PG 35 4 Jul 2019Summer Sums

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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 the following has the largest value?

  • 12 + 14
  • 12 - 14
  • 12 × 14
  • 12 ÷ 14
  • 14 ÷ 12

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.

2.

1 mark

2.1 Heidi is 2.1 m tall, while Lola is only 1.4 m tall. What is their average height?

  • 1.525 m
  • 1.6 m
  • 1.7 m
  • 1.725 m
  • 1.75 m

The average height, in metres, is 2.1+1.42=3.52=1.75 m.

3.

1 mark

3.1 What is the value of x?

  • 25
  • 35
  • 40
  • 65
  • 155

Let the other angles be as marked in the diagram. Then y=65 as the angles marked 65° and y° are the base angles of an isosceles triangle.

Because the angles in a triangle sum to 180°,z=1806565=50.

Now, w=z, as the angles marked w° and z° are vertically opposite. Therefore w=50. So, the angle marked x° is the third angle of a triangle in which the other two angles are 50° and 90°.

Therefore x=1805090=40.

4.

1 mark

4.1 Gill went for a five-hour walk. Her average speed was between 3 km/h and 4 km/h.

Which of the following could be the distance she walked?

  • 12 km
  • 14 km
  • 19 km
  • 24 km
  • 35 km

Walking at 3km/h for five hours, Gill would walk 5 × 3 km = 15 km, and at 4km/h she would walk 5× 4 km = 20 km.

As her average speed is between 3 km/h and 4 km/h, she walks between 15 km and 20km. Of the options given, only 19 km is in this range.

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.

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. All old Mother Hubbard had in her cupboard was a Giant Bear chocolate bar. She gave each of her children one-twelfth of the chocolate bar. One third of the bar was left. How many children did she have?

  • 6
  • 8
  • 12
  • 15
  • 18

One third of the bar was left, so Mother Hubbard’s children ate two-thirds of the bar. Since they ate one-twelfth of the bar each, Mother Hubbard had

7.

2 marks

7.1. What is the sum of the marked angles in the diagram?

  • 90°
  • 180°
  • 240°
  • 300°
  • 360°

The six marked angles are the interior angles of 2 triangles. The interior angles of 1 triangle add up to 180°. So the marked angles add up to 2x180° = 360°.

8.

2 marks

8.1. Peter Piper picked a peck of pickled peppers. 1 peck = 1/4 bushel and 1 bushel = 1/9 barrel. How many more pecks must Peter Piper pick to fill a barrel?

  • 12
  • 13
  • 34
  • 35
  • 36

We have 1 barrel = 9 bushels and 1 bushel = 4 pecks, and so 1 barrel = 9 × 4 = 36 pecks. Peter Piper has already picked 1 peck, so he needs to pick 35 more pecks to fill a barrel.

9.

2 marks

9.1. A square is divided into three congruent rectangles. The middle rectangle is removed and replaced on the side of the original square to form an octagon as shown.

What is the ratio of the length of the perimeter of the square to the length of the perimeter of the octagon?

  • 3:5
  • 2:3
  • 5:8
  • 1:2
  • 1:1

Suppose that the square has side length 3d. Then the square has perimeter of length 4 × 3d = 12d.

Each rectangle has size 3d × d. So we see that the octagon has a perimeter length:

4d + 3d + 4d + d + 3d + d + 3d + d = 20d

Hence the ratio of the lengths is 12d:20d = 3:5

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.

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
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
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

11.

3 marks

11.1. On standard dice the total number of pips on each pair of opposite faces is 7. Two standard dice are placed in a stack, as shown, so that the total number of pips on the two touching faces is 5.

What is the total number of pips on the top and bottom faces of the stack?

  • 5
  • 6
  • 7
  • 8
  • 9

The total number of pips on the top and bottom faces of the two dice is 7 + 7 = 14. As there is a total number of 5 pips on the touching faces, there are 14 - 5 = 9 pips altogether on the top and bottom faces of the stack.

12.

3 marks

12.1. Usain runs twice as fast as his mum. His mum runs five times as fast as his pet tortoise, Turbo. They all set off together for a run down the same straight path. When Usain has run 100 m, how far apart are his mum and Turbo the tortoise?

  • 5 m
  • 10 m
  • 40 m
  • 50 m
  • 55 m

When Usain has run 100 m his mum has run half this distance, that is, 50 m and Turbo has run one-fifth of his mum’s distance, that is, 10 m. So the distance between his mum and Turbo is (50 − 10) m = 40 m.

13.

3 marks

13.1 Beth, Carolyn and George love reading their favourite bedtime stories together. They take it in turns to read a page, always in the order Beth, then Carolyn, then George. All twenty pages of the story are read on each occasion. One evening, Beth is staying at Grandma's house but Carolyn and George still read the same bedtime story and take it in turns to read a page with Carolyn reading the first page.

