagram, what is the value of x?

  • 16
  • 36
  • 64
  • 100
  • 144
  • (Not answered)

The unmarked interior angle on the right of the triangle =360324°=36°. So, by the exterior angle theorem, x=10036=64.

5. Airplanes taking off

3 marks

5.1 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.

6.

2 marks

6.1. At Spuds-R-Us, a 2.5kg bag of potatoes costs £1.25. How much would one tonne of potatoes cost?

  • £5
  • £20
  • £50
  • £200
  • £500
  • (Not answered)

The cost of 1 kg of potatoes is £1.25 ÷ 2.5 = 50p. So the cost of 1 tonne, that is 1000 kg, is 1000 × 50p = £500.

7.

2 marks

7.1. The diagram shows a single floor tile in which the outer square has side 8cm and the inner square has side 6cm.

If Adam Ant walks once around the perimeter of the inner square and Annabel Ant walks once around the perimeter of the outer square, how much further does Annabel walk than Adam?

  • 2 cm
  • 4 cm
  • 6 cm
  • 8 cm
  • 16 cm
  • (Not answered)

Adam Ant walks 24 cm, while Annabel Ant walks 32 cm, so Annabel walks 8 cm further.

8.

2 marks

8.1. Mr Owens wants to keep the students quiet during a Mathematics lesson. He asks them to multiply all the numbers from 1 to 99 together and then tell him the last-but-one digit of the result.

What is the correct answer?

  • 0
  • 1
  • 2
  • 8
  • 9
  • (Not answered)

As 2, 5 and 10 are all factors of the correct product, this product is a multiple of 100. So the last digit and the last-but-one digit are both zero.

9.

2 marks

9.1. In a triangle which has the angles x°, y°, z° the mean of y and z is x.

What is the value of x?

  • 90
  • 80
  • 70
  • 60
  • 50
  • (Not answered)

If the mean of y and z is x, then y+z=2x. So the sum of the interior angles of the triangle is x+y+z°=3x°. So 3x=180, that is x=60.

10. Is it Better to Walk or Run in the Rain?

MinutePhysics is a great YoutTube channel, so dive in and watch loads of them. But, for now, take a look about this video about running in the rain.

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

2 marks

10.1 If you are outside in the rain and you want to minimise how wet you get, you should…

  • Stand still
  • Walk slowly
  • Run fast
  • Use an umbrella
  • (Not answered)
2 marks

10.2 If you are outside in the rain and don’t have an umbrella and you want to minimise how wet you get, you should…

  • Stand still
  • Walk slowly
  • Run fast
  • Borrow an umbrella
  • (Not answered)
2 marks

10.3 If you are outside in the rain and don’t have an umbrella and can’t borrow an umbrella and you want to minimise how wet you get, you should…

  • Stand still
  • Walk slowly
  • Run fast
  • (Not answered)

11.

3 marks

11.1. 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.

12.

3 marks

12.1. Sir Lance has a lot of tables and chairs in his house. Each rectangular table seats eight people and each round table seats five people.

What is the smallest number of tables he will need to use to seat 35 guests and himself, without any of the seating around these tables remaining unoccupied?

  • 4
  • 5
  • 6
  • 7
  • 8
  • (Not answered)

There are 36 people to be seated so at least five tables will be required. The number of circular tables must be even.

However, five rectangular tables will seat 40 people and three rectangular and two circular will seat 34. So at least six tables are needed.

Two rectangular and four circular tables do seat 36 people: so six is the minimum number of tables.

13.

3 marks

13.1 One of the examination papers for Amy’s Advanced Arithmetic Award was worth 18% of the final total. The maximum possible mark on this paper was 108 marks.

How many marks were available overall?

  • 420
  • 480
  • 540
  • 560
  • 600
  • (Not answered)

As 108 marks represented 18% of the final total, 6 marks represented 1% of the final total. So this total was 600.

14.

3 marks

14.1 The letters J, M, C represent three different non-zero digits, where JM + JM + CC = JMC.

What is the value of J+M+C?

  • 19
  • 18
  • 17
  • 16
  • 15
  • (Not answered)

The hundreds column shows us that J=1 or 2. [We can’t carry more than 2 from the units to the tens; and 2 plus the biggest feasible values 7, 8, 9 for the three letters is only 26.]

The units column shows that J+M is a multiple of 10 and it can’t be 0 (or else J+M=0); so J+M=10 and M=9 or 8 respectively.

Also, the sum of the units column is 10+C, so there is exactly 1 to carry to the tens column. The tens column now tells us that J+C+1=10J. So J=2 is not possible and therefore J=1, C=8 and M=9.

15. What if the Earth was made of Blueberries?

Liv Boeree is a physics graduate who has an insatiable curiosity, which includes asking questions about what would happen if the Earth was made of blueberries?

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

3 marks

15.1 After a while, what would the atmosphere consist of?

  • Oxygen
  • Nitrogen
  • Carbon dioxide
  • Helium
  • Steam
  • (Not answered)

16.

