Parallelogram 35 Year 10 9 Jul 2020Summer 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 Intermediate Maths Challenge papers, 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 the new term starts.

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

1. QI mathematics

Here is a neat video from the BBC show “QI”, which demonstrates a lovely maths trick.

(This trick was mentioned in a Year 9 Parallelogram, but the video clip is a lovely and funny demonstration of the trick, so it is worth watching and following the explanation again.)

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

To summarise...

  • write down a three-digit number whose digits are decreasing and different.
  • reverse the digits to create a new number.
  • subtract this new number from the original number to get a SUB-RESULT.
  • Add the SUB-RESULT to the reverse of itself.
  • The final RESULT will always be 1089.

I like how QI turned it into a magic trick, predicting the 10th word on page 89 of a book. You might like to try this trick.

The big mystery is this – why does it work?

If the person picks the number ABC, then the reverse would be CBA, and result of the subtraction would be (100A + 10B + C) - (100C + 10B + A) to get 99(A - C).

Because the digits were decreasing, (A - C) is between 2 and 9, so the result must be 198, 297, 396, 495, 594, 693, 792, or 891.

When you add any one of these numbers to the reverse of itself, the result is always 1089.

2.

3 marks

2.1 30 ÷ 0.2 equals

  • 1.5
  • 6
  • 15
  • 150
  • 600

30 ÷ 0.2 = 300 ÷ 2 = 150, or 30 ÷ 0.2 = 30 ÷ 1/5 = 30 × 5 = 150.

3.

3 marks

3.1 In Britain in 1996 we consumed on average 9.6kg of bananas per person per year (that is, around 60 bananas each). In some parts of Africa, the consumption of bananas is as high as 250kg of bananas per person per year. Roughly how many bananas is that?

  • 4 or 5 a day
  • 1 or 2 a day
  • 4 or 5 a week
  • 1 or 2 a week
  • 4 or 5 a month

250 kg is approximately 25 times 9.6 kg. The consumption in some parts of Africa is therefore approximately 25 × 60, ie 1500, per person per year. 4 bananas per day are equivalent to 4 × 365, ie 1460, per year and the best answer is therefore 4 or 5 a day.

4.

3 marks

4.1 Which is smallest?

  • 2+34+6
  • 2÷34÷6
  • 2346
  • 2346
  • 2×34×6

The values are respectively 12, 1, 12, 12, 14.

5.

4 marks

5.1 In the diagram PQ=PR=RS. What is the size of angle x?

  • 54°
  • 72°
  • 90°
  • 108°
  • 144°

Triangle PQR is isosceles and therefore PQR = PRQ = 72°.

Triangle PSR is also isosceles therefore RPS = RSP = 36° (using the external angle theorem or by calculating that PRS = 108°) and x = 180° - 36° - 36° = 108°.

6.

4 marks

6.1 It is evening, and Meg, who is 1m tall, casts a shadow of length 3m. If Meg stands on her brother's shoulders, which are 1.5m above the ground, how long a shadow will she and her brother cast?

  • 3m
  • 4.5m
  • 5.5m
  • 6.5m
  • 7.5m

The ratio of height to shadow length is 1:3. Their shadow will therefore be 3 × (1 + 1.5) m = 7.5m long.

7.

4 marks

7.1 In March 1998, a book called "The Shadow of the East" was returned to a library in Sussex. It had been borrowed on January 3rd 1924! The library charges a fine of 60p per week for overdue books. Approximately how big a fine should the person who returned the book have paid?

  • £45
  • £180
  • £230
  • £2200
  • £16000

The fine should be approximately 74 × 52 × 60p ≈ 75 × £30 = £2250. Although in reality, the person who returned the book disappeared without paying a fine at all!

8. Adjective order

This video by Tom Scott is nothing to do with maths, but it is interesting, so it’s worth two minutes of your time.

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

9.

4 marks

9.1 A pencil AB lying on a table is given a half-turn about the end B (so that A moves to A') and then a half-turn about A' (so that B moves to B').

The point C on the pencil is one third of the way from A to B.

What is the ratio of the total distances moved by A and by C?

  • 3:1
  • 3:2
  • 1:1
  • 2:3
  • 1:3

Let AB be of length 3r. The distanced moved by A is then the circumference of a semicircle of radius 3r, ie 3πr. C moves along the circumference of a semicircle of radius 2r, ie 2πr, followed by the circumference of a semicircle of radius r, ie πr. The total distance moved by C is therefore also 3πr.

10.

