Parallelogram 31 • Level 2 • 4 Apr 2024Testing tankards
Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/
- 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.
- Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.
1.
1.1 Tommy Thomas's tankard holds 480 ml when it is one quarter empty. How much does it hold when it is one quarter full?
- 120 ml
- 160 ml
- 240 ml
- 960 ml
- 1440 ml
- (Not answered)
When Tommy's tankard is one quarter empty it is three quarters full. So 480 ml is three quarters of the capacity of the tankard. So when it is one quarter full it holds
2.
2.1 Which of the following is correct?
- 0 × 9 + 9 × 0 = 9
- 1 × 8 + 8 × 1 = 18
- 2 × 7 + 7 × 2 = 27
- 3 × 6 + 6 × 3 = 36
- 4 × 5 + 5 × 4 = 45
- (Not answered)
The values of the left-hand sides of the expressions are 0, 16, 28, 36 and 40 respectively.
3.
3.1 Gill is 21 this year. When she was born, her weight was calculated to be 5kg. She now weighs 50kg.
What has been the percentage increase in Gill's weight from when she was born to now?
- 900%
- 1000%
- 5000%
- 9000%
- 10,000%
- (Not answered)
The increase in Gill's weight is 45 kg, which is 9 times her birth weight.
So the percentage increase in weight is 900%.
4.
4.1 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
- 2
- 3
- 5
- 7
- 8
- (Not answered)
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
To complete the solution we should check that with
5.
5.1 The diagram shows a regular octagon with sides of length 1.
The octagon is divided into regions by four diagonals.
What is the difference between the area of the hatched region and the area of the region shaded grey?
- 0
1 8 1 4 1 2 - 1
- (Not answered)
The octagon is divided into the central square, which has sides of length 1, four congruent rectangles and four congruent right-angled triangles. The hatched region is made up of two of the rectangles, and the square. The grey area is made up of two of the rectangles and three of the triangles. Therefore the difference between their areas is the difference between the area of the square and the area of three of the triangles.
The square has area 1. Each of the triangles is a right-angled isosceles triangle with hypotenuse of length 1. Let x be the length of each of the other two sides. By Pythagoras’ Theorem,
Therefore the difference between the area of the square and the area of three of the triangles is
We can also see this geometrically. The diagram on the right shows that four of the triangles fit together to make a square whose side has length 1. So the difference between the area of the square and that of three of the triangles is the area of one-quarter of the square.
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.
- The next Parallelogram is next week, at 3pm on Thursday.
- 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.