In total, how many pages that Carolyn and George read are the same as the pages that they would read if Beth was there?

  • 1
  • 2
  • 4
  • 6
  • 7

One way to do this is to write down a list of 20 pages and then label how they would be read usually and then on the night that Beth is away.

Usual B C G B C G B C G B
Page 1 2 3 4 5 6 7 8 9 10
Beth C G C G C G C G C G
Usual C G B C G B C G B C
Page 11 12 13 14 15 16 17 18 19 20
Beth C G C G C G C G C G

By comparison, you can see that pages 5, 6, 11, 12, 17 & 18 are read by the same person, regardless of Beth’s absence. Next is a more mathematical approach to the problem.

When all three children are present, Carolyn reads pages 2, 5, 8, ... , that is those pages whose number leaves remainder 2 when divided by 3. George reads pages 3, 6, 9, ... that is, those whose number is a multiple of 3. When Beth stays at Grandma's, Carolyn reads pages 1, 3, 5, ... that is the odd numbered pages, and George reads the even numbered pages. So the pages that Carolyn reads both normally and also when Beth is away are those with numbers which have remainder 2 when divided by 3, and are odd. That is, the three pages 5, 11 and 17.

The pages that George reads both normally and also when Beth is away are those with numbers that are multiples of 3 and are even, that is, the three pages 6, 12 and 18. So there are altogether 6 pages which are read by the same person normally and when Beth is away, namely pages 5, 6, 11, 12, 17 and 18.

14.

3 marks

14.1 Peri the winkle leaves on Monday to go and visit Granny, 90m away. Except for rest days, Peri travels 1m each day (24-hour period) at a constant rate and without pause.

However, Peri stops for a 24-hour rest every tenth day, that is, after nine days travelling. On which day does Peri arrive at Granny’s?

  • Sunday
  • Monday
  • Tuesday
  • Wednesday
  • Thursday

Until Peri reaches Granny’s, he travels 9m in 9 days and then rests for a day. So he travels 9m in 10 days. So it takes him 90 days to travel the first 9m x 9 = 81m.

Then after a further 9 days he has travelled the final 9m and has reached Granny’s. So the journey takes him 99 days. Now 99 = 14 x 7 + 1, so the journey takes him 14 weeks and 1 day. Therefore he arrives on a Tuesday.

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?

3 marks

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

  • Oxygen
  • Nitrogen
  • Carbon dioxide
  • Helium
  • Steam

16.

4 marks

16.1 The numbers 2, 3, 4, 5, 6, 7, 8 are to be placed, one per square, in the diagram shown such that the four numbers in the horizontal row add up to 21 and the four numbers in the vertical column add up to 21.

Which number should replace x?

  • 2
  • 3
  • 5
  • 7
  • 8

If we add all the numbers in the horizontal column and all the numbers in the vertical row, we get a total of 21 + 21 = 42 . In doing this sum we add in all the numbers 2, 3, 4, 5, 6, 7, 8 once except for x which is added in twice. So the total we get is 2 + 3 + 4 + 5 + 6 + 7 + 8 + x = 35 + x. Since this equals 42, we must have x = 7.

To complete the solution we should check that with x = 7, it is possible to place the remaining numbers in the other squares so that the four numbers in the horizontal row add up to 21, and so also do the four numbers in the vertical column.

17.

4 marks

17.1 Weighing the baby at the clinic was a problem. The baby would not keep still and caused the scales to wobble. So I held the baby and stood on the scales while the nurse read off 78 kg. Then the nurse held the baby while I read off 69 kg. Finally I held the nurse while the baby read off 137 kg. What was the combined weight of all three?

  • 142 kg
  • 147 kg
  • 206 kg
  • 215 kg
  • 284 kg

We let the weights of the baby, the nurse and myself be x kg, y kg and z kg, respectively. The information we are given implies that x+z=78, x+y=69 and y+z=137.

Adding these three equations gives x+z+x+y+y+z=78+69+137, that is, 2x+2y+2z=284.

It follows x+y+z=142. So the combined weight of all three was 142 kg.

18.

4 marks

18.1. A list is made of every digit that is the units digit of at least one prime number. How many of the following numbers appear in the list?

  • 1
  • 2
  • 3
  • 4
  • 5

The numbers 2, 3 and 5 are themselves prime numbers so they occur in this list. Also 1 is the units digit of the prime number 11 (and also 31, 41 and many more). However, a number with units digit 4 is an even number greater than 2 and so is not prime. Thus, there are 4 numbers in the list, namely 1, 2, 3 and 5 which are the units digits of at least one prime number. Therefore the solution is D.

19.

4 marks

19.1 In rectangle PQRS, the ratio of PSQ to PQS is 1:5.

What is the size of QSR?