4 marks

10.1 Here are three statements:

(i) 310 is even
(ii) 310 is odd
(iii) 310 is square

Exactly which ones are true?

  • (i) only
  • (ii) only
  • (iii) only
  • (i) and (iii)
  • (ii) and (iii)

All powers of 3 are odd and 310=35×35 which means that 310 is also square.

11.

4 marks

11.1 The three circles in the diagram have the same centre and have radii 3cm, 4cm and 5cm.

What percentage of the area of the largest cirlce is shaded?

  • 20%
  • 25%
  • 28%
  • 30%
  • 3313%

The area of the largest circle = 25π cm2.

The shaded area = (16π - 9π) cm2 = 7π cm2.

The percentage which is shaded = 725 × 100% = 28%.

12.

4 marks

12.1 Seventy pupils (37 boys and 33 girls) are divided into two groups, with forty pupils in Group I and thirty pupils in Group II. How many more boys are there in Group I than there are girls in Group II?

  • 4
  • 7
  • 8
  • 9
  • more information needed
Show Hint (–1 mark)
–1 mark

Let the number of boys in Group I be x. The number of girls in Group I is therefore 40 - x.

Let the number of boys in Group I be x. The number of girls in Group I is therefore 40 - x and the number of girls in Group II is 33 - (40 - x) = x - 7.

Therefore there are 7 girls fewer in Group II than there are boys in Group I.

13.

5 marks

13.1 Four wiggles are the same as three woggles; two woggles are the same as five waggles, and six waggles are the same as one wuggle. Which is smallest?

  • 1 wuggle
  • 2 woggles
  • 3 waggles
  • 4 wiggles
  • two have the same value
Show Hint (–1 mark)
–1 mark

Try converting everything to waggles.

1 wuggle = 6 waggles;
2 woggles = 5 waggles;
4 wiggles = 3 woggles = 7.5 waggles.

All are therefore greater than 3 waggles.

14.

5 marks

14.1 Inspector Remorse estimates that he can solve the average murder in x hours, a bank robbery in half that time, and a car theft in one third of the time he takes to solve a bank robbery. How many hours would he expect to take in solving two murders, six car thefts and four bank robberies?

  • 3x
  • 5x
  • 6x
  • 7x
  • 12x

Two murders: 2x hours;
Six car thefts: 6x6 = x hours;
Four bank robberies: 4x2 = 2x hours.

Total = 2x+x+2x hours = 5x hours.

15. Ice ages

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

2 marks

15.1 What is the current tilt of the Earth?

  • 7.2°
  • 23.5°
  • 45°
  • 90°
2 marks

15.2 What helps to stabilise the tilt of the Earth?

  • The moon
  • The Earth’s spin
  • The Earth’s mass
  • The Earth’s flatness
  • The Earth’s ice

16.

5 marks

16.1 A rectangular sheet of paper with sides 1 and 2 has been folded once as shown, so that one corner just meets the opposite long edge.

What is the value of the length d?

  • 12
  • 21
  • 716
  • 32
  • 23
Show Hint (–2 mark)
–2 mark

The fold is made along BE. A folds onto A.

AB=AB=2, therefore AC=1 (by Pythagorus).

The fold is made along BE. A folds onto A.

AB=AB=2, therefore AC=1 (by Pythagorus).

Triangle ABC is therefore a right-angled isosceles triangle, therefore BAC=45°, therefore EAD=45°.

Thus ED=AD=21.

17. Bluetooth

Here is Bluetooth’s origin story, courtesy of Tom Scott.

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

2 marks

17.1 Who did Harald Bluetooth unite?

  • Denmark and Germany
  • Norway and Sweden
  • Denmark and Norway
  • Denmark and Sweden
  • Norway and Germany

18.

3 marks

18. 567 is multiplied by 3489. What is the units digit of the answer?

  • 1
  • 3
  • 5
  • 7
  • 9

Take the units digits of any two numbers and multiply them together. The units digit of the answer is also the units digit of the product of the original numbers. As 7 × 9 = 63, the units digit of 567 × 3489 must also be 3.

19.

3 marks

19. An ice cream stall sells vanilla, strawberry and chocolate ice creams.

The pie chart illustrates the sales of ice cream for last Saturday.

The number of vanilla and the number of chocolate ice creams sold were the same.

The stall sold 60 strawberry ice creams.

How many chocolate ice creams were sold?

  • 90
  • 99
  • 100
  • 120
  • 135

The angle occupied by the 'chocolate' sector is 12360°90°=135°. This is 112 times bigger than the 'strawberry' sector and hence the number of chocolate ice creams sold is 112×60=90.

