Parallelogram 35 Year 7 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. What is the value of 0.1+0.2+0.3×0.4?

  • 0.24
  • 0.312
  • 0.42
  • 1.0
  • 1.5
  • (Not answered)

0.1+0.2+0.3×0.4=0.3+0.12=0.42.

2.

1 mark

2.1. My train was scheduled to leave at 17:40 and to arrive at 18:20. However, it started five minutes late and the journey then took 42 minutes. At what time did I arrive?

  • 18:21
  • 18:23
  • 18:25
  • 18:27
  • 18:29
  • (Not answered)

The train arrived 5 + 42 = 47 minutes after 17:40, that is at 18:27.

3.

1 mark

3.1 What is the smallest four-digit positive integer which has four different digits?

  • 1032
  • 2012
  • 1021
  • 1234
  • 1023
  • (Not answered)

Here it is easy just to check the options that are given. A, D and E are the only options in which all four digits are different. Of these, clearly, E is the smallest.

For a complete solution we need to give an argument to show that 1023 really is the smallest four digit positive integer with four different digits. It is easy to do this.

To get the smallest possible number we must use the four smallest digits, 0, 1, 2 and 3. A four digit number cannot begin with a 0. So we must put the next smallest digit, 1, in the thousands place, as a four-digit number beginning with 2 or 3 is larger than one beginning with a 1. For similar reasons the hundreds digit must be the smallest remaining digit, 0. Similarly the tens digit must be 2 and the units digit must be 3.

So the required number is 1023.

4.

1 mark

4.1 Each letter in the abbreviation UKMT is rotated through 90° clockwise.

Which of the following could be the result?

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

In A, the letter T is incorrect; in B it is U which is incorrect; in C and D the incorrect letters are M and K respectively.

5. QI : Curious Sporting Loopholes

Mathematics is about rules (e.g., odd plus odd equals even), but it is also about understanding the rules in such detail that you know how to gain an advantage in solving problems. In that spirit, here is a clip from the BBC show “QI” about taking advantage of loopholes in sporting rules.

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

3 marks

5.1 Because of cricketer Thomas White, cricket bats can no longer be infinitely wide. Today, what is the maximum permitted width of a cricket bat in first class cricket? (You might have to googly this.)

  • 4.25 mm
  • 38 mm
  • 108 mm
  • 965 mm
  • (Not answered)
2 marks

5.2 In the previous question, why did I write that you might have to “googly” the answer, rather than writing “google” the answer?

  • Googly was a typo
  • Googly is a way of bowling a cricket ball
  • Thomas Googly invented cricket
  • Googly is the Indian word for Google
  • (Not answered)

6.

2 marks

6.1 What is the remainder when 354972 is divided by 7?

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

Note that 7 divides 35, 49 and 7, so it divides 354970. So the remainder is 2.

7.

2 marks

7.1. Which of the following numbers is three less than a multiple of 5 and three more than a multiple of 6?

  • 12
  • 17
  • 21
  • 22
  • 27
  • (Not answered)

Of the options given, only 27, which is three less than a multiple of 5, namely 30, and three more than a multiple of 6, namely 24, has both of the properties in the question.

8.

2 marks

8.1 Which of the following has exactly one factor other than 1 and itself?

  • 6
  • 8
  • 13
  • 19
  • 25
  • (Not answered)

The factors of 6 are 1, 2, 3 and 6; the factors of 8 are 1, 2, 4 and 8; the factors of 13 are 1 and 13; the factors of 19 are 1 and 19; and the factors of 25 are 1, 5 and 25. We see from this that, of the numbers we are given as options, only 25 has exactly one factor other than 1 and itself. It is worth noticing that 25 is the only square number.

9.

2 marks

9.1 If the net shown is folded to make a cube, which letter is opposite X?

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

Let X be on the top face of the cube. If the base is placed on a horizontal surface, then A, B, C, E will all be on vertical faces of the cube and D will be on the base, opposite X.

10. Bananas

Here is a tweet that caught my eye recently:

It turns out a major new study recently found that humans eat more bananas than monkeys.

I can't remember the last time I ate a monkey.

(from Charina91)

3 marks

10.1 How many bananas do humans eat each year?

  • 100 thousand
  • 100 million
  • 100 billion
  • 100 trillion
  • (Not answered)

11.

