Parallelogram 41 Level 5 13 Jun 2024A concrete transfer

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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.
  • 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. An unusual football transfer

3 marks

1.1 A number of years ago, a newspaper reported that Turkish football team Sarigol Municipality transferred four of its players in return for a fee of 225 sacks of cement, needed to repair their stadium. At the same rate of exchange, how many sacks of cement would the transfer fee for a full team of eleven players and one reserve?

  • 233
  • 450
  • 675
  • 900
  • 2700
  • (Not answered)

The transfer fee for a total of 12 players would be three times that for 4 players.

2. Liar Numbers

Here's a video by the great James Grime:

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

4 marks

2.1 Numbers that pass this prime number test, but which are not actually prime, are known as what?

  • Carmichael numbers
  • Vanmichael numbers
  • Articulatedlorrymichael numbers
  • Wagonmichael numbers
  • Busmichael numbers
  • (Not answered)
4 marks

2.2 The third number that passes the prime test, but which is not a prime, is 1,729. It is known as what?

  • An uber number
  • A taxicab number
  • A minicab number
  • A chauffer number
  • A bus number
  • (Not answered)

3.

4 marks

3.1 After a year's training, Minnie increased her average speed in the London Marathon by 25%.

By what percentage did her time decrease?

  • 50
  • 30
  • 20
  • 10
  • 5
  • (Not answered)
Show Hint (–1 mark)
–1 mark

It helps to think about this problem in terms of fractions, rather than percentages. If the new speed is 5/4 of the original speed, then what is the new time?

Minnie's average speed this year is 54 of last year's value. Hence her time this year will be 45, ie 80% of last year's time.

4.

5 marks

4.1 What is the area of the pentagon shown?

  • 12abc
  • 12ba+c
  • 12ab+c
  • 12bca
  • 12ca+b
  • (Not answered)

The area of the pentagon is that of a rectangle of length b and breadth a plus that of a triangle of base b and height ca, ie ab+12bca=12b2a+ca=12ba+c.

The area may also be considered to be the sum of the areas of two trapezia which have parallel sides a and c and whose heights have total length b.

5.

5 marks

5.1 The width:height ratio of television screens is changing from the traditional 4:3 to the widescreen 16:9. If a traditional screen and a widescreen have the same area, then what is the ratio widescreen width:traditional width?

Assume that television screens are rectangles.

  • 2:3
  • 3:2
  • 3:2
  • 4:3
  • none of these
  • (Not answered)
Show Hint (–2 mark)
–2 mark

Let the widescreen width and traditional width be w and W respectively. Then the respective heights are 9w16 and 3W4.

As the areas are equal:

w×9w16=W×3W4, ie w2=43W2. Hence w : W = 2:3

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.