20.

3 marks

20.1 Which is the largest of these fractions?

  • 715
  • 37
  • 1123
  • 49
  • 611

611 is the only one of these fractions which is greater than 12.

21.

3 marks

21.1 In Worcestershire, Wyre Piddle is 12km south of the village of North Piddle, and Abbotts Morton is 12km east of North Piddle. What is the direction of Abbotts Morton from Wyre Piddle?

  • South East
  • South West
  • North East
  • North West
  • West

22.

3 marks

22.1 In a magic square, each row, each column and both main diagonals have the same total.

In the partially completed magic square shown, what number should replace x?

  • 15
  • 18
  • 21
  • 24
  • 27
Show Hint (–1 mark)
–1 mark

The leading diagonal sums to 58. So every row and column should add to 58.

The numbers along the leading diagonal total 58 and this is therefore the sum of each row and column. We can now calculate that the number to the left of the 10 must be 20, and the number below that is 7. Hence x=5816+14+7=21.

23.

3 marks

23.1 Granny has been having a smashing time. Yesterday she had 12 cups and 10 matching saucers, but this morning she dropped a tray holding one third of the cups and half the saucers, breaking all of those on the tray. How many of her cups are now without saucers?

  • 1
  • 3
  • 4
  • 5
  • 6

Granny dropped 4 cups and 5 saucers, leaving her with 8 cups and 5 saucers. Therefore 3 cups did not have matching saucers.

24. Telescopes

Space scientist Maggie Aderin-Pocock talks about telescopes.

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

2 marks

24.1 Which type of telescope is less reliant on the quality of the glass?

  • Refractive
  • Reflective

25.

4 marks

25.1 My bargain settee cost me £240 in a sale offering 25% reductions on all items. How much did I save?

  • £25
  • £40
  • £60
  • £80
  • £100

The sale price is 75% of the original price. Therefore the amount I saved, 25% of the original price, is one third of £240, ie £80.

26.

4 marks

26.1 Timmy, Tammy and Tommy all have tummy ache! They all set off separately to visit their doctor, leaving their homes at exactly the same time. Timmy cycles the 8km to the surgery at an average speed of 20 km/h; Tammy walks the 1.2km to the surgery at an average speed of 4 km/h and Tommy drives the 16.5km to the surgery at an average speed of 45 km/h. In what order do they arrive at the surgery?

  • Tommy, Timmy, Tammy
  • Timmy, Tommy, Tammy
  • Tommy, Tammy, Timmy
  • Tammy, Timmy, Tommy
  • Tammy, Tommy, Timmy

Timmy takes 24 minutes (8/20 of 1 hour) to reach the surgery; Tammy takes 18 minutes (12/40 of 1 hour) and Tommy takes 22 minutes (33/90 of 1 hour). The order, therefore, is Tammy, Tommy, Timmy.

27.

4 marks

13.1 The diagram shows two rectangles which enclose five regions. What is the largest number of regions which can be enclosed by any two rectangles drawn on a sheet of paper?

  • 10
  • 9
  • 8
  • 7
  • 6
Show Hint (–1 mark)
–1 mark

A rectangle could be a square.

Show Hint (–1 mark)
–1 mark

Imagine 2 squares on top each other and rotate one of them slightly.

28. The Battery That's Lasted 176 Years

Sally Le Page tells the story behind the world’s longest running battery!

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

2 marks

28.1 Sally explains that the bell requires 1 nanoamp (nA) of current. How much is a nanoamp?

  • A trillionth of an amp.
  • A billionth of an amp.
  • A millionth of an amp.
  • A thousandth of an amp.
  • A hundredth of an amp.

29.

4 marks

29.1 The ratio a:b=2:3 and the ratio a:c=3:4. What is the ratio b:c?

  • 1 : 8
  • 1 : 2
  • 8 : 9
  • 9 : 8
  • 2 : 1

bc=ac×ba=34×32=98.

30.

4 marks

30.1 In how many whole numbers between 100 and 999 is the middle digit equal to the sum of the other two digits?

  • 28
  • 36
  • 45
  • 50
  • 55

There are 9 such numbers whose first digit is 1 : 110, 121, 132, ..., 187, 198. Similarly there are 8 such numbers, beginning with 220 and ending with 297, whose first digit is 2; 7 such numbers, beginning with 330 and ending with 396, whose first digit is 3 and so on. Lastly there is only 1 such number whose first digit is 9: 990.

The answer, therefore, is 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45.

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 11 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... 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.
  • 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.