3 marks

11.1 Which of the following has the largest value?

  • 6÷12
  • 5÷13
  • 4÷14
  • 3÷15
  • 2÷16
  • (Not answered)

The values of the expressions are as follows:

  • 6÷12=12
  • 5÷13=15
  • 4÷14=16
  • 3÷15=15
  • 2÷16=12

12.

3 marks

12.1. The equilateral triangle XYZ is fixed in position. Two of the four small triangles are to be painted black and the other two are to be painted white. In how many different ways can this be done?

  • 3
  • 4
  • 5
  • 6
  • More than 6
  • (Not answered)

If the top triangle is painted black, then any one of the three remaining triangles may also be painted black. Similarly, if the top triangle is painted white, then any one of the three remaining triangles may also be painted white. So there are six different ways.

13.

3 marks

13.1. Amy, Ben and Chris are standing in a row. If Amy is to the left of Ben and Chris is to the right of Amy, which of these statements must be true?

  • Ben is furthest to the left
  • Chris is furthest to the right
  • Amy is in the middle
  • Amy is furthest to the left
  • None of statements A, B, C, D is true
  • (Not answered)

From the information, we see that Amy is to the left of both Ben and Chris. So the three are in the order Amy, Ben, Chris or the order Amy, Chris, Ben.

So Amy is certainly furthest to the left, and the others are all false either in one case or in both.

14.

3 marks

14.1. In the diagram on the right, ST is parallel to UV.

What is the value of x?

  • 46
  • 48
  • 86
  • 92
  • 94
  • (Not answered)

As ST is parallel to UV, PRT=132° (corresponding angles).

So PRQ=48° (angles on a straight line).

From the exterior angle of a triangle theorem, SQP=QPR+PRQ, so x=13448=86°.

15. Can you light a match with water?

Watch the video below - there will be a question to follow it:

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

3 marks

15.1 The Statue of Liberty is coated in...

  • copper oxide
  • chlorophyll
  • malachite
  • shredded Shrek
  • jade
  • (Not answered)

16.

4 marks

16.1. Which of the following has the largest value?

  • 12 + 14
  • 12 - 14
  • 12 × 14
  • 12 ÷ 14
  • 14 ÷ 12
  • (Not answered)

The values of the five expressions are:

  • 12 + 14 = 34
  • 12 - 14 = 14
  • 12 × 14 = 18
  • 12 ÷ 14 = 2
  • 14 ÷ 12 = 12

So 12 ÷ 14 has the largest value.

17.

4 marks

17.1. A station clock shows each digit by illuminating up to seven bars in a display. For example, the displays for 1, 6, 4 and 9 are shown.

When all the digits from 0 to 9 are shown in turn, which bar is used least?

  • (Not answered)

The number of times each bar is used is: A 4; B 6; C 8; D 7; E 7.

  • = 4
  • = 6
  • = 8
  • = 7
  • = 7

18.

4 marks

18.1 At the Marldon Apple-Pie-Fayre bake-off, prize money is awarded for 1st, 2nd and 3rd places in the ratio 3 : 2 : 1.

Last year Mrs Keat and Mr Jewell shared third prize equally.

What fraction of the total prize money did Mrs Keat receive?

  • 14
  • 15
  • 16
  • 110
  • 112
  • (Not answered)

Third prize is worth one-sixth of the total prize money, so Mrs Keat received half of that amount, that is one-twelfth of the total.

19.

4 marks

19.1 In the following expression each ▲ is to be replaced with either + or - in such a way that the result of the calculation is 100.

123 ▲ 45 ▲ 67 ▲ 89

The number of + signs used is p and the number of - signs used is m. What is the value of p - m ?

  • -3
  • -1
  • 0
  • 1
  • 3
  • (Not answered)

One approach is trial and error, as there are only a few combinations. For example, if it is (+ 45) then it is hard to see how we obtain a result of 100. So, we can then explore (- 45). Following this approach quickly leads you to the correct answer, but there is also a more rigorous and mathematical approach.

The sum is made up of 123 and ± 45, ± 67and ± 89. Suppose that the total of the positive terms in the calculation is x and the total of the negative terms is y. So x>0 and y<0. We need to have that:

x+y=100.