4 marks

16.1 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.

17.

4 marks

17.1 Sam's 101st birthday is tomorrow. So Sam's age in years changes from a square number (100) to a prime number (101).

How many times has this happened before in Sam's lifetime?

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

The other times that this has happened previously are when Sam's age in years went from 1 to 2; from 4 to 5; from 16 to 17 and from 36 to 37.

Note that since primes other than 2 are odd, the only squares which need to be checked, other than 1, are of even numbers.

18.

4 marks

18.1. In the diagram on the right, PT=QT=TS, QS=SR, PQT=20°.

What is the value of x?

  • 20
  • 25
  • 30
  • 35
  • 40
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Since PT=QT, PTQ is an isosceles triangle, so QPT=20°.

Show Hint (–1 mark)
–1 mark

Since QS=SR, QSR is an isosceles triangle as well, so QRS=x°. Now consider the angles which form triangle PQR.

As QS=SR. SRQ=SQR=x°.

So QST=2x° (exterior angle theorem). Also TQS=2x° since QT=TS.

As PT=QT, TPQ=TQP=20°.

Consider the interior angles of triangle PQR: 20+20+2x+x+x=180.

So 4x+40=180, hence x=35.

19.

4 marks

19.1 Pat needs to travel down every one of the roads shown at least once, starting and finishing at home.

What is the smallest number of the five villages that Pat will have to visit more than once?

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

Villages which have more than two roads leading to them (or from them) must all be visited more than once as a single visit will involve at most two roads. So Bentonville, Pencaster and Wytham must all be visited more than once.

The route Home, Bentonville, Greendale, Wytham, Bentonville, Pencaster, Home, Wytham, Horndale, Pencaster, Home starts and finishes at Home and visits both Greendale and Horndale exactly once so the minimum number of villages is three.

20. Can You Solve This Chess Conundrum?

This puzzle was featured on the BBC TV show “QI”. First, you need to know that the knight in chess moves in an L-shape, as shown in the image below. “L” means 2 squares left/right and 1 up/down, or 2 up/down and 1 left/right.

4 marks

20.1 What is the maximum number of knights that you can place on a chess board (8 x 8 squares), so that none of the knights can take any other knight?

Correct Solution: 32

Show Hint (–1 mark)
–1 mark

When a knight starts on a black square, what colour squares can it attack? Only black? Only white? Black and white?

This QI clip explains why the answer is 32 knights.

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

21.

5 marks

21.1 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=1802×69°=42°.

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

Finally, from triangle FJI, JFI=18011142°=27°.

22.

5 marks

22.1. In the diagram on the right, the number in each box is obtained by adding the numbers in the two boxes immediately underneath.

What is the value of x?

  • 300
  • 320
  • 340
  • 360
  • More information needed
  • (Not answered)
Show Hint (–2 mark)
–2 mark

You do not have any two entries given which lie in the same three-box stack, so instead of working out a particular value, label the box next to 12 as a. Now label the box next to 78 with an expression, using the fact that this box plus a=90.

Let the numbers in the boxes be as shown in the diagram.

Then b=90a; c=12+a; d=b+78=168a.

Also, e=90+c=102+a; f=90+d=258a.

So x=e+f=102+a+258a=360.

23.

5 marks

23.1 Nicky has to choose 7 different positive whole numbers whose mean is 7.

What is the largest possible such number she could choose?

  • 7
  • 28
  • 34
  • 43
  • 49
  • (Not answered)
Show Hint (–1 mark)
–1 mark

Since the mean is 7, the total must be 7×7=49. She can have one large number, if she makes the others as small as possible.

The seven numbers must total 49 if their mean is to be 7. The largest possible number will occur when the other six numbers are as small as possible, that is 1, 2, 3, 4, 5, 6.

So the required number is 49 - 21 = 28.

24.

5 marks

24.1 Kiran writes down six different prime numbers, p, q, r, s, t, u, all less than 20, such that p+q=r+s=t+u. What is the value of p+q?

  • 16
  • 18
  • 20
  • 22
  • 24
  • (Not answered)
Show Hint (–2 mark)
–2 mark

The primes less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. The number 2 is the only even prime, so you cannot use it, because otherwise one sum will be (even + odd) and the two others will be (odd + odd) (

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. It is not possible for 2 to be one of the six numbers Kiran wrote down, since that would give one of the pairs an odd sum, whereas both of the other pairs would add up to an even number.

The sum of the remaining 7 primes is 75 which is a multiple of 3. The sum of the six primes making up the three pairs must also be a multiple of 3 since each pair has the same total.

So the odd prime not used in the six pairs must be a multiple of 3 too. Therefore 3 is the odd prime not used. So each pair totals 72 ÷ 3, that is 24, and the pairs are 5 + 19 7 + 17 11 + 13.

25. 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

25.1 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 our brand new Year 9 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.