  • 15°
  • 18°
  • 45°
  • 72°
  • 75°

Let QSR =x°. Since PS is parallel to QR, the alternate angles PQS and QSR are equal. So PQS =x°. Therefore, as PSQ:PQS =1:5, PSQ =15x°. Therefore, from the right angled triangle PQS we deduce that x+15x+90=180 and so 65x=90. Therefore x=56×90=75.

So QSR is 75°.

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 night?

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.

21.

5 marks

21.1 A swimming club has three categories of members: junior, senior, veteran. The ratio of junior to senior members is 3:2 and the ratio of senior members to veterans is 5:2.

Which of the following could be the total number of members in the swimming club?

Show Hint (–1 mark)
–1 mark

The 5:2 ratio suggests that we might have 5 seniors and 2 veterans, but if we have only 5 seniors then the 3:2 ratio would mean 7.5 juniors, which is clearly impossible. So, instead of 5 seniors and 2 veterans, what other numbers would agree with a 5:2 ratio?

  • 30
  • 35
  • 48
  • 58
  • 60

The 5:2 ratio suggests that we might have 5 seniors and 2 veterans, but if we have only 5 seniors then the 3:2 ratio would mean 7.5 juniors, which is clearly impossible.

However, the 5:2 ratio also suggests that we might have 10 seniors and 4 veterans, and if we have only 10 seniors then the 3:2 ratio would mean 15 juniors, which is a much more sensible answer, with a total of 10 + 4 + 15 = 29 members. But this answer is not one of the choices, so we need to look for a multiple of 29, which is 58.

22.

5 marks

22.1. One cube has each of its faces covered by one face of an identical cube, making a solid as shown. The volume of the solid is 875cm3.

What, in cm2, is the surface area of the solid?

  • 750
  • 800
  • 875
  • 900
  • 1050
Show Hint (–1 mark)
–1 mark

If the object is one cube surrounded by cubes then we have (1 + 6 = 7 cubes), so you can now work out the volume of each cube... and therefore the length of any side of any cube. When it comes to looking at the surface area, we have no surface area from the cube in the middle, and 5 faces from each of the other 6 cubes. So, just be careful when you work this out. Take it step by step, lay our your calculation clearly and check your answer before you submit it.

The solid is made up of 7 cubes. So each of them has volume 8757 = 125 = 53cm3. Hence the side length of each of the cubes is 5cm. Each of the 6 faces of the central cube is covered by one face of one of the other 6 cubes, each of which has 5 faces showing. So the surface of the figure is made up of 6 x 5 = 30 squares, each with side length 5 cm and hence area 25 cm2. Hence the total area of the figure is 30 x 25 = 750cm2.

23.

5 marks

23.1 Aaron says his age is 50 years, 50 months, 50 weeks and 50 days old. What age will he be on his next birthday?

  • 56
  • 55
  • 54
  • 53
  • 51

50 months is 4 years and 2 months. 50 weeks is around 11 12 months and 50 days is about 1 23 months.

So Aaron's age is approximately (50 + 4) years + (2 + 11 12 + 1 23) months = 54 years + 15 16 months = 55 years and 3 16 months. So he will be 56 on his next birthday.

24.

5 marks

24.1 Two numbers in the 4 × 4 grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).

What is the sum of these two numbers?

Show Hint (–1 mark)
–1 mark

Start by sketching the initial magic square. Next add up all the rows, columns and diagonals, and you will notice that most of them sum to 34. If you are going create a magic square by just swapping two numbers, then it is probably going to be easier to make everything add up to 34, rather than making all the 34s add up to something else.

Show Hint (–1 mark)
–1 mark

If a row, column or diagonal already sums to 34, then you probably don’t need to adjust anything, so cross out all the numbers in those columns, rows and diagonal. The numbers that you need to swap will be among whatever numbers are left. (As a bonus hint, you will notice that one of the diagonals adds to less than 34, so you will need to replace a number on the problematic diagonal with a larger number.)

  • 12
  • 15
  • 22
  • 26
  • 28

Start by sketching the initial magic square. Next add up all the rows, columns and diagonals, you will notice that most of them sum to 34. If you are going create a magic square by just swapping two numbers, then it makes sense to force everything else to add up to 34.

If a row, column or diagonal already sums to 34, then you probably don’t need to adjust anything, so cross out all the numbers in those columns, rows and diagonal. The numbers that you need to swap will be among whatever numbers are left.

One of the remaining numbers is the 13 on the diagonal that goes from top left to bottom right, which totals to only 32. So this will need to be swapped with the 15 on the bottom row. As well as correcting the faulty diagonal, this swap also fixes the other problematic rows and columns.

So, the two swapped numbers are 13 + 15 = 28, which means the answer is 28.

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

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.