We also have that:

xy=123+45+67+89=324.

Adding these equations, we obtain 2x=424. So x=212 and hence y=112. It is readily seen that 45+67=112 and that no other combination of 45, 67 and 89 gives a total of 112. So the correct calculation must be 1234567+89=100 with 1 plus sign and 2 minus signs. So p=1 and m=2, giving pm=1.

20. Remainder riddle

3 marks

20.1 What is the remainder when you divide 2100 by 10?

Correct Solution: 6

21=2, therefore the last digit = 2
22=4, therefore the last digit = 4
23=8, therefore the last digit = 8
24=16, therefore the last digit = 6
25=32, therefore the last digit = 2
26=64, therefore the last digit = 4

The pattern for the last digit (which is also the remainder when dividing by 10) is 2, 4, 8, 6, ... repeated.

Every 4th remainder is 6.

As 100 is a multiple of 4, then the remainder for 2100 will be 6.

21.

5 marks

21.1. The sculpture ‘Cubo Vazado’ [Emptied Cube] by the Brazilian artist Franz Weissmann is formed by removing cubical blocks from a solid cube to leave the symmetrical shape shown.

If all the edges have length 1, 2 or 3, what is the volume of the sculpture?

  • 9
  • 11
  • 12
  • 14
  • 18
  • (Not answered)

Consider the sculpture to consist of three layers, each of height 1. Then the volumes of the bottom, middle and top layers are 5, 2, 5 respectively. So the volume of the sculpture is 12.

Alternatively: the sculpture consists of a 3×3×3 cube from which two 2×2×2 cubes have been removed. The 2×2×2 cubes have exactly one 1×1×1 cube (the cube at the centre of the 3×3×3 cube) in common. So the volume of the sculpture =272×81=12.

22.

5 marks

22.1 Points P and Q have coordinates (1, 4) and (1, −2) respectively. For which of the following possible coordinates of point R would triangle PQR not be isosceles?

  • (−5, 4)
  • (7, 1)
  • (−6, 1)
  • (−6, −2)
  • (7, −2)
  • (Not answered)

If R is (−5, 4) then PQ=PR=6.

If R is (7, 1) or if R is (−6, 1) then R lies on the perpendicular bisector of PQ (the line Y=1), so in both cases PR=QR.

If R is (7, −2), then QP=QR=6.

However if R is (−6, −2), then PQ=6, QR=7 and PR>7, so triangle PQR is scalene.

23.

5 marks

23.1 Each side of an isosceles triangle is a whole number of centimetres. Its perimeter has length 20 cm. How many possibilities are there for the lengths of its sides?

  • 3
  • 4
  • 5
  • 6
  • 7
  • (Not answered)

Let the length of the two equal sides of the isosceles triangle be a cm. Since the triangle has perimeter 20cm, the third side will have length 202a cm. Since this must be a positive length, 202a>0 and so a<10.

In a triangle, the length of one side must be less than the sum of the lengths of the other two sides. So 202a<2a.

This gives 20<4a and hence 5<a. So, we have 5<a<10.

Therefore, as a is a whole number, there are just four possible values for a, namely 6, 7, 8 and 9.

So there are four possibilities for the side lengths of the triangle:

  • 6, 6, 8;
  • 7, 7, 6;
  • 8, 8, 4; and
  • 9, 9, 2.

24.

5 marks

24.1 A 6 by 8 and a 7 by 9 rectangle overlap with one corner coinciding as shown.

What is the area (in square units) of the region outside the overlap?

  • 6
  • 21
  • 27
  • 42
  • 69
  • (Not answered)

The two shaded regions measure 3 by 7 and 1 by 6, so the total area outside the overlap is 27 units2.

25. How to See Without Glasses

The video channel “minutephysics” has some great short explanations of scientific mysteries. Take a look at this one about a simple way to fix your eyesight without glasses.

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

3 marks

25.1 What is the disadvantage of a pinhole when trying to see clearly?

  • You need to have a pin
  • You need to have a hole
  • The image is relatively dark
  • The image is stripy
  • The image seems further away
  • (Not answered)
3 marks

25.2 What is another word for a hole?

  • Aperture
  • Apparition
  • Apartment
  • Aperitif
  • Appendicitis
  • (Not answered)

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.